COURANT 10 THIERRY C A ZEN AVE LliCTURh NOTES Semilinear Schrodinger Equations American Mithetniticil Society \A C Contents Preface.............................. vii Notation..............................ix 1. Preliminary Results......................1 1.1. Functional Analysis.....................1 1.2. Integration.........................3 1.3. Sobolev Spaces.......................6 1.4. Sobolev and Besov Spaces on RN.............. 12 1.5. Elliptic Equations..................... 15 1.6. Semigroups of Linear Operators............... 18 1.7. Some Compactness Tools.................. 20 2. The Linear Schrödinger Equation............... 29 2.1. Basic Properties ..................... 29 2.2. Fundamental Properties in RN............... 30 2.3. Strichartz's Estimates................... 33 2.4. Strichartz's Estimates for Nonadmissible Pairs......... 40 2.5. Space Decay and Smoothing Effect in RN .......... 41 2.6. Homogeneous Data in RN................. 43 2.7. Comments........................ 51 3. The Cauchy Problem in a General Domain.......... 55 3.1. The Notion of Solution .................. 55 3.2. Some Typical Nonlinearities ................ 58 3.3. Local Existence in the Energy Space............. 63 3.4. Energy Estimates and Global Existence............ 74 3.5. The Nonlinear Schrödinger Equation in One Dimension .... 77 3.6. The Nonlinear Schrödinger Equation in Two Dimensions .... 78 3.7. Comments........................ 80 4. The Local Cauchy Problem.................. 83 4.1. Outline ......................... 83 4.2. Strichartz's Estimates and Uniqueness............ 84 4.3. Local Existence in H1^)................. 92 4.4. Kato's Method...................... 93 4.5. A Critical Case in H1(RN)................. 103 4.6. L2 Solutions....................... 109 4.7. A Critical Case in L2(RN)................. 115 4.8. H2 Solutions....................... 119 4.9. Hs Solutions, s < N/2................... 125 4.10. Hm Solutions, m > N/2.................. 134 v vi CONTENTS 4.11. The Cauchy Problem for a Nonautonomous Schrodinger Equation........................ 138 4.12. Comments........................ 143 5. Regularity and the Smoothing Effect............. 147 5.1. Hs Regularity, 0 < s < min{l,./V/2}............. 147 5.2. H1 Regularity....................... 149 5.3. H2 Regularity....................... 152 5.4. Hm Regularity, m > N/2.................. 154 5.5. Arbitrary Regularity.................... 155 5.6. The C°° Smoothing Effect................. 156 5.7. Comments........................ 160 6. Global Existence and Finite-Time Blowup.......... 163 6.1. Energy estimates and Global Existence........... 163 6.2. Global Existence for Small Data.............. 165 6.3. Global Existence for Oscillating Data............ 167 6.4. Global Existence for Asymptotically Homogeneous Initial Data . 171 6.5. Finite-Time Blowup.................... 178 6.6. The Critical Case: Sharp Existence and Blowup Results .... 196 6.7. The Pseudoconformal Transformation and Applications..... 204 6.8. Comments........................ 208 7. Asymptotic Behavior in the Repulsive Case......... 211 7.1. Basic Notions of Scattering Theory.............. 211 7.2. The Pseudoconformal Conservation Law........... 213 7.3. Decay of Solutions in the Weighted L2 Space......... 215 7.4. Scattering Theory in the Weighted L2 Space......... 220 7.5. Applications of the Pseudoconformal Transformation...... 224 7.6. Morawetz's Estimate.................... 233 7.7. Decay of Solutions in the Energy Space ........... 239 7.8. Scattering Theory in the Energy Space............ 246 7.9. Comments........................ 249 8. Stability of Bound States in the Attractive Case....... 255 8.1. Nonlinear Bound States.................. 255 8.2. An Instability Result ................... 269 8.3. A Stability Result..................... 274 8.4. Comments........................ 280 9. Further Results......................... 283 9.1. The Nonlinear Schrodinger Equation with a Magnetic Field . . 283 9.2. The nonlinear Schrodinger Equation with a Quadratic Potential . 287 9.3. The Logarithmic Schrodinger Equation ........... 291 9.4. Existence of Weak Solutions for Large Nonlinearities...... 299 9.5. Comments........................ 303 Bibliography........................... 305 Preface This book presents various mathematical aspects of the nonlinear Schrodinger equation. It is based on the notes of three courses, the first one given at the Federal University of Rio de Janeiro and at the IMPA in 1989 [55], the second given at the Federal University of Rio de Janeiro in 1993 [56], and the third one given at the Courant Institute in 1997. The nonlinear Schrodinger equation received a great deal of attention from mathematicians, in particular because of its applications to nonlinear optics. Indeed, some simplified models lead to certain nonlinear Schrodinger equations. See Berge [27] and C. Sulem and R-L. Sulem [330] for the modelization aspects. Nonlinear Schrodinger equations also arise in quantum field theory, and in particular in the Hartree-Fock theory. See, for example, Avron, Herbst, and Simon [5, 6, 7], Bialinycki-Birula and Mycielski [31, 30], Combes, Schrader, and Seiler [93], Eboli and Marques [109], Gogny and Lions [149], Kato [202], Lebowitz, Rose, and Speer [223], Lieb and Simon [229], Reed and Simon [301], B. Simon [313], and C. Sulem and P.-L. Sulem [330]. The nonlinear Schrodinger equation is also a good model dispersive equation, since it is often technically simpler than other dispersive equations like the wave or KdV. From the mathematical point of view, Schrodinger's equation is a delicate problem, and possesses a mixture of the properties of parabolic and hyperbolic equations. Particularly useful tools are energy and Strichartz's estimates. We study in this book both problems of local nature (local existence of solutions, uniqueness, regularity, smoothing effect) and problems of global nature (finite-time blowup, global existence, asymptotic behavior of solutions). The methods presented apply in principle to a large class of dispersive semilinear equations. On the other hand, we do not study quasilinear Schrodinger equations (with nonlinearities involving derivatives of the solution). They require in general the use of specific linear (and nonlinear) estimates, and most results of global nature are limited to small initial data. The book is organized as follows. In Chapter 1, we recall some well-known properties of functional analysis concerning integration, Sobolev and Besov spaces, elliptic equations, and linear semigroups that we use throughout the text. We also introduce some useful compactness tools. In Chapter 2, we establish some fundamental properties of the (linear) Schrodinger equation. The case of the whole space M.N is studied in detail. Chapter 3 contains a few partial results of local existence for the nonlinear Schrodinger equation in a general domain of WN. The rest of the book is concerned with the case Q = RN. Chapter 4 is devoted to the study of the local Cauchy problem in various spaces, and in Chapter 5 we study the regularity properties and the smoothing effects. Chapter 6 is devoted to the study of global existence and finite-time blowup of solutions. In Chapter 7, we vii viii PREFACE study the asymptotic behavior of solutions in the repulsive case. The main results are the construction of the scattering operator in a weighted Sobolev space and in the energy space. In Chapter 8, we study the stability and instability properties of standing waves in the attractive case. We establish the existence of standing waves, and in particular of ground states, and we show that ground states are stable or unstable, depending on the growth of the nonlinearity. Chapter 9 is devoted to some further results concerning certain nonlinear Schrodinger equations that can be studied either by the methods used in the previous chapters or else by different methods. Bibliographical references are given in the text. In order to be informed of the latest news, it is advised to have a look at the web page "Local and global well-posedness for non-linear dispersive and wave equations'1''1 maintained by J. Collian-der, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. Let us also mention a few monographs specialized in the nonlinear Schrodinger equation: Berge [27], Bour-gain [38], Ginibre [128], Kato [204], Strauss [326], and Sulem and Sulem [330]. I am grateful to my colleagues who reported misprints (and more serious mistakes) in previous versions of these notes, and in particular to P. Begout, F. Castella, J. Ginibre, T. Kato, and G. Velo. I thank my friend Jalal Shatah, who invited me to publish these notes in the Courant Lecture Notes series. Finally, it was a pleasure to collaborate with Paul Monsour and Reeva Goldsmith in their beautiful editing work. http: //www. math, ucla. edu/~t ao/Dispersive Notation a.a. almost all a.e. almost everywhere iff if and only if Re z real part of the complex number z Im z imaginary part of the complex number z E closure of the subset E in the topological space X C(E,F) space of continuous functions from the topological space E to the topological space F 1# characteristic function of E defined by 1e(x) = 1 if x £ E and Ie(x) = 0 if x & E CC(E, F) space of continuous functions E —> F compactly supported in E C(E,F) Banach space of linear, continuous operators from the Banach space E to the Banach space F, equipped with the norm topology £(E) =C{E,E) X* (topological) dual of the (topological) space X {x',x)x*,x duality product of x' £ X* and x £ X (also "(x',a;)x*,x") A* adjoint of the operator A X <—*Y if X C Y with continuous injection Q open subset of RN ft closure of ft in RN d£l boundary of Q, i.e., dfl = £7 \ Q, oj eg ft if To C and uJ is compact BR = {x £RN : \x\ < R}, ball of radius R and center 0 of RN du du = dtU=Tt = Tt du dxi IX x NOTATION ur - dru — ^ = -x ■ Vw, where r = \x\ or r Da — 0 a, ... „ QiV for a multi-index a £ N"^ OXi OXjf Vu = (<9i«,..., 8nu) N d2 A = ^ T Fourier transform1 Tu{^) = J e~2rKlx'^u(x)dx ~F = JT-1 given by 7v(x) = j e2^'xv{^)d^ u = Tu Cc(ft) = Cc(ft,lR) (or Cc(ft,C)) C(ft) space of continuous functions ft —* K. (or ft —► C). When ft is bounded, C(ft) is a Banach space when equipped with the L°° norm Cb,u(^) Banach space of uniformly continuous and bounded functions ft —> M (or ft —► C) equipped with the topology of uniform convergence C™u(fl) Banach space of functions u £ Cb,u(ft) such that J9aw € Cb,u(ft) for every multi-index a £ such that |a| < to. The space Cj^fft) is equipped with the norm of Wm,(Q,). Co(ft) closure of Z>(ft) in L°°(ft) Cm'Q(ft) for 0 < a < 1, the Banach space of functions u £ C™ (CI) such that ||u]]Cm,a = ||u||h/"<,~ + sup {\x - y\~a\D0u(x) - D0u(y)\) < oo 10! = ".. V(Cl) = C£°(f2), the Frechet space of C°° functions ft -> R (or ft -> C) compactly supported in ft, equipped with the topology of uniform convergence of all derivatives on compact subsets of ft V(Cl) space of distributions on ft, i.e., the topological dual of £>(ft) K (or ft —+ C) such that Z)Qu 6 Lp(0) in the sense of distributions, for every multi-index a with \a\ < m. Wm,p(£l) is equipped with the norm ||u||Wm.P = ^ \\DaU\\LP ■ \a\(fi) in VFm'p(fi) W"m'p'(fi) (m e N, 1 < p < oo) dual of W0m'p(ft) Hm{9) = Wm>2(Q). Hm(Q) is equipped with the equivalent norm 1/2 |u||#m = ^ £ J \Dau{x)\2 dx \a\2{Cl) HS'P(RN) (s e K, 1 < p '< oo) Banach space of elements u e S'{RN) such that •F-1[(l + |£|2)s/2«] e H^R*) is equipped with the norm Hiff-.p = H-?r~1[(i + ^r)*«liu»'- iP(RN) ^tf^R*) ifs'p(MN) (5 € R, 1 < p < 00) homogeneous version of the Sobolev space HS,P(RN) HS(RN) = Hs>2 NOTATION BP,<3 (s G R, 1 < p,q < oo) Banach space of elements u € S'{RN) such that < oo with \u\\B^=\\F-\m)\\Lr> + { , oo v \/q (^(2«'||^-1(^2)!Up)M if?<°o Vj=i ' sup2SJ|j^r_1((/9ju)|jz/p if g = oo, where ^^(ipju) is the jth dyadic block of the Littlewood-Paley decomposition of u ) (s (I) —> X, where X is equipped with the weak topology Cb,u(^, X) Banach space of uniformly continuous and bounded functions I —> X, equipped with the topology of uniform convergence C™ (I, X) Banach space of functions u : I —* X whose derivatives of order j belong to Cb,u(I,X), for all 0 < j < m. C™U(I,X) is equipped with the norm of Wm'°°(I,X). Cm'a(l, X) for 0 < a < 1, the Banach space of functions u € C^u(7, X) such that \u\\Cr |u||iym,« + SUp < \t — s\ a ——it)---—is) dtm w dtm w < oo C(I,X) space of continuous functions / —> X. When J is bounded, C(I,X) is a Banach space with the norm of L°°(I, X), LP(I, X) Banach space of (classes of) measurable functions u : I —> X such that llulkp < oo, with \u\\lp = < \u(t)\\pxdt ess sup ||u(t)||x I i/p if p < oo if p = oo Wm,p(I, X) Banach space of (classes of) measurable functions u : I —► X such that d?u — e LP(I,X) for every 0 < j < m. Wm>p(I,X) is equipped with dP the norm ii ii d u j'=i dP Lv NOTATION xiii except when otherwise specified, the group of isometries on L2(Q) generated by the skew-adjoint operator iA, where A is the Laplacian with Dirichlet boundary condition on d£l CHAPTER 1 Preliminaries In this chapter we recall some basic properties of functional analysis, complex and vector integration, Sobolev spaces, elliptic equations, and linear semigroups that we use in the next chapters. 1.1. Functional Analysis See, for example, Brezis [43], Brezis and Cazenave [44], Cazenave and Haraux [64, 65], Rudin [304], Strauss [320], and Yosida [366]. We recall that if X and Y are two Banach spaces such that X c—+ Y with dense embedding e, then Y* > X* with embedding e*. Moreover, if X is reflexive, then the embedding Y* <-> X* is dense. We will use repeatedly the following elementary properties of weak topologies. (i) Let X Y be two Banach spaces. Consider x oo, then xn —x in Y as n —► oo. (ii) Let X <—> Y be two Banach spaces. Assume X is reflexive and consider y e Y and a bounded sequence (xn)nE^ C X. If xn —y in F as n —+ oo, then y £ X and xn —^ y in X as n —► oo. (iii) Let X t-> V be two Banach spaces and let 7 be a bounded, open interval of R. Let u : I —► Y be weakly continuous. If X is reflexive and if there exists a dense subset E of I such that u(t) e X for all £ e and sup{||it(£)||x) £ £ £} = K < oo, then w(£) e X for all £ e J and u : 7 -» X is weakly continuous. (iv) Let X be a uniformly convex Banach space; let 7 be a bounded, open interval of K; and let u : 7 —+ X be weakly continuous. If the function 11—► is continuous 7 R, then u € C(7, X). (v) Let X be a Banach space, let I be a bounded, open interval of R, and let u : 7 —»■ X be weakly continuous. If there exists a Banach space i? such that X B with compact embedding, then u € C(7,73). We will construct solutions of the nonlinear Schrodinger equation either by a fixed point argument, or by a compactness technique. For the first method, we will use Banach's fixed point theorem and for the second, we will use Proposition 1.1.2 below. Theorem 1.1.1. (Banach's fixed point theorem) Let (X, d) be a complete metric space and F : X —» X. If there exists a constant L < 1 such that d(F(x)>F(y)) < Ld(x, y) for all x,y € X, then F has a unique fixed point xq € X; i.e., there exists a unique xq € X sitc/i i/iai F(xo) = xq. 1 2 1. preliminaries Proposition 1.1.2. Let X <—> Y be two Banach spaces and let I be a bounded, open interval of R. Let (fn)n^N be a bounded sequence in C(I,Y). Assume that fn{t) e X for all (n,t) G N x I and that sup{||/n(i)[|x, (n,t) 0,38 > 0, Vn, s, t G N x 7 x 7, \\fn(t) ~ fn(s)\\Y < £ if \t ~ s\ < 5). If X is reflexive, then the following properties hold: (i) There exists a function f £ C(7, Y) which is weakly continuous I —> X and a subsequence nk such that fnk it) f(t) in X as k oo, for all t £ I. (ii) If there exists a uniformly convex Banach space B such that X <—► B <—* Y and if (/n)n€N C C(I,B) and \\fnk(t)\\B ||/(*)||b as k oo, uniformly on I, then also f € C(I, B) and fnk —>■ / in C(I, B) as k —> oo. Proof, (i) Let (tn)neN be a representation of Q D I. Using the reflexivity of X and the diagonal procedure, we see easily that there exist a subsequence and a function / : Q n 7 —► X such that fnk{tj) f{tj) in X (hence in Y) as k —> oo, for all j £ N. By the uniform equicontinuity of (/n)neN and the weak lower semicontinuity of the norm, / can be extended to a function of C(I,Y). Furthermore, / : I —> X is weakly continuous and sup{|j/(t)[|x, t € 1} < K. Consider now tel. Let (tj)j^ c Q h I converge to t and let y' € Y*. We have \{y'Jnk (t) - /(«)>y* ,V| < l(y',/nh(t) - fnk(tj))Y*,Y\ + \(y\ fit) - /(t^y.vl + 1<2/', /»„&) - /(t,-))y*,y|- Given e > 0, it follows from the uniform equicontinuity that the first and second terms of the right-hand side are less than e/3 for j large enough. Given such a j, the third term is less than e/3 for k large enough; and so Kx',/nfc(i)-/(t))y.,y|->0 asfc-^OO. Thus /nfc(t) /(t) in Y; and so /„,(£) /(£) in X. Hence (i). (ii) Note first that / : I —► B is weakly continuous. Also, : 7 —> K is continuous; and so / € C(I,B). It remains to prove that fnk —> / in C(I,B). We argue by contradiction, and we assume there exist a sequence (£fc)j:eN C 7 and £ > 0 such that \\fnk{tk) - /(*fe)||s > £5 f°r every fcGN. We may assume that t^ —> t e 7 as A; —> oo. It follows from (i) and the uniform continuity that fnic(tk) f(t) in Y as fc —> oo. Since (/n)neN is bounded in C(I,B), we obtain as well that fnk(tk) ~^ f(t) in 7? as fc —* oo. Furthermore, |||/n*(t*)b - ll/(0IU| < |||/»t(**)b - + \\\m)\\B - • Therefore, ||/nfe(*Jfc)||B -» and so /nfc(*fc) ~> /(*) in B as A; —> oo, which is a contradiction. □ Finally, we will use some properties of the intersection and sum of Banach spaces. Consider two Banach spaces X\ and X2 that are subsets of a Hausdorff topological vector space X. Let Xi nX2 = {x e X : x € Xx, x e X2} and Xi + X2 = {x e X : 3x! € Xi, 3x2 e X2, x = xx + x2} . 1.2. integration 3 Set lkllxinx2 = \\x\\xi + \\x\\xi for x e X1nX2, and ll^llxi+x2 = infills\\x1 + \\x2Wx2 :x = x1+ x2} for x € Xx + X2 . We have the following result (see lemma 2.3.1 and theorem 2.7.1 in Bergh and Lofstrom [28]). Proposition 1.1.3. (Xx n X2, || JUinxJ and (Xx + X2, || \\Xl+x2) are Banach spaces. If furthermore Xi D X2 is a dense subset of both X\ and X2, then (Xx n X2)* =X$+ XI and (Xi + X2)* = n JT|. 1.2. Integration For real and complex integration, consult Brezis [43], Dunford and Schwartz [108], Rudin [305], and Yosida [366]. For vector integration, see Brezis and Caze-nave [44], Cazenave and Haraux [64, 65], Diestel and Uhl [105], Dinculeanu [106], Dunford and Schwartz [108], J. Simon [314], Yosida [366], and the appendix of Brezis [42]. Throughout these notes, we consider Lp spaces of complex-valued functions. Q, being an open subset of RN, Lp(£l) (or Lp, when there is no risk of confusion) denotes the space of (classes of) measurable functions u : fl —» C such that \\u\\lp < 00 with / \\u(x)\\pdx) P if P€ [1,00) ess sup if p = 00. n Lp(£l) is a Banach space and L2(Q) is a real Hilbert space when equipped with the scalar product {u,v)l2 = Re J u(x)v{x) dx. Below is a useful result of Strauss [321j! proposition 1.2.1. Let Q be an open subset of RN and let 1 < p < 00. Consider u : Q —+ R and a bounded sequence («n)neN of Lp(Cl). If un —> u a.e. in Q. as n —*■ 00, then u € LP(Q) and un —» u as n —> 00 in LQ(Q,'), for every O' C of finite measure and every q € [l,p). in particular, un —* it as n —> 00, m Lp(fi) it>ea& if p < 00, and in L°°(Q) weak-* if p — 00. Consider now an open interval / cl and a Banach space X equipped with the norm j| ■ ||. A function / : I —> X is measurable if there exist a set N C I of measure 0 and a sequence (/n)neN C CC(/,X) such that lim /n(t) = /(*) for all t X is measurable, then |[/|| : i" —> R is also measurable. Also, if / : I —> X is measurable and if F is a Banach space such that X <—> F, then / : / —> F is measurable. More generally, if / : I —> X is 4 1. preliminaries measurable, V is a Banach space, and $ : X —> Y is continuous, then (xf, f(t))x*,x is measurable /->!) and there exists a set TV c / of measure 0 such that /(/ \ N) is separable. One deduces the following properties: (i) If / : I —» X is weakly continuous (i.e., continuous from I to X equipped with its weak topology), then / is measurable. (ii) Let {fn)n£N be a sequence of measurable functions I -* X and let f : I —> X. If fn(t) —* f(t) in X as n —* oo, for a.a. t € I, then / is measurable. (iii) Let X <—> Y be two Banach spaces and let / : I —> Y be a measurable function. If f(t) £ X for a.a. t £ I and if X is reflexive, then / : I —» X is measurable. A measurable function / : I —+ X is integrable if there exists a sequence (/«)neN C CC(7,X) such that (1.2.1) lim / ||/n(t)-/(t)||dt = 0. If / : I —> X is integrable, then there exists € X such that for any sequence (/n)neN C CC(I, X) satisfying (1.2.1), one has lim f fn(t)dt - x(f) n—»00 J the above limit being for the strong topology of X. The element x(f) is called the integral of / on I. We write *(/) = ff = ff = f f{t)dt-i i If I = (a, b), we also note *(/)= /*/= /6 As for real-valued functions, it is convenient to set f/3 / f{t)dt=~ r men Ja J/3 if /3 < a. Bochner's theorem asserts that if / : I —> X is measurable, then / is integrable if and only if : I —* K is integrable. In addition, < / \\f(t)\\dt. f(t)dt Bochner's theorem allows one to deal with vector-valued integrable functions like one deals with real-valued integrable functions. It suffices in general to apply the usual convergence theorems to ||/||. For example, one can easily establish the 1.2. integration 5 following result (the dominated convergence theorem). Let (/n)nGN be a sequence of integrable functions / —> X, let g £ L1^), and let / : / —* X. Assume that ||/n(*)ll < 9{t) for a.a. t <= I and all n £ N lim /n(t) = /(£) for a.a. (e/. n—coo It follows that / is integrable and J f(t)dt=]imo J fn{t)dt. i i For p £ [l,oo], one denotes by LP(I,X) the set of (classes of) measurable functions f : I —* X such that the function t h-> ||/(£)j| belongs to LP(I). For / £ LP(I,X), one defines II/I!lp(/,x) = 4 I n/(o ir /p ifp X belongs to LP(I, X) if and only if there exists a function g £ LP(I) such that ||/|| < g a.e. on I. (iii) Suppose / : I —» X is measurable. If / £ LP(J,X) for all J € / and if I!/||lp(jx) < O for some C independent of J, then / £ LP(I,X) and II/IUp(/,x) < c. (iv) If / £ Lp{I, X) and tp £ Lq(I) with I + I = I < i, then tpf £ Lr(I,X) and \\||L.(7) < WfWzuwWghw) ■ (vi) If / £ Lp(I,X)DLQ(I,X) withp < g,then / £ Lr(I,X) for every r £ [p,g], and < II/IIl^/^II/IIl^/^) where ; = j + ^T' 6 1. preliminaries (vii) If \I\ < oo and p < q, then Li(i,x) for all / £ Lq(I, X) (viii) If Y is a Banach space and if A £ £(X, Y), then A/ e I X such that f\j £ LP(J, X) for all open, bounded intervals J C I. We end this section by two useful criteria. \ . Theorem 1.2.4. Le\l X such that for a.a. t £ I, fn(t) —^ f{t) in X as n —* oo, then f £ Lp(I,X) and ||/||lp(/,x) < Hminfn_f00 \\fn\\Lp(i,x)- Theorem 1.2.5. Consider two Banach spaces X <—> Y and 1 < p, q < oo. Let (fn)n>o be a bounded sequence in Lq(I,Y) and let f : I —► Y be such that fn(t) —1 f(t) inY as n —> oo, for a.a. tel. If (fn)n>o is bounded in LP(I,X) and if X is reflexive, then f € LP(I,X) and \\f\\Lp{i,x) < liminf^oo \\fn\\Lp(i,x)- 1.3. Sobolev Spaces For Sobolev spaces of real- (or complex-) valued functions, see, for example, Adams [3], Bergh and Lofstrom [28], Brezis [43], Gilbarg and Trudinger [127], J.-L. Lions [231], Lions and Magenes [232], and Triebel [338]. For vector-valued Sobolev spaces, see the appendix of Brezis [42], Brezis and Cazenave [44], Cazenave and Haraux [64, 65], J.-L. Lions [231], and Lions and Magenes [232]. Consider an open subset Q of RN. We recall that V(Q) (— T>(Q, C)) is equipped with the topology induced by the family of seminorms dK,m, where K is a compact subset of ft and m € N, defined by dK,m{'(ft), is the dual space of V(Q). If T £ £>'(ft) and if a £ NN is a multi-index, one defines the distribution dai daN by (DaT, ip) = (-1)IQI (T, Datp) for all

(x) dx^j for all y> £ £>(ft). ft It is well known that if Tf = Tg, then / = g a.e. A distribution T £ P'(ft) is said to belong to Lp(ft) if there exists / £ Lp(Cl) such that T = Tf. In this case, / is unique. For ?7i £ N and 1 < p < oo, the Sobolev space Wm,p(ft) is defined by Wm'p(fL) = {ue Lp(ft) : £>Qu £ Lp(ft) for |a| < m} . Wm>p(ft) is a Banach space when equipped with the norm || ||wm.p = || ||wm'p(fi) defined by ||u||Wm,P = \\Dau\\Lp(n) ■ 0<|a|(ft). When p = % set Wm>p(n) = Hm{Q) and W™>V{Q) = H^{Q) and equip Hm(ft) with the equivalent norm ll«llff- 1 and 1 < p < oo. If (wn)n€N is a bounded sequence of Wm,p(£l), then there exist u g Wm'p(£l) and a subsequence (unk)ksf^ such that unk —> u a.e. as A; —» oo, and ||u||vk^.p < liminf ||tin||vvm'p • n—>oo If p < oo, then also wnjb -^ain iym-p. If p < oo and (un)n£K C W0m,p(ft), thenu£ W0w'p(ft). (iv) Let to > 0 and 1 < p < oo. Consider a bounded sequence (wn)neN of Wm,p(Q) and assume that there exits u : ft —* M such that un —> u a.e. as n -> co. It follows that u g iym'p(ft) and |w||wrm-p < liminf ||unJ|vvto'P 8 1. PRELIMINARIES If p < oo, then also un u in Wm'p. If p < oo and (u„)n€N C W™'p(£l), then u e W^p(n). (v) Let F : C -+ C be a Lipschitz continuous function such that F(0) = 0. We may consider F as a function E2 —► 3R2, so that F'(u) — DF(u) (which is defined for a.a. u € C) is a 2 x 2 real matrix, hence a linear operator C C. Let p e [l,oo]. For every u e W^itt), F(u) e W1*^) and \diF(u)\ < L\diu\ a.e. for every 1 < % < iV, where L is the Lipschitz constant of F. In particular, \\VF(u)\\LP < L\\Vu\\LP. If p < oo and if u e W01,p(ft), then F{u) e W01,p(fi). If we assume furthermore that F is Cl except at a finite number of points, then VF(u) = DF(u)Vu a.e. for every u G W1,P(Q) and the mapping u i-» F(u) is continous Wl,p(Q) —> W1,P(Q.) for every p < oo. On these questions, see Marcus and Mizel [237, 238, 239] and the appendix of Brezis and Cazenave [44]. (vi) In particular, if p e [l,oo] and u 6 W1'P(Q.), then \u\ e W1*^) and |V|u|| < |Vu| a.e. If p < oo and u g W01,p(ft), then |u| e W01,p(fi). Moreover, the mapping u k-> |w| is continuous W1,P(Q) —> W1,p(fi) if p < oo. (vii) Let F : C —> C satisfy F(0) = 0, and assume that there exists a > 0 such that \F(v)-F(u)\ < L(\v\a + \u\a)\v-u\ forallu,v 6 C. Let 1 1, Men VT1^) ^ L9(ft) /or every qr 6 [p,oo). (hi) Ifp = N=\, then W^P(Q) ^ L*(n) for every q e [p, oo]. (iv) Ifp > N, then W^ify ^ L°°(Q). If Q has a uniformly Lipschitz continuous boundary, then: (v) Ifp > N, then W^P(Q) <-> C0>a(U), where a = Theorem 1.3.4. (Rellich's compactness theorem) If Q is bounded and has a Lipschitz continuous boundary, then the following properties hold: 1.3. sobolev spaces 9 (i) If 1 < p < N, then the embedding Wl'p(£l) <-> Lq(£l) is compact for every (ii) If p > N, then the embedding W1,p(Cl) «-» L°°(Q) is compact. If we assume further that Q, has a uniformly Lipschitz continuous boundary, then: (iii) If p > N, then the embedding W1,p(fl) <^-+ C0,A(f2) is compact for every A 6(0,^). Theorem 1.3.5. The conclusions of Theorems 1.3.3 and 1.3.4 remain valid without any smoothness assumption on Q if one replaces Wl>p{£t) by Wq'p(CI) (note that Q still needs to be bounded for the compact embedding). remark 1.3.6. If p = N > 1, then Wl>p(Q) <-+ Lq(fl) for every p < q < oo, but W1,p(f2) 0 and K < oo such that J(eM|u|2 -\) 1 and m — j — =0), £/ien i/iere e:m£s C(N,m,j,a,q,r) such that Y \\Dau\\LPp'(Q) as the (topological) dual of W™'P{Q). One defines #-m(£l) = W~m'2(n), so that H~m(n) = (f/£»(fi))\ Remark 1.3.8. Here are some useful properties of the spaces W_7ri'p'(0). (i) From the dense embedding V(Q) W™'p(tt), we deduce that W~m^p'' (£1) is a space of distributions on O. Furthermore, it follows from the dense embedding W0m'p(ft) LP(Q) that Lp'{ft) ^ V^"m'P'(fi). If p > 1, then the embedding is dense. In particular, V(Q) is dense in W~m'p (Q). (ii) Assume that that 1 < q < oo is such that W™'p{tt) ^ Lq(n). It follows that LQ'(Cl) <-► W-T,l'p'(ft). Furthermore, if p,q > 1, then the embedding is dense. (iii) Even though H™^) is a Hilbert space, one generally, does not identify H~m(Q) with H^(n). One rather identifies L2(Q) with its dual, so that 10 1. preliminaries H~m(Q) becomes a subspace of T>'(Q) containing L2(Q). In particular, if u g ^(fi) and v g £2(ft), then = Re y u(a:)v(a;) . It follows that ||w||^2 < for all w € H™^). (iv) Like any distribution, an element of H~m(Q) can be localized. Indeed, if T g iJ_m(fi) and Qf is an open subset of f2, then one defines T\w as follows. Let (Q) be equal to

'(£)'). Since < IMItf™^), it follows that ^ g if~m(f2'), and one sets T\q> = ^. It is clear that the operator is linear and continuous, and is consistent with the usual restriction of functions. (v) For every multi-index a of length j, Da is a bounded operator from H~m(Q,) to i7~m~J(fi) for every m g N. Since also Da is bounded from Hk(Q) to Hk~i(Q) for every > j, it follows easily that for every k g Z, Da is bounded from to tffc^(ft). (vi) In particular, A defines a linear, continuous operator —> H_1(fi). Note that for u g H1^), the linear form Ait € on H£ (Q) is defined by "_ (Au, v) = - Re J Wu(x)Vv(x) dx for v g i?o(^) • n This is clear for v g V(Vi) and follows by density for v g Hq(Q). Consider now an open interval / Cl and a Banach space X, equipped with the norm || |j. We denote by V'(I,X) the space of linear, continuous mappings V{I) —> X, where X is equipped with the weak topology. It is called the space of X-valued distributions on /. An element / g L\oc(I, X) defines a distribution Tf g V'(I, X) by the formula {Tf, p{I,X) and j|/'||Lp(/,x)' < C. (iii) If Y is a Banach space and if A € C(X, Y), then for every / £ W^P(I, X), Af £ W^P(I,Y), and In particular, if X <-* Y and ii f £ W^P(I,X), then / € Wx>p{I,Y) (let A be the embedding). If / is an interval of R, one defines the space W^P(I, X) as the set of functions f : I —> X such that f\j £ W1,P(J, X) for all open, bounded intervals J C I. Theorem 1.3.10. // 1 < p < oo and f £ LP(I,X), then the following properties are equivalent. (iii) / is weakly absolutely continuous (hence weakly differentiable a.e.) and f (in the sense of the a.e. weak derivative) is in LP(I,X). In addition, if f satisfies these properties, then the derivatives of f in the senses of V'(I,X) and almost everywhere coincide and one may let g = f in (ii). Remark 1.3.11. It follows easily from the above result that and that if p > 1, then W^P(I,X) C°'a(I,X) with a=^. The following result is also quite useful. Proposition 1.3.12. Assume X is reflexive and let f £ LP(I,X). It follows that f £ W1,P(I, X) iff there exist ip £ LP(T) and a set N of measure 0 such that In this case, \\f'\\Lp(i,x) < IMIlp(J)- Remark 1.3.13. Applying Proposition 1.3.12, one can show the following results: (i) Assume that X is reflexive and let / : I —» X be Lipschitz continuous and bounded. It follows that / £ Wl>GO(I,X) and ||/'||l«(/,x) < L, where L is the Lipschitz constant of /. H4f||wi.J'(J,Y) < ||^li£(X,y)||/||tV1.p(7,X) • (i) f£W^p(I,X). (ii) There exists g £ LP(I, X) such that f(t) — f(s) + j* g(a)du for a.a. s,t £ I. W1'1(/,X)-Cb,u(/1X) 12 1. preliminaries (ii) Assume that X is reflexive and that 1 < p < oo. Let (/n)n€N be a bounded sequence of W^P{I,X) and let / : 7 -> X be such that /„(*) f{t) in X as n —► oo for a.a. t € I. It follows that / € W1,P(/,X) and ||/||wi.p(j,:v) < liminfn^oo j[/n||vi/i.p(7,X)- (in) Assume that X is reflexive, 1 < p < oo, and let / e LP(7,X). If 3K such that for all J oo, uniformly on I, then also fnk —► / in C(7,L2(ft)) as k -> oo. (hi) 7/(/n)neN c C(7,^(n)) anrf ||/njt(t)||Hi -* ||/(i)|[#i as fc -> oo, uniformly on I, then also f € C(7, iJg(Sl)) and /njt —> f in C(I,Hq(Q)) as k —> oo. PROOF. Part (i) follows from Proposition 1.1.2(1) applied with X = 77q(£7) and y = 77-m(f2) and from Remark 1.3.13(h) (note that (fn)n€N is uniformly equicon-tinuous in Y by Remark 1.3.11). Part (ii) follows from Proposition 1.1.2(h) applied with X = 77* (ft), Y = H-m(Q), and B = L2{£l). Part (hi) follows from Proposition 1.1.2(ii) applied with X = B = H^(Q) and Y = H-m(Q). □ One can define higher-order vector-valued Sobolev spaces as follows: For m€N, set Wm*(I,X) = {/ € 77(7, X) : ^ 6 77(7, X) for all j e {1,..., m}|. It is clear that Wm*(I,X) = {/ e W1'P(7,X) : ^ e W^(7,X) for all j e {1,..., m - 1}|, so that Wm'l{I,X) ^ C^il.X) and Wm*(I, X) <-* Cm-l'a(l,X) with a = ^i,ifp>l. 1.4. Sobolev and Besov Spaces on RN For more detail, see, for example, Adams [3], Bergh and Lofstrom [28], the appendix to Ginibre and Velo [140], Lemarie-Rieusset [225], Shatah and Struwe [312], and Triebel [337, 338]. It is convenient to consider a function r\ £ C£°(RN) such that f 1 if l£l < 1 (1.4.1) r/(£) = { \0 if ICI > 2, and to define the sequence (xpj)j£Z c S(RN) by (1-4.2) m) = v(~)-v( * ^ 23 J \23'-1 1.4. sobolev and besov spaces on r- 13 in order to define the Littlewood-Paley decomposition. We see that supp^j C {2j-1 < |£| < 2j+1} and that Y, m) = (1 if^° j-oo \0 if« = 0, where the above sum contains at most two nonzero terms. Given sGM, one defines HS(RN) = S'(RN) : (1 + e L2(RN)} and NI^HI(i + I£|¥£||l- It is clear that HS(RN) is a Hilbert space and it follows easily from Plancherel's formula that the above definitions are consistent with those of Section 1.3. More generally, one defines HS'P(RN) = {ue S'(RN) : + |^|2)tu] e LP(RN)} and \\u\\Hs,P = \\^l[(l + \^u]\\Lp for 1 < p < oo and s € K, so that HS>P(RN) is a Banach space (reflexive if 1 < p < oo). Remark 1.4.1. Here are some fundamental properties of the space HS'P(RN). (i) HS'2(RN) = HS(RN) and H°>P(RN) = LP(RN) (same norms). (ii) HSl'p(RN) HS2'P(RN) if Sl > s2. (iii) If p < oo, then [#*>(R^)]* = H~s'p'(RN) (see corollary 6.2.8 in [28]). (iv) It follows from Mihlin's multiplier theorem (see theorem 6.1.6 in [28]) that if m is a nonnegative integer and 1 < p < oo, then Wm>p(RN) = Hm>p(RN) with equivalent norms. By (iii), we also have Wm'p(RN) = Hm'p(RN) when m is a negative integer. (v) Sobolev's embedding: HS'P(RN) ^ HSl'p*(RN) if s - N/p = sx - N/pi and 1 < p < pi < oo, si, 52 € R (see theorem 6.5.1 in [28]). In particular, if 1 < p < oo and 0 < s < N/p, then HS'P(RN) L^(RN). Moreover, HS,P(RN) ^ L°°(RN) if p > 1 and s > N/p. See remark 2, p. 206 in [337]. We now define the Besov space B^g(RN) for 1 < p, q < oo and sGlby BspJRN) = {ue S'(Rn) : \\u\\B.g < co} , with - oo .l/g ^(2^'|j^-1(^)||LP(RJv))9 if gq(RN) is a Banach space. Moreover, Bpq(RN) is isomorphic to a closed subspace of £g(N,Lp(RN)), so that B^q(R*) is reflexive if 1 < p,q < oo. (iii) B*(RN) <- B%q(RN) if Sl > s2- and B^qi(RN) ^ B^Q2(RN) if 1 < 9i < 92 < oo- (iv) If p,q < oo, then [B°tq{RN)]* = B~^(RN) (see Corollary 6.2.8 in [28]). (v) Sobolev's embedding (see theorem 6.5.1 in [28]): B^g(RN) if s — N/p = si - N/pi and 1 < p < pi < oo, 1 < q < qi < oo, si, S2 e K. remark 1.4.3. Here are some relations between the spaces HS,P(RN) and ) (see theorem 6.4.5 in [28]). (i) If 1< p < 2, then B*jP(RN) ^ HS,P(RN) (ii) If 2 < p < oo, then B«2(RW) ^ HS'P(RN] (iii) In particular, ^(R*) = Jfs'2(R") = tfs ^(R^ We now introduce the homogeneous Sobolev spaces HS(RN) and HS,P(RN) and the homogeneous Besov space B^q(RN). In fact, they are rather delicate to define, since they can be considered either as seminormed spaces or as quotient spaces. It will be sufficient for our purpose to define only the (semi-) norms. Let r) and ipj satisfy (1.4.1)-(1.4.2) and let 1 < p < oo and s G R. Given u e S'(RN), we set \u\ Hs>r> +00 f-Hl^ju) j=-oo if the series EjJ-oo ?~ (If ls^«) is convergent in S'{R ) to a function of LP(RN), and [|u||^a,p = oo otherwise. We define \\u\\hs ~ \u\ii^2 • We note that \£\s 0, then \\u\\B^q « \\u\\Lp + NIb« (see theorem 6.3.2 in [28]). (iii) If 0 < s < 1, then / /*°° dt\1/q / (t s sup \\u(--y) -u(-)\\LPimN))q—) ifgo l»l 1 (see theorem 6.3.1 in [28]). 1.5. Elliptic Equations Consult, for example, Agmon, Douglis, and Nirenberg [4], Brezis [43], Brezis and Cazenave [44], Gilbarg and Trudinger [127], J.-L. Lions [231], Lions and Ma-genes [232], and Nirenberg [272]. We recall below some of the results that we will use in the following sections. In all this section we consider an open subset f2 c RN. We equip jr7_1(fi) with the dual norm, that is \\u\\H-i = bvlp{{u,v),v G H£(Q), \\v\\hi = 1} . We recall that (by Lax-Milgram's lemma) for every / G H~1(Q)1 there exists a unique element u G Hq(Q) such that -Au + u = f inH'1^). In addition, 11/11*- = ■ It follows in particular that A — I defines an isometry from Hq(Q) onto H~1(Q). By the same method, one shows also that for every A > 0 and every / G H~1(Q), there exists a unique element u G Hq(VL) such that -Au + Xu = f inH-\Q,). ill/Ill = defines an equivalent norm on i7_1(q) and A||u||t/-i < If / G L2(Q), then Au G L2(Q), the equation makes sense in L2(£l), and A||u||jr,a(n) < ll/IUw One shows also that if Q has a C2 boundary and if / G L2(£l), then u G H2(£l) and \\u\\h2 < C||/||i,2. In particular, — A+I is an isomorphism from H2(Q)C\Hq (£2) onto L2(f!,). Concerning LP estimates, we have the following result. Proposition 1.5.1. Let A>0? ueH&(Q), and f eH~l(Vt) satisfy -Au+Xu = f. If f G Lp(fi) for some p G [1, oo), then u G LP(Q) and A||w||i> < ||/[|lp- 16 1. preliminaries PROOF. Let (p : (0, oo) —* [0, oo) be smooth. Assume further that ip(s) and sip'(s) are bounded on (0,oo) and that p,ip' > 0. It follows that ip(\u\)u 6 i?o(fi) for all Hq(Q) (see Remark 1.3.1 (v)). Moreover, one easily verifies the following identity. Re (Vu • V(v?((w|)il)) = 0. Taking the H 1 — H1 product of the equation —Au + Xu = f with <^(|u|)u, we deduce that a J HV(WI) < J\f\\uM\u\) Xp J lufie + lu]2)1^ < J |/P If ip(s) < sp~2 for some p £ [l,oo), then |u|<£>(|u|) < dulVC^!))1^-; and so, by Holder's inequality, (1.5.1) A" j |u|V(l«l)< / i/r- For p < 2 and e > 0, let -For 2 < p < oo and £ > 0, let It follows from (1.5.1) that >? f-^-T7 < /l/lP- 7 (1+eH2)^ ~7 ' ' Letting e i 0 and applying Fatou's lemma, we obtain ueLP and A[|u||iP < ||/||l»>- □ Next, we recall some convergence results. Given e > 0, we define the operator J£ on if-1 (ft) by Jeu=(/-eA)~1. In other words, for every / € H~1(Ct), us = Je/ € #o(H) is the unique solution of u£ — £:Aiis = /, We deduce from what precedes that ||Je/j|x < whenever X = i^(ft),L2(ft),tf_1(^)> or X - Lp(tt) for 1 < p < oo. In particular, Je can be extended by continuity to an operator of C(X) with || Je\\c{x) < 1- Furthermore, we have the following result. Proposition 1.5.2. IfX is either of the spaces H^(Q)r L2(Q), H'1^), orLp(tt) for 1 < p < oo, then: (i) (JJ,g)x,x* = (/, Jeg)x,x* for allfeX,ge X*. (ii) Jef —> / in X as e I 0 for every / el. (iii) If f£ is bounded in X as e j. 0, then Jef£ — f£ —^ 0 in X as e [ 0. 1.5. elliptic equations 17 PROOF, (i) Let f,g € X>(ft) and let u = J£f, v - Jeg. It follows that (Jef,g)x,x* = {Jef,g)L* = (u,-eAv + V)L2 = (-eAu + u, v)L2 = {/, Jeg)L2 = (f, J£g)x,x* ■ (i) follows by density of V(Vt) in X. (ii) By density, we may assume / g £>(ft). Let u€ = J£f. One easily verifies that u£ - f = JE(f - (I - eA)f) = eJ£Af, and so (I - A)/ in #-1(ft), which implies that u£ —* u in #o(ft). Hence (ii) is established for X = Hq(CI). Finally, let 1 < q < p < r < oo. It follows from Holder's inequality that IK - /lU, < IK - /||£(rq) IK - /u*-" . Note that ||ue - f\\Lr < \\ue\\Lr + \\f\\Lr < 2||/||Lr. If p > 2, we let q = 2 and we obtain 11«, - IU* < IK - /IIeHwHlOS^ ^o. If p < 2, we let r = 2 and we obtain a similar conclusion. This completes the proof of (ii) (iii) Let u£ — J£f£. We know that ue is bounded in X; and so it suffices to show that u£ - f£ -> 0 in P'(ft). Given ip g £>(ft), (ue - /e, v7)p',D = e(u£, Aip)w,v —► 0. Hence the result follows. □ Suppose now ft = RN. Applying the Fourier transform, we see that u = Jsf, being the solution of u — eAu = f, is given by (1 + 4s7r2\^\2)u = /. In particular, J£ can be extended to an operator ^'(R^) —> S'flR^). We have the following result. Proposition 1.5.3. Suppose ft — RN and let Js = (I - eA)'1 for e > 0. Given any s £ R, it follows that Je is a contraction of HS(RN) and that J£ g £(HS(RN),HS+2(RN)) with \\Je\\C(H;H>+2) < max{l, (to2)"1}. Proof. Let / g S'(Rn) and w = Je/, so that u = (1 + 4£ir2\£\2)-lf. We see that |«| < |/| and (1 + |£|2)|S| < (1 + K|2)(l +4£7r2|e[2)-1|/| < max{l, {4eTT2)~1}\f\. The result follows from the definition of HS(RN) (see Section 1.4). □ 18 1. preliminaries 1.6. Semigroups of Linear Operators Consult, for example, Brezis [43], Brezis and Cazenave [44], Cazenave and Haraux [64, 65], Haraux [158], and Pazy [294]. Let X be a complex Hilbert space with norm \\-\\x and sesquilinear form (-,-)x-We consider X as a real Hilbert space with the scalar product (x, y)x = Re{x, y)x- Let A : D(A) C X —> X be a C-linear operator. Assume that A is self-adjoint (so that D(A) is a dense subset of X) and that A < 0 (i.e., (Ax,x) < 0 for all x g Z?(^4)). .4 generates a self-adjoint semigroup of contractions (S(t))t>o on X. -D(A) is a Hilbert space when equipped with the scalar product (x,v)d(A) = (Ax,Ay)x + (x,y)x corresponding to the norm IMIl,^) = H^Hx + \\u\\x and D(A) t-> X (D(A))*, all the embeddings being dense. We denote by Xa the completion of D(A) for the norm H^Hx^, = \\x\\x — {Ax, x)x- Xa is also a Hilbert space with the scalar product defined by (x, y)A — {x,y)x - {Ax,y)x for x,y g D(A). It follows that D(A) ^XA^X^XA^ (D(A))*, all the embeddings being dense. Furthermore, it is easily shown that A can be extended to a self-adjoint operator A on (D(A))* with domain X. We have A\D{A) = A, A\D(A) 6 C(D(A),X), A\Xa g ^d A|x g {D(A))*). Since ^4 is self-adjoint, iA : D(A) c X —> X defined by («A)ar = iAr for x g D(A) is also C-linear and is skew-adjoint. In particular, iA generates a group of isometries {7{t))teR on X. We deduce easily from the skew-adjointness of iA that T(t)* = 7(-t) for every teR. We know that for every x g £>(^4), u(t) = CT(t)x is the unique solution of the problem C u€ Ci(R,D(A))nC1(R,A"), < + Au = 0 for all t g R, at ^ u(0) = x. Moreover, ||u(i + h) - u(t)\\ < \h\\\Ax\\ for all t,x g R. Next, it follows easily from the preceding observations that (7(t))tern can be extended to a group of isometries {T(t))t£U on (D(A))*, which is the group generated by the skew-adjoint operator iA. 7{t) coincides with T(t) on X, and (T(t))tgK restricted to any of the spaces XA, X, Xa, D(A) is a group of isometries. For convenience, we use the same notation for 7(t) and 7(t). We know that for every x g X, u(t) = 7{t)x is the unique solution of the problem ' u g C(R, X) n C^R, {D(A))*), _ < i— + Au = 0 for all t g R, at k u(0) = x. 1.6. semigroups of linear operators 19 In addition, the following regularity properties hold: IfxeXA, thenu£C(R,XA)r\C\R,X*A); if x g D(A), then u g C(R, D{A)) n C1^, *) • Concerning the nonhomogeneous problem, we recall that for every x g X and every / g C([0, T], X) (where T g R), there exists a unique solution of the problem (1.6.1) ( ueC([Q,T},X)nC1([0,T},(D(A)r), + ~Äu + f = 0 for all t g [0, T], u(0) = x. Indeed, w g C([0,T], X) is a solution of the above problem if and only if u satisfies (1.6.2) u{t) = 7{t)x + i J 7{t- s)f{s)ds for all t g [0, T]. Jo Formula (1.6.2) is known as DuhamePs formula. It is well known that if, in addition, / g W1'1((0,r),X) or / g Ll((Q,T),D(A)), then Remark 1.6.1. For every x g (D(A))* and / g L1((0,T), (D(A))*), (1.6.2) defines a function u g C([0,T], (D(A))*). A natural question to ask is under what additional conditions u satisfies an equation of the type (1.6.1). Here are some answers. (i) If, in addition, x g X and u g W1,l((0, T), (D(A))*)) or u g ^((O^X), then u satisfies (1.6.2) if and only if u satisfies ' u g ^((O, T), X) n W1'1^, T), (D(A)T), ckub ■_ < i—+-Au + / = 0 a.e. on [0,T], (J/L k u(0) = x. (ii) If x g X and / g C([0,r], (Z>(>1))*), and if u g C1([0,T], (D(^4))*)) or u g C([0,T],X), then « satisfies (1.6.2) if and only if u satisfies (1.6.1). (iii) Similarly, if x g XA, f g L1([0,T],XA) and if u g ^((O.T),^)) or if u g L1((0, T),X,4), then u satisfies (1.6.2) if and only if u satisfies ' u g L^nx^) n W^ttO.T),^), < ij+Iw + / = 0 a.e.on[0,T], tit ^ u(0) = x. 20 1. preliminaries (iv) Let x g XA and / g C([0,T],X^). Hue C^^T],^)) or if u g C([0, T], Xa)-> then u satisfies (1.6.2) if and only if u satisfies (ueC([0,T},XA)nC1([0,TlX*A), + Au + f = 0 for all t g [0,T], < u(0) = x. (v) Also, if x € D(A) and / g jL1([0,T,],X), and if u g W1'1((0,X,),X)) or if u g L1((0, T), I?(^4)), then u satisfies (1.6.2) if and only if u satisfies f u g Ll{{0,T),D{A)) n W1,:l((0,T),X), du . i— + Au + f = 0 a.e. on [0,T], u(0) (vi) Suppose i g Z>(A) and / € C([0,T],X). If u g C1([0,T],X)) or « g C([0,T], £>(J4)), then ti satisfies (1.6.2) if and only if u satisfies UEC{%TlD{A))nC\[^T],X), du i — +Au + f = 0 for all t g [0,T], (0) x. 1.7. Some Compactness Tools It is well known that the embedding i?1(MAr) t—> L2(RN) is not compact. In order to pass to the limit in certain problems, we will use some specific tools that take into account the lack of compactness. The first one is due to W. Strauss [323] (see also Berestycki and Lions [25]). Proposition 1.7.1. Let (un)n>o c H1(RN) be a bounded sequence of spherically symmetric functions. If N > 2 or if un(x) is a nonincreasing function of \x\ for every n > 0, then there exist a subsequence (unk)k>o and u g H1^™) such that unk —► u as k —> oo in LP(RN) for every 2 < p < 2N/(N - 2) (2 < p < oo if N = 1). Proposition 1.7.1 is an immediate consequence of the following two lemmas. Lemma 1.7.2. Let («n)n>o be a bounded sequence in Hl(RN). Suppose un(x) —> 0 as \x\ —» oo, uniformly in n > 0. It follows that there exist a subsequence (unk)k>o and u g i71(MJV) such that unk —> u as k —> oo in LP(RN) for every 2 < p < #*5.(2o such that unk —1 u as n —+ oo in H1{RN). Fix s > 0 and let R > 0 to be chosen later. Given p as in the statement, we have \\Unk - U\\LP^N) - \\unk - u\\Lp(Br) + \\unk - u\\lp({|x|>.R}) j>~2 < IK* - u\\lp(Br) + \\Unk ~ u\\L~({\xl>R})\\Unk ~ tt||z,2(r")- 1.7. some compactness tools 21 Let S > 0. We first fix R large enough so that (by uniform convergence) llunfc - wlll,»({|x|>fl})llwnfc -U||l2(R") < g - Next, since (wnfc|sH)fc>o is bounded in Hl(Bn)1 it follows from Rellich's compactness theorem that unJsR —> u|sn in Lp(Br). Therefore for k large enough we have ll«nfe - < g ' and so \\unk — w||/>(rjv) < e. This proves the result. □ Lemma 1.7.3. Ifue H\RN) is a radially symmetric function, then (1.7.1) sup jarl^lu^)! < Cj|w|||a||Vu||Ja . If, in addition, u(x) is a nonincreasing function of \x\, then (1.7.2) sup \x\%\u{x)\ < C\\u\\L2 . x€RN proof. Suppose first u e C%°(RN). We have /oo j poo ^-s{sN^u{s)2)ds < 2 J sN'lu(s)u'{s)ds, and (1.7.1) follows from the Cauchy-Schwarz inequality. If u(x) is a nonincreasing function of then \u\\h > J \u(x)2\dx > \{\x\ < R}\\u(r)\2 , {\x\ 0 and let the concentration function p(u, •) be defined by (1.7.3) p{u,t) = sup / \u{x)\2dx for.t>0. {\x-y\ 0, and lim^oo p(u, t) = a. 22 1. preliminaries (ii) There exists y(u,t) € RN such that {\x-y(u,t)\ 2, then \p(u,t) - p(u,s)\ 0 with C = C{N,r). Proof. Property (i) is immediate. Next, given t > 0, there is a sequence (yn)neN c MN such that = lim / \u(x)\2dx > 0. n—*oo / {|*-»n|<*} We claim that the sequence (yn)n>o is bounded. Otherwise, there exists a subsequence {ynj)j>o such that {\x - ynj\ < t} n {\x - yne\ < t} = 0 for j ^ £, and so \u\2>^2 J \u(x)\2dx = +oo, ffiw J-°{|a:-v„J.|o has a convergent subsequence, and its limit y{u,t) satisfies (ii). Finally, consider 0 < s < t < oo. We have \p(u,t)-p(u,s)\ = / H2- / M2 {\x-y(u,t)\o c H1^1*) be such that (1.7.4) ||un||l2 = a > 0, (1-7.5) SUp [[VUnH^a < CO , and let p(un,t) be defined by (1.7.3). Set (1.7.6) ji = lim liminf p(un,t). t—*oo n—*oo 1.7. some compactness tools 23 Then there exist a subsequence (unk)k>o, a nondecreasing function j(t), and a sequence tk —► oo with the following properties: (i) p(unk,-) —+ 7(-) e [0,a] as k —> oo uniformly on bounded sets of [0,oo). (ii) p = limf^007(i) - limfe-^oo p(unkrtk) - lim^oo p(unk,tkf2). Proof. We deduce from (1.7.6) that there exist tk —► co such that (1.7.7) p ~ lim p(unk,tk). k~*oo Note that p(un,t) < \\un\\2L2 < a. Since H1(RN) ^ Lr(RN) for some r > 2, it follows from (1.7.4), (1.7.5), and property (iii) of Lemma 1.7.4 that p{un,-) is uniformly Holder continuous. Therefore, (i) follows from Ascoli's theorem (after renaming the sequence nk). Note that we do not loose property (1.7.7) by passing to a subsequence. Since p(un, •) is nondecreasing, it follows from (1.7.7) that (1.7.8) limsupp( unk, ^ ] < hmsupp(unk,tk) = p. Next, for every t > 0, lim inf p(unk, t) > lim inf p(un, t), k—>oo n—>oo so that, by letting t —► oo and using (1.7.6) and (i), (1.7.9) lim7(t)>/x. t—yoo Finally, given t > 0, we have tk/2 > t for k large, so that Letting £ —> oo, we obtain (1.7.10) Uminfp^nfc,|^ >p. Part (ii) follows from (1.7.8), (1.7.9), and (1.7.10). □ Proposition 1.7.6. Let (un)n€K C i71(EiV) aaiis/y (1.7.4)-(1.7.5),let p(un,t) be defined by (1.7.3), and let p be defined by (1.7.6). There exists a subsequence (unk)k>o that satisfies the following properties. (i) If p = a, then there exists a sequence (yk)k>o C RN and u € H1(RN) such that unk (• - yk) —► u as k —> oo in LP(RN) for all 2

co for all 2 < p < (2 < p < oo if N = l). 24 1. preliminaries (iii) There exist (vk)k>o,(wk)k>o C H1^1^) such that (1.7.11) supp vk n supp wk = 0 , (1.7.12) \vk\ + \wk\ < Kfc|, (1.7.13) \\vk\\Hi + IJWfclUi < C\\unk\\Hx , (1-7.14) K||2,2 —> /i, IkjtlLa —» a-At, (1.7.15) hminf [J\Vunk\2 - J\Vvk\2 - J |Vtofe|2j >0: (1.7.16) j KJP~ J\vk\p- J\wk\p —► 0 k—>oo for all2 0, where p is defined by (1.7.3). Proof. Let (Qj)j>o be a sequence of open, unit cubes of R such that Qj n Qk = 0 if j 7^ k and \Jj>oQj ~ It follows that /oo ~ qq /. |u|«+* = £ / |«r+2 and ||u|| = £ / dVwl2 + M2) • We now proceed in two steps. Step 1. There exists a constant C independent of j such that 2 (1.7.18) j \uf^ < c(^j\u\2j ^ (^J\Vu\2 + \u\2^j for all u £ H^Qj). Qj Qj Qj Indeed, suppose first N > 3. It follows from Sobolev's embedding that and (1.7.18) follows by using Holder's inequality. Suppose now N — 2. It follows from Sobolev's embedding that ' \\u\\L»(Qi) < C(\\Vu\\Li{Qj) + \\u\\LHQj)) . Changing u to \u\2 and using the estimate |V|u|2| < 2|u||Vw| together with Holder's inequality, we obtain (1.7.18). Suppose now N = 1. Sobolev's embedding yields I|w||l~(q,-) < C(\\Vu\\LHq.)) + ||u||li(q.)) . 1.7. some compactness tools 25 Therefore, changing u to \u\2, we deduce as above that Hioo(Qj.) < ^IMIz^q^OIVuII^q.)) + ||wjjL2(Q.)), and (1.7.18) follows from Holder's inequality Finally, the fact that in the above calculations the constant is independent of j follows from translation invariance. Step 2. Proof of (1.7.17). Summing on j the inequality (1.7.18), we obtain _2_ J \uf^ < C^sup j \u\2^j " J\Vu\2 + \u\2 , and (1.7.17) for t = 1 follows easily. The general case of (1.7.17) is obtained by changing u(x) to u(tx). □ Proof of Proposition 1.7.6. We use the functions 7(-) and y(-,-) and the sequences (unk)k>o and (tk)k>o constructed in Lemmas 1.7.4 and 1.7.5. Fix T sufficiently large so that 7(T) > a/2 and let = y(unk,T). By possibly extracting a subsequence, we may assume that there exists u e H1 (RN) so that (1.7.19) unk(--yk)--u in H1(RN) as fc —> oo. We now proceed in three steps. Step 1. Proof of (i). Suppose /i = a. We claim that if u is given by (1.7.19), then (1.7.20) !|u|||2 =a, from which (i) follows. We now prove the claim (1.7.20). Note that, since the embedding Hx(Br) ^ L2(Br) is compact, (1.7.21) / \u(x)\2dx= lim / \unk(x)\2 dx for every R>0. J k^J{\x-yk\ a/2 for k large. Fix e < a/2 and let r be large enough so that p(unk, r) > a — e for k large. Since J \unh\2 + J \unk.\2 > ~ + a - e > a for k large, {\x-yk\ J Wnk\2 > a — e for k large. {{x-yk\ j \u(x)\2 dx > a - £. {\x\ 0 as k —> oo, and (ii) follows from (1.7.17). Step 3. Proof of (hi). We fix 0, y e C°°([0, oo)) such that 0 < 0,

~ , 3 <^(t) = 0 for 0 < t < - , 1, and we set where ftw = g("-yy»/2)l), ^)^(l'-^;"/2)l). Properties (1.7.11)—(1.7.13) are then immediate. Next, Hu-2 r y kj2< y kp {|*-y{ti„fc,W2)| ^ by Lemma 1.7.4(ii). Set now zk = unk —v^— Wk, so that in particular \zk\ < \unk\ We have KJ2 {tk/2<\x-y(unk,tk/2)\ |2 1.7. some compactness tools 27 so that (1.7.23) ||*fc||2a —♦ 0 K—*00 by Lemma 1.7.4(ii). We deduce from (1.7.11), (1.7.22), and (1.7.23) that \\u>k\\h J—* a~ A, k—»oo which proves (1.7.14). Next, one easily verifies that ,r-Mp-Kii -S|Wnfc|2 - 7^Kfe||Vunfc| from which (1.7.16) follows. □ CHAPTER 2 The Linear Schrodinger Equation This chapter is devoted to the study of some fundamental properties of the (linear) Schrodinger equation. We study in particular the dispersive properties and the smoothing effect of the equation in RN. 2.1. Basic Properties Let Q be an open subset of RN (fi does not need to be smooth or bounded). We define the operator A on L2(Q) by D{A) = {uE Hi{Q), Au £ L2(Q)}, Au = Au for u £ D(A). Evidently, D(A) = H2(£l) D Hq(Q) if Q is smooth enough. It is well known that A is self-adjoint and < 0, and so we may apply the results of Section 1.6. Observe that the space XA is nothing other than Hq{Q). Indeed || • |U = |j • and V(Q) C D(A), so that H$(Q) C XA. Since also D(A) is a subset of H%(Q), we see that XA C Hq(Q), and so XA — Hq(£1) with equality of the norms. It follows that X*A = ff-1(fi). On the other hand, note that D(A) / Hq(Q), and so D(A)* ^ H~2{n). The operator A £ £(L2(n), (D(A))*) is simply defined by (A~u,v)(rnA)y,D(A) = (u,Av)L2 for u E L2(Q) and v £ D(A). Let us denote by (T(i))teR the group of isometries generated by %A in any of the spaces D(A), Jf i, then ^(tn)^n 7(t)

(^T(-*)«)5^ = (rr(t)¥'^>5',5 n—►oo by theorem 2.17 of [304]. (ii) It follows from (2.2.1) that \F{7{t)ip){£)\ = |v>(£)l f°r all £ £ R and £ £ RN■ In particular, we see that for all s £ R and £ £ R, IIWvIIjt- = IN*- • Since S(RN) is dense in H*(RN) for all s £ R, we deduce that for any s £ R, (T(t))teR can be extended to a group of isometries in HS(RN), which we still denote by (CT(£))t6K. The generator of (7(t))t€R in HS(RN) is the operator As defined by D(AS) = HS+2(RN), and Asu = iAu for u £ D{AS). It follows easily that if

0CJ'(R,JHr*-2J(RAr)). (iii) Let 7 be a bounded, open interval of R with 0 £ 7. Let s £ R, <^ £ HS(RN), f £ L1(7,77s(RiV)), and « £ C(7,ifs(R*)). We deduce from (ii) above and the results of Section 1.6 that u satisfies u(t) = 7{t)ip + i I 7(t- s)f(s)ds Jo for t £ 7 if and only if u £ Wl'l{I, HS~2(RN)) and J mt + Au + f - 0 for t £ 7, I «(0) = v- 2.2. fundamental properties in RN 31 If, in addition, / € C(I,Ha~2(RN)), then u e C^I.H3-2^)). Remark 2.2.2. Here are some comments on the scaling properties of 0, and set ip{x) = tpfrx) so that e L2(RN). It follows that for all t G R, (2.2.2) [7(t)^}(x) = [7(j2t)ip}(jx) a.e. Indeed, let u(t) = CT(t)t^ and set v(t, x) = u(72£, 7a;). Since iut + Au = 0, it follows that ivt 4- Av = 0; and since v(0,x) = ip(yx) = ip{x), we see that = 7{t)tp, Similarly, let / G £1(R,I<2(R7V')), 7 > 0, and set g(t,x) = -ff^H^x). If u(t) = i f 7{t- s)f(s)ds, v(t) = i f T(t - s)g(s)ds, Jo Jo then for all t € R, (2.2.3) v(i,x) = w(72i,7x) a.e. Indeed, both v and w defined by w(t, x) = w(72i,7^) are solutions of the equation izt + Az + f — 0 with the initial condition z(0) = 0, so that v = w. These calculations are justified when ip and / are sufficiently smooth ( G H2(RN) and / G L1(R,i/2(MN)), say). Then (2.2.2) and (2.2.3) follow in the general case by a density argument. The following well-known result is the fundamental estimate for 7(t). Proposition 2.2.3. If p G [2,00] and t ^ 0, then 7(t) maps LP'(RN) continuously to LP(RN) and (2.2.4) ||T(tMUp(R«) < (4»r|*l)"N(*"')[|y|li^(Ri») for all tp G LP'(RN). The proof of Proposition 2.2.3 relies on the following lemma. Lemma 2.2.4. Given t 0, define the function Kt by N_ Kt(x) = (-^~] 2 for x G RN. \ Aitit J It follows that 7(t)(p = Kt * (x/47ni), then 7{t)ip = MtDtT{Mty). In particular, estimate (2.2.4) is optimal in the sense that if 7(t) £ C(Lq, Lp), then necessarily 2 < p < oo and q = p''. Corollary 2.2.6. Ift^O, then W(t)P(RN) and HS'P'(RN). Moreover \\ntMB*iq < (47r\t\rN^'^y\\B;rti for all

,q(RN). PROOF. Fix t ^ 0 and let u = 7(t)tp. Given w £ S{RN), it follows from (2.2.1) that J^\wu(t))=J:'1(we-4ir2i^2tip) (2.2.6) 2 2 = T~x{e-^ ^ 'fr1^)) = 7(t)(J=-\w^)). In particular, it follows from (2.2.4) that W^-^wu)]]^ < (47r|il)-JV{^p)||^'-1(w^)||LP/ for any 2 < p < oo. The result follows immediately from the above estimate and the definitions of the various Sobolev and Besov norms (see Section 1.4). □ 2.3. strichartz's estimates 2.3. Strichartz's Estimates Estimate (2.2.4) is remarkable but it is not quite handy for solving the nonlinear problems, since the Lp spaces are not stable by 7(t). However, we will derive from (2.2.4) space-time estimates that are essential for solving the nonlinear Schro-dinger equations. The first estimates of that kind were obtained by Strichartz [327] as a Fourier restriction theorem. Strichartz's estimates were generalized by Ginibre and Velo [136], who gave a remarkable, elementary proof. Strichartz's estimate for the nonhomogeneous problem was generalized by Yajima [364] and by Cazenave and Weissler [68]. Finally, the endpoint estimates were established by Keel and Tao [210]. We begin by introducing the notion of admissible pair. Definition 2.3.1. We say that a pair (o,r) is admissible if and 2N (2.3.2) 2 3. Theorem 2.3.3. (Strichartz's estimates) The following properties hold: (i) For every

2, there exists a dense subset Ev of 1? such that 7{t)ip & Lp for every

3. Note that by the preceding observation, the restriction "for a.a. t g R" cannot be reduced to "for all t ^ 0" in general. Concerning property (ii), note that the definition of <]>/ makes sense. Indeed, Lp' H~l, and so / g Ll(I',H~l) for every bounded interval /' c I. In particular, g C{I'^H~x). Evidently, properties (i) and (ii) give an estimate of the solution of the nonhomogeneous Schrodinger equation in terms of / and , the operators # and 6t (where t g (0,T) is a parameter) by and ^ */(*) = J\{s-t)f{t)dt Vsg[0,T) t = / 1{s-a)f(a)da Vs g [0,T) It is clear that both * and 0t are continuous L}oc([0, T),if"1) -» C([0, T), H'1). Step 1. For every admissible pair (q,r), the mappings t are continuous l/jj0.T).Lr'jRN)) -> I/?((0, T), Lr(RN)). We only prove the estimate for 4>7"the other ones being obtained similarly. By density, we need only 2.3. strichartz's estimates 35 consider the case / G Cc([0,T),Lr ). In this case, Proposition 2.2.3 shows that $/ E C([0,T),Lr), and that ||*/(t)||Lr < / \t-s\-N(i-$)\\f(S)\\Lr,ds< [ \t-8\?\\f(8)\\Lsd8. Jo Jo It follows from the Riesz potential inequalities (cf. Stein [319], theorem 1, p. 119) that ^— — ||*/IIl«((0,T),Z,'-) < C\\f\\Li'((0,T),L<-') where C depends only on q. Step 2. For every admissible pair (q, r), the mappings and Qf are continuous L«'((O.T).Lr'(RN)) -» <^[n.Tl.TWv)V We only prove the estimate for L*((0,T),Lr{RN)). Let / G L1((0,T),L2) and consider

{(0,T),L*) (2.3.6) Jo < C\\f\\m(0,T),L2)\\') ■ Thus $ is continuous ^'(,(0,1),^'^)) -+ L«((0,T), Z7(RN)). Let now (g, r) be an admissible pair for which p < r, and let p 6 [0,1] be such that 1 p 1 — p , ■ 1 u 1 — a 7' 1 q' p' 2 r> By Steps 1 and 3, $ is continuous L*1'((0,T), Lr'(RN)) -> L9((0,T),Z7(RN)) and L1((0,r),X2(RJV)) Z,9((0,T),i7(R*)). By interpolation (see Bergh and Lof-strom [28], theorem 5.1.2, p. 107), <& is continuous IT ((0, T), L5 (RN)) Lq ((0, T), U (RN)) for every pair (cr, 5) such that for some 9 £ [0,1], 1 9 1-9 1 9 1-6 - = - + —— and - = - + —— . a 1 q' 0 2 r' The result follows by choosing 9 = p. Step 5. Proof of (i). The proof is parallel to the proof of (ii), and we describe only the main steps. Let /+00 /-+oo T(t-s)f(s)ds and Tf= 7(-t)f(t)dt. -00 J—00 One shows (see Step 1) that \\^j\\l"((0,T),L') < C||/IL<7'((0,T),L'') for every admissible pair (g,r). Deduce (see Step 2) that ||r/||L2 < CH/IIl^cT),^-') > 2.3. strichartz's estimates 37 from which one obtains that ■+oo /+oo / p+oo \ (7{t)if^(t))L,dt = U,J 7{-t)1>{t)dt) < C\\\\L«'({0,T),Lr') for every ip 6 L2(RN) and ^ e Cc{[0,T),V(RN)). Assertion (i) follows (see Step 3). This completes the proof. □ Corollary 2.3.6. Let I = (T, oo) for some T > -oo (respectively, I = (-oo, T) for some T < oo) and let J = I. Let (7, p) be an admissible pair, and let f £ L1' (I.L^' (RN)). It follows that the function ti->$f(t) = J 7(t~s)f(s)ds (respectively, $/(t) = J 7(t - s)f (s)ds^ for every t G J, makes sense as the uniform limit in L2(RN), as m —> +00 (respectively, as m —> -co), of the functions 7(t - s)f(s)ds for every teJ. In addition for every admissible pair (q,r), $/ e Lq(I,Lr(RN)) n C(J,L2(RN)). Furthermore, there exists a constant C such that < C|l/ilLV(/,L,') for every f € LP1'(I,Lp'(RN)). L2 proof. We consider, for example, the case / = (T, 00). Let j, m be two integers, T < j < m. For every t € J, ||*?(t)-*}(t)||L2 - ||T(m-t)($7(t)-^(i))||L2 = / 3*(m - *)/(*)<* By (2.3.5), there exists a constant C(7) such that jj$7(i)-^(t)||L2 T, define £ e i7' (/, i/(RN)) by Since /_,- -+ / in L1'(I, L?' (RN)) as j 00, we deduce from (2.3.7) that (2.3.9) inL2(Rw) uniformly in t € J. 38 2. THE LINEAR SCHRODINGER EQUATION Note that for m > j, <3>™ is independent of m. It follows from (2.3.8) that € L7'(^5^'(KiV)) Furthermore, letting T < j < k, we deduce from (2.3.8) that II*/; - ®h \\l»(1,L-) < 0\\fj - fk\\Li'{I,Lp') ^ CWf\\Lt'(ti,k),Lp') ■ In particular, <&/., is a Cauchy sequence in Lq (I, Lr (RN)), which possesses a limit ip such that, by (2.3.8), (2-3.10) HhHI,Lr) 0 as i ±oo. proof. Let g be such that (g, r) is an admissible pair. It follows from Gagliardo-Nirenberg's inequality that there exists C such that for every t, s g R, ||u(t) - u(s)\\L, < C\\u{t) - t \\u(t) - u(s)\\^f . Since

■ H'1^1*) (see Theorem 1.3.10). Therefore (see Remark 1.3.8(iii)), there exists C such that \\u{t) - u(s)\\L2 0, we may replace ip by Da

rt2) i2 for every ip 6 HS(RN). (ii) If (q,r) is an admissible pair, then there exists a constant C such that for every (p g S'(RN) such that \\ip\\gs < oo. (iii) Let I be an interval of R (bounded or not), let J = I, and let to 6 J. 7/(7, p) and (q, r) are admissible pairs, then there exists a constant C independent of I such that \\*fh*(i,B;a), j for some constant C independent of I. Proof. We only prove the homogeneous estimates (ii) and (iv), the proofs of (i) and (iii) being similar. Also, we assume q < oo, the necessary modifications to treat the case q = oo are obvious. Let rj and ipj satisfy (1.4.1)-(1.4.2). Step 1. Proof of (ii). We set u(t) = 7{t)

1, we obtain !>(■) j Lf(R) < E Haj(')llLP(R) j = ^22sj'||0'(-)(^-1(|?|^^))"2 40 2. THE LINEAR SCHRÓDINGER EQUATION Applying (2.3.3), we deduce that Lit = C\\i/(t)). 3. , 2 ll*(/^ia)=(/(Ell^(OII^)adi i 3 Arguing as in the proof of (ii) above, we obtain Ml <£|| 1- It follows that i ). Concerning (2.3.3), the answer is no. One sees easily that the condition (2.3.1) is necessary. Indeed, assume (2.3.3) holds for some pair (g,r) with q,r > 1. Fix 6 e L2(RN), 6 ^ 0 and, given 7 > 0, let tp{x) = 6(^x). Setting w(t) = 7(t)6 and u(t) = 7(t) 0, we obtain (2.3.1). In particular, we see that r > 2 (otherwise q < 0). If N = 2 and ( 3, r > 2N/(N - 2) is more easily eliminated (see Keel and Tao [210]). Assume now that (2.3.5) holds for some pairs (q,r) and (7, p). Changing / to a2f(a2t,ax) and applying (2.2.3), we obtain by arguing as above the necessary condition This is clearly satisfied if (q, r) and (7, p) are admissible, but (2.4.1) allows many more choices. We present here a simple case where (2.3.5) holds for nonadmissible pairs. This case corresponds to p — r in (2.4.1). proposition 2.4.1. Let I be an interval of R (bounded or not), set J = I, let t0 e J, and consider defined by (2.3.4). Assume 2 < r < 2N/(N-2) (2 < r < 00 if N = 1) and let 1 < a, a < 00 satisfy (2.4.2) It follows that € La(I,Lr(RN)) for every f 3, r > 2N/(N — 2) is more easily eliminated (see Keel and Tao [210]). 2.5. Space Decay and Smoothing Effect in TSLN We still assume in this section that ft = RN. We have seen in Proposition 2.2.3 and Theorem 2.3.3 that 7(t) has a smoothing effect in some LP spaces. On the other hand, one easily verifies with the formula of Lemma 2.2.4 that for every tp e ^(RN) supported in a compact subset ft of RN, the function (t,x) t—► 7(t) £ L2(RN). Ifu(t) = T(t)

(RN)). 0, 7(—t)Pa^ — xOL'J(-t)ip. In particular, (x + 2itV)7(t) = 7(t)x, or equivalently 7(-t)(x + 2itV) = x7(-t). corollary 2.5.4. Let tp e L2(RN) be such that \ ■ \ 0. In fact, much more can be said. 44 2. the linear schrodinger equation Theorem 2.6.1. Let ijj(x) = \x\~p with p e C and 0 < Rep < N. It follows that 7{t)if> £ Lr(RN) for allt>0 and all r such that ( N N 1 (2.6.1) r > max Rep1 N - Rep J ' For such r, (2.6.2) \\7{t)nL^t^-^\\7{l)nLr fort>0. Moreover, if u(t,x) = 7{t)ip(x), then u £ C°°((0,oo) x RN). Before proceeding to the proof, we make some simple observations. Given A > 0, let D\ be the dilation operator Dxw{x) = Xpw{Xx). Dx is defined on S(RN) and is extended by duality to S'(RN) by for all w G S'(RN) and all 4> e S(RN). It is immediate that (2.6.3) \\Dxw\\Lr = \R*p-Z\\w\\Lr whenever w £ Lr(RN). Moreover, it is easy to check, first by applying (2.2.2) for w € S(RN), then by duality for w £ S'(RN)} that "J(t)w = D\J(\2t)D\w. In particular, letting A = t 2, (2.6.4) T(t)w = Dj.T(l)D^w for all t > 0. Now if ip = \x\~p, then = ijj for all A > 0. Therefore, it follows from (2.6.4) that (2.6.5) 7(t)ij) = D_L.7{\)il>. vt In view of (2.6.5) and (2.6.3), all the conclusions of Theorem 2.6.1 follow if we show that (2.6.6) T(l)^ € C00^) and that (2.6.7) € Lr{RN) for all r satisfying (2.6.1). We next establish some notation and recall some well-known facts. The gamma function satisfies the following relation /■OO (2.6.8) c~zT(z)= / e~cttz~ldt, Jo valid for c > 0 and z £ C with Re 2 > 0. Also, if O denotes the domain of the standard branch of the logarithm; i.e., O = {z £ C : z is not a negative real number or 0} , then for a fixed complex number p, the function f(z) — zp — ep]ogz is analytic in O. Note that if r > 0, then (rz)p = rpzp for all z £ O. Also, \rp\ = r^p if r > 0. 2.6. homogeneous data in R 45 Another function that plays a central role in the analysis is given by (2.6.9) H(y;a,b)= f e^V^l - r)6"1 dr, Jo where a, b £ C with Re a > 0 and Refr > 0, and y E 3R (or C). Note that #(y; a, b) is separately analytic as a function of y, a, and 6 in the domains just specified. Lemma 2.6.2. Let ip{x) = \x\~v with 0 < Rep < N. For t > 0 and x £ RN, (2.6.10) [TW](x) = (4tt)"«r(p/2)-1i7^; |, ^) , where the function H is defined by (2.6.9). PROOF. The basic idea is to express \x\~p using the gamma function, then change variables so that the Gauss kernel Gs (x) — (4tts) 2 e 4, appears in the integral. It will then be possible to apply the operator ezA. By formula (2.6.8), if x ^ 0 poo \x\-p = T(p/2)-1 / e-WV^dt Jo = 4"2r(p/2)-1 / e-^s-i-^ds Jo pOO = 4-f(4Tr)Tr(p/2)-1 / Gs(x)s^-2-Us. Jo This integral, in addition to being absolutely convergent for each x ^ 0, is an absolutely convergent Bochner integral in Z^QR^) + Co(RN). In other words, poo V, = 4-2(47r)fr(p/2)-1 / GstfsVS-ids. Jo Next, we apply the heat semigroup, etA for t > 0, which gives (since etAGs = Gt+S) pOO e^V = 4-t(47r)^r(p/2)-1 / G^O^-?"1^. Jo This integral now is absolutely convergent in Co(RN), where pointwise evaluation is a bounded linear functional. Making the change of variables r = we see that for all x £ RN 1 Jtf-p-2 T{p/2){e^){x) = 4-f (4tt)t jf (a;) ' ^ (2-6.11) =(4£)-"(47rf)£ /'cx^r^a-O^dr = (4£)-§ / e—^-r5_1(l -r)~^dr. We next claim that formula (2.6.11) is valid not only for t > 0, but for all t E C with Rei > 0. Indeed, if n £ S(RN), then (etAip,r}} is an analytic function of t 46 2. the linear schrodinger equation on the open half plane Ret > 0, and continuous on the closed half plane Ret > 0. Next, if we integrate the right side of (2.6.11) against rj(x) over RN, the result is also an analytic function of t on the right half plane Ret > 0, continuous at least on the closed half plane with t = 0 removed. By the identity theorem, these two functions are equal on the open half plane. By continuity, they are equal also for t = ir, t e R, r ^ 0. Since n is an arbitrary Schwartz function, (2.6.11), as an identity between two tempered distributions, has been proved for all complex t ^ 0 with Ret > 0. This establishes the proposition. □ Corollary 2.6.3. Let ^(x) = \x\~p with 0 < Rep < N. It follows that T(l) € C00{RN)DL00(RN). Lemma 2.6.4. If y > 0, Re a > 0, and Refe > 0, and if n and m are nonnegative integers such that n + 2 > Re a and m + 2 > Re b, then m H(y; a, b) = y~a £ Ck(a, b)e~^y~k -a-m-1 m + 1 + Cm+1(a,b)y~ r(m + 2-b) (2.6.12) f-oo rl / , \ —a—m—1 St y x / [ (i-s)m(-i--) dse-Hm+l-bdt Jo Jo + eiVy-*±Ck(b,a)e-i^y-k fc=0 + Cn+1(b, ajjyy-"-"-1 " + 1 r(n + 2 - a) coo pi / nJ.\ — b—n— 1 x J J (1-sWi-^J ds e~Hn+1-a dt, where r9 6i^ ci (a h\ - r(a + k)r(k + i-b) (2.6.13) c*(a,&)-—--r(i _ 6) PROOF. For the moment, we assume that 0rdr$-ae-*t-6dsdi Jo Jo Jo = / / ~-—s-ae-H-bdsdt Jo Jo (2.6.14) iy - s + t rOO pOG /■oo /-oo -a = / / —-e~H-bdsdt Jo Jo -ty-t + s n Jo Jo n Jo Jo fOO /-oo J) + e*s/ / / -e-ss~adtds iy - s + t OO /-OO ^-a -iy — £ + s coo /•oo -6 iy - t + S We therefore consider the integral /•OO - + eW / .-r^dse-tt-adt. Jo Jo p(w) = Jo VJ + s ds where w E O and 0 < Re a < 1. It is known (by changing variables in the beta function) that p(l) = T(l — a)T(a). Next, if w is a positive real number, we set s — wt, and so f°° (wt)~a (2.6.15) P(w)= ~—-—wdt = w~aT(l - o)r(o), w>0. Jo w + wt Since p(w) and w~a = e_alog™ are both holomorphic in O, (2.6.15) is true for all weO. Substituting (2.6.15) back into (2.6.14) with w — ±iy — t, we see that H(y;a,b) = -^- H\-iy - t)~ae~H~bdt (2-6.16) + r7i~\e*W / (iy~trbe-H-adt. r(l - «) Jo The next step is to replace (—iy — t)~a and (iy — t)~b in (2.6.16) by their finite Taylor formulas around t = 0 with integral remainder terms. If f(t) = (—iy — i)~a and g(t) = (iy - t)~b, then /<*>(*) = o(o + 1) • • • (a + k - l)(-ty - t)-a~k , = 6(6 + 1) • ■ ■ (6 + fc - l)(-iy - t)~b~k . Since 48 2. THE LINEAR SCHRODINGER EQUATION and similarly for g{t), we see that T{a) ^ 1 '* + /* 1 r / — / (t - s)m/ 0, pOC pi / / (t-src-iy-s)-0-™-1^-'*-6^ Jo Jo poo pi / o+\ m—1 and so we obtain the formulation (2.6.12)-(2.6.13). Formula (2.6.12) has been proved only for y > 0, 0 < Re a < 1, and 0 < Re 6 < 1. On the other hand, the right-hand side is an analytic function in a for 0 < Re a < n + 2, with y > 0 and b such that 0 0 and a such that 0 < Re a < n + 2 fixed. (Recall that l/r(z) is an entire function.) It follows that (2.6.12) holds for all y > 0, and all a, b in the region stated in the lemma. □ Proposition 2.6.5. Let ip(x) = \x\~p where 0 < Rep < N. It follows that 7(l)tp e Lr(RN) for all r satisfying (2.6.1). Moreover, 7(l)ip(x) is given by the explicit formula (2.6.17) below for x ^ 0. 2.6. homogeneous data in R- 49 Proof. We apply the asymptotic expression from Lemma 2.6.4 to formula (2.6.10) in Lemma 2.6.2 with a = p/2, b = (N - p)/2, n > 5|£ - 2, and m > N~*ep - 2. We see that if x ^ 0, then (2.6.17) where and pr(i)VK*) m , | |2 \ -A: , /» / isI |_p (wC\ (m+ l)e 2 4 y r(m + 2-6) + e " |x| ^(4)2 P2^Bfc(6,a)e * (4 J xri1(i-s)"(i-s"6"""ldse",("+w*' Cjfe(a, 6) _ T(a + fc) r(fc + 1 - b) Ak{a,b) Bk{b,a) r(o) r(a)A:! T(l - b) Ck(b,a) _ F(b + k) r(fc + l - a) T(a) r(a)fc! T(l-a) ' By Corollary 2.6.3, T(l)^ € C°°(R^). Therefore, to determine whether 7{l)ip e Lr(RN), it suffices to consider |x| large. Proposition 2.6.5 now follows immediately from formula (2.6.17). □ Proof of Theorem 2.6.1. As observed before, we need only establish properties (2.6.6) and (2.6.7). They follow from Corollary 2.6.3 and Proposition 2.6.5, respectively. □ Remark 2.6.6. Here are some comments on Theorem 2.6.1. (i) Note that Ao(a,b) = Bo(a,b) = 1. Therefore, the term with slower decay in (2.6.17) is either of order \x\~p or of order \x\N~p, depending on p. Thus, if r does not satisfy the condition (2.6.1), then 7(l)ip (£ Lr{RN). If Rep < N/2, then T{l)ip behaves like \x\~p as |ar| -» 00. If Rep > N/2, then T(1)V> behaves like ce^2^\x\~N+p as \x\ 00. And if Rep = N/2, then 7(l)ip behaves like \x\~p + ce1^ ^4\x\~N+p as \x\ —> 00. In particular, jT(l)^j « |a;|-min{ReP^-Rep}j so that the decay ig at most \x\~N/2, This is justified by the fact that 7(l)ip cannot be in any Lq space with q < 2 for otherwise we would have %p € Lq . (ii) The conclusions of Theorem 2.6.1 hold for the more general homogeneous function xj)(x) ~ oo(x)\x\~p with 0 < Rep < N and u) homogeneous of degree 50 2. the linear schrodinger equation 0 and "sufficiently smooth." See Cazenave and Weissler [73, 74], Oru [286], Planchon [298], and Ribaud and Youssfi [302]. (iii) In view of (2.6.5), u(t) = 7(t)ip is a self-similar solution for the group of transformations ux(t,x) = Xpu(X2t, Xx). This can also be seen independently by observing that tp is homogeneous of degree —p. Corollary 2.6.7. Let ip(x) = \x\~p with 0 < Rep < N. Suppose

min ^ -—, ——-—V. v ; [Rep'iV-Repj It follows that (2.6.19) \\7(t)(

||0*(l)^[|Lr as t —> oo. Proof. By (2.2.4), \\m()hr <*-^iiv-^'- Hence (2.6.19) follows by using the assumption (2.6.18). The result (2.6.20) now follows from (2.6.19), (2.6.18), and (2.6.2). □ Remark 2.6.8. In view of (2.6.2) and (2.6.20), we can determine the possible decay rates as t —* oo of ||3"(£)y|U'- for 2 < r < oo. (i) The decay rate (as t —» oo) given by (2.2.4) is optimal, since (2.6.21) liminft^^HT^IlLr. > 0 t—»oo for every

0 is also a solution of Schrodinger's equation in ^'(R^). By duality, the same holds for

£ S'(RN). In particular, v(t) = "J(t)ift for some i/j G S'(Rn). Now, assuming by contradiction that JV(r—2) tn 2r \\T(tn)ip\\Lr —► 0 for some tn —+ oo , we deduce, setting sn = l/tn, that 11^(5^)^11^ = i^21 \\7{tM\Lr -» 0 as n oo . 2.7. comments 51 In particular, 7(sn)i> —> 0 in ^'(R^). By Remark 2.2.1 (i), we conclude that i/> = 0, thus (f = 0, which proves the claim. Note also that the maximal decay rate is indeed achieved if

G Lr'(RN). Thus |j7{t) N^2'> • ^n Pai"ticular, all possible decay rates between the maximal N(r-2) N(r-2) ~ decay t ^ and t tr are achieved by L solutions. On the other hand, the lower limit JV^~2^ is optimal (in fact it is not even achieved), at least for r < 2N/{N - 2) (r < oo if N = 1, r < oo if N = 2). Indeed, it follows from Strichartz's estimate that for such r's, IIWVII oo 2.7. Comments As we will see in Chapter 4, Theorem 2.3.3 is an essential tool for the study of the nonlinear Schrodinger equation in MN. Therefore, it is natural to ask if Theorem 2.3.3 can be generalized to a wider class of equations. In fact, a careful analysis of the proof shows that it uses only two properties. The first one is the identity T(t)* = 7(—t), which is valid for every skew-adjoint generator. The second one is the estimate (2.2.4), which itself follows from Lemma 2.2.4. Therefore, such an inequality holds whenever 7(t) has a kernel K(t) whose L°°-norm behaves like (at least near 0). In particular, we have the following result (see Keel and Tao [210] for more general results). Theorem 2.7.1. Let A be a self-adjoint, < 0 operator on X — L2(Q). Assume that there exists to > 0 such that for every t G (—to,0) U (0,£o)? 9"(0 = etiA maps Ll(Q) to L°°(Q), with a norm less than K\t\~~. The following properties hold; (i) For every

2, and there exists a constant C, depending only on K and q, such that \\n-) 0. 52 2. THE LINEAR SCHRODINGER EQUATION (ii) Let 0 < \T\ < oo. If (7,p) is an admissible pair with 7 > 2 and f £ Li ((0,T),Z/> (fi)), tfien /or efen/ admissible pair (q,r) with q > 2, the function t»$f(t)= [ 7(t-s)f(s)ds Jo belongs to L9((0, T), Lr(Q,)). Furthermore, there exists a constant C, depending only on K, 7, and q, such that '1 + |2T to |$/IU«((0,T),L'-) < C\ " ' ' ' ) \\f\\Ly'{{0,T)M) for every f £ L7'((0,T), Lp'(Cl)). In addition, $f £ C([0,T], L2(Q)). proof. Repeating the proof of Theorem 2.3.3, one shows that estimates (i) and (ii) hold for T = to- In particular, assuming q < 00, n° \\7{tMirdt 2. The reason is that in this case, L2(Q) <-»■ Lp' (Q), and so if such an estimate held, then I = 7(t)7(—t) would map L2(ft) -> L2(n) LP'(Q) -» Lp(n). This is absurd, since this would mean that L2(£l) <^-> LP(Q). However, note that estimate (2.2.4) might hold if, for example, Q is the complement of a star-shaped domain. Unfortunately, such a result is apparently unknown (see Hayashi [160]). On the other hand, estimate (2.2.4) (hence those of Theorem 2.3.3) hold in certain cones of UN. For example, they hold if Q = Rf. To see this, consider tp £ V(R%), and let Tp be defined by ip{xu ... ,xn) if xn > 0 fp{xlt...,xn) = <| -(p(xi,...,-xn) ifxn<0. It follows that tp £ T>(RN). Let u = 7(t)(p, where 7{t) is the group of isometries generated by iA in RN. One easily verifies, by uniqueness, that w|kn = 7(t) 0,0 < 9 < ir/2m} for some nonnegative integer m. Remark 2,7.3. The estimates of Theorem 2.3.3 fail in a bounded domain ft c RN. In the case where ft is a cube of RN there are, however, substitutes to these estimates that can be used to solve the Cauchy problem for the nonlinear equation. (In fact, this holds in the more general case of periodic boundary conditions.) On these questions, see Bourgain [35, 38]. Remark 2.7.4. Estimate (2.2.4) holds when one replaces the Laplacian by a more general pseudodifferential operator on RN (see Balabane [11, 12], and Balabane and Emami Rad [13, 14]). Remark 2.7.5. We note that the results of this chapter have been stated for the equation iut + Au = 0. It is clear that similar results hold for the equation iut + aAu = 0, where a 6 K, o ^ 0, which is equivalent by an obvious scaling. Remark 2.7.6. Consider the operator A = A - V, where V : R^ —> R is a given potential. If the negative part of V is not too large, then A defines a self-adjoint operator on L2(RN) (see for example, Kato [202]). If V is small enough in L1nL°°, then it follows from a perturbation method that 7(t) = ettA satisfies (2.2.4) (see Schonbek [308]). More general cases are considered in Journe, Soffer, and Sogge [200]. If V e C°°(RN) is nonnegative and if DaV e L°°(RN) for all \a\ > 2 (the model case is V(x) = \x\2), then also 7(t) = eitA satisfies (2.2.4) (see Fujiwara [115, 116], A. Weinstein [355], Zelditch [368], and Oh [277, 278]). On the other hand, such estimates do not hold in general for several reasons. First of all, A may have eigenvalues. Therefore, if A is an eigenvalue of A and if

0; i.e., consider the model equation 3.1. The Notion of Solution (3.1.1) (3.1.2) 55 56 3. THE CAUCHY PROBLEM IN A GENERAL DOMAIN We first observe that, multiplying the equation by u, integrating over Q, and taking the imaginary part, we obtain formally the conservation of charge (3.1.3) — / \u(t,x)\2dx = 0. n Therefore, the L2 norm of the solution is constant. Next, multiplying the equation by ut, integrating over fi, and taking the real part, we obtain formally the conservation of energy (3.1.4) jtE(u(t)) = 0, where the energy E is defined by >)r2} E(w)= I lhvw(x)\2--^— \w(x)\a+2 \dx, T x " a + 2' Finally, multiplying the equation by Vu, integrating over £7, and taking the real part, we obtain formally the conservation of momentum d f (3.1.5) dilm u(t>x)Vu(tix)dx = 0- When N — 1 and a = 2, equation (3.1.2) is completely integrable and there are infinitely many conservation laws. When a = 4/iV and Q, = RN, there is the pseudoconformal conservation law (see Section 7.2). In general, however, the only known conservation laws for (3.1.2) are (3.1.3), (3.1.4), and (3.1.5). Since (3.1.5) does not involve any positive quantity, only (3.1.3) and (3.1.4) can possibly provide useful estimates of the solutions. The above conservation laws suggest two possible "energy spaces," namely, L2(£l) associated with (3.1.3), and Hq(Q,) associated with (3.1.4). The point in working in an energy space is that, if there is a "good" local existence result, then the global existence of solutions follows from a priori estimates. These in general follow from the conservation of energy under some relevant assumptions on the nonlinearity. We will study the local Cauchy problem in L2 in Chapter 4. For the moment, we restrict our attention to solutions in Hq(Q), and we make the following definition. Definition 3.1.1. Consider g e C(H$(Q), H-1 (Q,)),

H~1(Q) is bounded on bounded sets), then Au + g(u) G L°°(I, i7_1(ft)). Therefore, if u satisfies iut + Au + g{u) = 0 in the sense of distributions, then u G W1'°°(7,H~l{Sl)). In addition, if u € C(I, Hq (ft)) satisfies iut + Au + g(u) = 0 in the sense of distributions, thenueC1^,^-1^)). (iv) We gave the definitions of weak and strong #o-solutions °f the Cauchy problem at t = 0, i.e., with the initial condition u(Q) = ip. Of course, given any to £ R, one can give similar definitions for the Cauchy problem at t = to, i.e., with the initial condition u(to) = u in C(I,Hq(Q)). 58 3. THE CAUCHY PROBLEM IN A GENERAL DOMAIN Remark 3.1.6. Here are some comments on Definition 3.1.5. (i) The property that (—Tmm, Tmax) is the maximum interval of existence means that if I 3 0 is an interval such that there exists a strong Hq -solution of (3.1.1) on /, then / C (-Tmin,Tmax). (ii) If 2max < oo (respectively, Tmin < oo), then by the blowup alternative limtTT,™. \\u(t)\\Hi = +oo (respectively, limu_Tin.n \\u(t)\\Hi = -t-oo). In this case, the solution u is said to blow up at jTmax (respectively, -Tmm). If Tmax = oo (respectively, Tmjn = oo), the solution is said to be positively (respectively, negatively) global. Note that in this case the blowup alternative does not say anything about the possible boundedness of ||u(t)||//i as t —> oo. (iii) Note that the continuous dependence property implies that the functions rmax and rmin are lower semicontinuous Hq(£1) —► (0,oo]. (iv) There are various notions of well-posedness in the literature. We adopted a quite strong notion of well-posedness by requiring uniqueness, the blowup alternative, and continuous dependence. 3.2. Some Typical Nonlinearities In this section, we introduce various classical models of nonlinearities. Example 3.2.1. The external potential. Consider a real-valued potential V : Q —» K. Assume that (3.2.1) V e Lp(Q) with (3.2.2) p>l, p>~. Let g be defined by (3.2.3) g{u) = Vu for all measurable u : Q —> C, and G be defined by (3.2.4) G{u)=l-jv{x)\u{x)\2dx for all measurable u : Q —► C such that V\u\2 € L1(S7). We have the following result. Proposition 3.2.2. Let V satisfy (3.2.1) and (3.2.2), let g and G be defined by (3.2.3) and (3.2.4), respectively, and set 2p (3.2.5) r = -~ . v ' p — 1 The following properties hold: (i) G € CHH^)^), g G C(ff0l(n),ff-andG'=g. (ii) 2 < r < ^ (2 < r < oo if N = 1). 3.2. SOME TYPICAL NONLINEARITIES 59 (iii) g £ C(Lr(fl),Lr'(fl)) and \\g(u)\\Lrf < \\V\\lp\\u\\lt for alluE Lr(fl). (iv) lmg(u)u = 0 a.e. in fl for all u £ Hq(Q). proof. Part (ii) follows from (3.2.2) and (3.2.5). (iii) is a consequence of (ii) and Holder's inequality. In particular, Hq(Q) <-> Lr(fl) and Lr> (fl) <-> H~1(fl) by Sobolev's embedding theorem, which implies that g £ C(Hq(fl), H~1(fl)). Next, it follows from Holder's inequality that 2||G(w|| < ||V"j|£,p[ju|||r, so that G is well defined on Hq(Q). Furthermore, n for all u,v £ Hq(Q), and one deduces easily that G £ C1(Hq(Q),R) and G' = g. Hence (i) is established. Finally, (iv) follows from the fact that V is real valued. □ Remark 3.2.3. Let V be a real-valued potential, V £ LP(Q) + L°°{Q). If p satisfies (3.2.2), then we may write V ~ V\ + V2, where V\ satisfies (3.2.2) and V2 satisfies (3.2.2) with p replaced by 00. In particular, we may apply Proposition 3.2.2 to both V\ and V2. Example 3.2.4. The local nonlinearity. Consider a function / : 0 x R —» K such that f(x, u) is measurable in x and continuous in u. Let F : Cl x R —> R be defined G(u + v)-G(u) - (g(u),v)H-iiHi = - by (3.2.6) for all u > 0. Assume that (3.2.7) f(x, 0) = 0 for a.a. x £ Q, and that for every K > 0 there exists L(K) < 00 such that (3.2.8) \f(x,u) - f(x,v)\ < L(K)\u - v\ for £ Q and all u, v such that |v| < K. Assume further that (3.2.9) Extend / to the complex plane by setting (3.2.10) f(x,u) u f(x,\u\) for all u £ C,u ^ 0. |u| Finally, set (3.2.11) g(u)(x) = f(x, u(x)) a.e. in fl for all measurable u : fl —* C, and (3.2.12) 60 3. THE CAUCHY PROBLEM IN A GENERAL DOMAIN for all measurable u : —> C such that F(-,u(-)) € L1^). We have the following result. proposition 3.2.5. Let f satisfy (3.2.7), (3.2.8), and (3.2.9); let g and G be defined by (3.2,10), (3.2.11), (3.2.6), and (3.2.12), and set Í2 ifN = l (3.2.13) r = I { a + 2 if N > 2. The following properties hold: (i) GeC1^1^)^), ge C{H£(Q), H-^Q)) and G'= g. (ii) If N > 2, then 2 < r < -jf^. (iii) geC(Lr(n),Lr'(n)). (iv) i*br a// M > 0, there exists C(M) < 00 such that \\g(u) — g(v)\\Lr> < C(M)\\u - v\\Lr for all u,v e H£(Q) with \\u\\Hi, \\v\\Hi < M. (v) lmg(u)u = 0 a.e. in Q, for all u e Hq(Q). Proof, (v) is an immediate consequence of (3.2.10). Let now K > 0, and let w, v € C be such that |u|, \v\ < K. Suppose for definiteness that \u\ > \v\. It follows from (3.2.7), (3.2.8), and (3.2.10) that (3.2.14) \f(x,v)\ 2. (ii) follows from (3.2.9) and (3.2.13). In particular, Hq{Q) ^ Lr(0.) by Sobolev's embedding theorem. Therefore, by (3.2.15), (3.2.9), and Holder's inequality, \\g(u) - g(v)\\Lr> < c(\\u\\aLr + ||u||£,.)i|u -v\\Lr. Hence (iii) and (iv) are proven. Next, it follows from (3.2.7), (3.2.8), and (3.2.6) that (3.2.16) |/(i,u)| 2. □ Remark 3.2.6. A typical / to which we may apply Proposition 3.2.5 is f(u) — 2N jv-2 \u\au with 0 < a < ^ (0 < a < oo if N = 1). Remark 3.2.7. Let g{u) = /(•,«(■)), where / satisfies (3.2.7) and (3.2.8) with C Lit) e C([0, oo)) if AT = 1 (3.2.17) < 4 yL(t) 2. If N > 2, define the functions f\ and /2 by f/(x,u) if0l and f 0 if 0 < u < 1 \/(z,u)-/(x,l) ifu>l. We have f = fi + J21 and /i and /2 both satisfy (3.2.7). Furthermore, one easily verifies that j\ satisfies (3.2.8) and (3.2.9) with a replaced by 0 and that /a satisfies (3.2.8) and (3.2.9) with a as in (3.2.17). In particular, we can write g = g\+g2 where both g\ and g^ satisfy the assumptions of Proposition 3.2.5. Example 3.2.8. The Hartree nonlinearity in Rr. Let Q — Rr and consider a real-valued potential W : RN —» R. Assume that (3.2.18) W e LP(RN) for some (3-2.19) p>l, P>^, 62 3. THE CAUCHY PROBLEM IN A GENERAL DOMAIN and (3.2.20) W is even. Let g be defined by (3.2.21) g(u) = (W * \u\2)u for all measurable u : RN —> R such that W * |u|2 is measurable, and let G be defined by for all measurable u : RN —> R such that (VT ★ |w|2)(a;)|u(a:)|2 is integrable. We have the following result. Proposition 3.2.9. Let W satisfy (3.2.18), (3.2.19), and (3.2.20), and let g and G be defined by (3.2.21) and (3.2.22), respectively. Set Then the following properties hold: (i) G G C1(H1(RN),R), g e C(H1(RN),H~1(RN)), andG' = g. (ii) 2 < r < ^ (2 < r < oo if N = 1). (iii) 0 eC(Lr(R*),Lr'(RN)). (iv) For Af > 0, there exists C{M) < oo such that \\g(u) - g(v)\\Lr' < C(M)\\u-v\\Lr for all ti, u G H1(RN) with \\u\\Lr, \\v\\Lr < M. (v) Img(u)u = 0 a.e. in RN for allueRN. proof. Part (ii) follows from (3.2.19) and (3.2.23). Next, we deduce from (3.2.19), (3.2.23), and Holder and Young's inequalities that Statements (iii) and (iv) follow easily. On the other hand, we deduce in particular from (ii) that H^R") ^ Lr(RN) and Lr' (RN) ^ H-1(RN) by Sobolev's embedding theorem, which implies that g G C(H1(RN),H~1(RN)). It also follows from Holder and Young's inequalities that (3.2.22) \\(W*(uv))w ■' < ||W/|Up||w||L^||t'||L'-||w||L'' • (3.2.24) so that G is well defined H1(RN) —» R. Since W is even, we see that 3.3. local existence in the energy space 63 Therefore, G(u + v) - G{u) - (g(u),v)H-i!Hi = Re{uv)J + J [W* Re{uv)) Re(uv) Applying now (3.2.24), we obtain \G(u + v)- G(u) - (g(u),v)H-itHi\ < C\\W\\Lv(\\u\\lr + ||t;||iP, and so G £ C1(H1(RN),R) and G' = g. This proves (i). Assertion (v) follows from Remark 3.2.10. Let W be an even, real-valued potential, W £ Lp(Q) + L°°(Q). If p satisfies (3.2.19), then we may write W = W\ + W2, where W\ satisfies (3.2.19) and W2 satisfies (3.2.19) with p replaced by oo. In particular, we may apply Proposition 3.2.9 to both Wi and W2. Example 3.2.11. Let where V, /, and W are as follows: • V is a real-valued potential, V £ LP(RN) + L00^) for some p > 1, p>N/2. • f : RN x M —> R is measurable in x £ RN and continuous in u £ R and satisfies (3.2.7), (3.2.8), and (3.2.17). / is extended to RN x C by (3.2.10). • W is an even, real-valued potential; W £ Lq(RN) + L°°(RN) for some q > 1, q > N/4. Applying Remarks 3.2.3, 3.2.7, and 3.2.10, we see that we may write 9 = gi +----h 9e , where each of the g^s satisfies the following conditions: (i) gj = G'j for some Gj £ C\Hl(RN),R), (ii) 9j£C{]7i(RN),Lr'i{RN)), (iii) for every M < oo, there exists C(M) < oo such that \\gj(v) - gj(u)\\ r'. < C(M)\\v - u\\Lri for all u,v£Hl(RN) such that \\u\\Hi + \\v\\Hi < M, (iv) Im{gj(u)u) = 0 a.e. in R^ for every u £ Hl{RN) for some Tj £ [2, ^~) (rj,pj £ [2, oo] if N = 1). We begin with an abstract result for which we use the notation introduced in Section 1.6. Theorem 3.3.1. Let X be a complex Hilbert space with the real scalar product (•,-)x- Let A be a C-linear, self adjoint, < 0 operator on X with domain D(A). the fact that W is real valued. □ g(u) = Vu + /(-,«(■)) + (W*\u\2)u, 3.3. Local Existence in the Energy Space 64 3. THE CAUCHY PROBLEM IN A GENERAL DOMAIN i— + Au + g(u) = 0 for all t G Let Xa be the completion of D(A) for the norm \\x\\x = ||x||^ — (Ax,x)x, XA = {XaY, and A be the extension of A to (D(A))*. Finally, let 7(t) be the group of isometries generated on (D(A))*, XA, X, Xa, or D(A) by the skew-adjoint operator %A. Assume that g : X —> X is Lipschitz continuous on bounded sets of X and that there exists G G C1(XA,M) such that G'(x) = g(x) for all x G XA. Assume further that (3.3.1) (g(x),ix)x=0 forallxtX. For x € XA, set (3.3.2) E(x) = ifllxll^ - \\xfx) -G(x) = -\{Ax,x)x - G{x) so that E G C1(XA,R) andE'(x) = -Ax-g(x) G XA for every x e XA- It follows that, for every x € X, there exists a unique solution u of the problem ' u G C(R, X) n C1 (R, (D(A)Y), (3.3.3) [ w(0) = x. In addition, the following properties hold: (i) ||m(£)||x = ||x||x for every t G R (conservation of charge). (ii) IfxeXA, then u G C(R,XA) n C1(R,XA) and E(u(t)) = E(x) for every t G R (conservation of energy). (iii) Ifx G D(A), then also u G C(R, D(A)) n C1(R,X). Proof. We proceed in five steps. Step 1. It is well known that for every x G X, there exists a unique, maximal solution u G C((T1,T2),X) of (3.3.3), T\ < 0 < T2. u is maximal in the sense that if |T;| < oo (for i = 1,2), then j|u(i)||x —> co as t —> T,. In addition, if x G D(A), then u G C((ri,r2),D(A))nC1((211,r2),X). Furthermore, u depends continuously on x in X, uniformly on compact subsets of the maximal existence interval. This follows essentially from Segal [309] (see Cazenave and Haraux [64, 65], Brezis and Cazenave [44], and Pazy [294]). Step 2. Assume x G D(A), and take the scalar product of the equation with iu. We obtain that (ut,u)x = {iuuiu)x = -{Au,iu)x - (g(u),iu)x ■ The first term of the right-hand side vanishes by self-adjointness, and the second by (3.3.1). Therefore, jt\\u(t)\\x=2(uuu)x=0. Hence the conservation of charge follows. Multiplying the equation by Ut, we obtain 0 = (iut,ut)x = (-Au,ut)x - {g(u),ut)x . Therefore, (3.3.4) jE{u(t))=0. 3.3. LOCAL EXISTENCE IN THE ENERGY SPACE 65 This establishes the conservation of energy. Step 3. By Step 2 and continuous dependence, we obtain conservation of charge when x £ X. Therefore ||w(£)IIat is uniformly bounded on the maximal existence interval, and so the solution exists on (—oo, oo). Hence (i) and (iii) follow. Step 4. Let x £ Xa, and let xn £ D(A) converge to x in Xa as n —* oo. We denote by un the solution of (3.3.3) with initial value xn. By (i), un is bounded in L°°(R, X), and so G(un) is uniformly bounded. We deduce from the conservation of energy (3.3.4) that un is bounded in L°°(R,Xa), and from the equation that (un)t is bounded in L°°(R, XA). On the other hand, it follows from continuous dependence that for every t e R, un(t) —► u(t) as n —► oo, strongly in X, hence weakly in XA. Therefore, u € L°°(R,XA) n W1'°°(R,XA) and E(u{t)) < E(x) for every t £ R. Step 5. Let t £ R, let y = u(i), and let u be the solution of (3.3.3) with initial value y. We deduce from Step 4 that E(v(—t)) < E(y). On the other hand, v(—t) — x by uniqueness so that E(u(t)) — E(x) for every t £ K. Hence there is conservation of energy. In particular, the function t »-* j]u(£)||;£ is continuous. Since u : R —+ is weakly continuous, we obtain w £ C(R,Xa), and so u £ CX(R,X^) by the equation. Hence (ii) is proven. □ Remark 3.3.2. Note that the assumption (3.3.1) is only needed to ensure conservation of charge, which implies that all solutions of (3.3.3) are global. Without that assumption, we would have a local version of Theorem 3.3.1 (without the conservation of charge). Theorem 3.3.1 is not applicable in general for solving the local Cauchy problem in the energy space for the nonlinear Schrodinger equation (3.1.1) for "large" nonlinearities. Indeed, we must take X = L2(£l), and so we need g to be locally Lipschitz continuous on L2(Q). If g is of the form g(u)(x) = f(u(x)) for some function / : C —> C, then / needs to be globally Lipschitz continuous, and in particular sublinear. Thus, we need to improve Theorem 3.3.1 under weaker assumptions on g. We now use the notation of Chapter 2. In particular, Q is an open subset of RN, A is the Laplacian with Dirichlet boundary conditions, and so X — L2(Cl), Xa = i?o(^)j an<3 XA = H~l(fl). We want to go as far as possible under fairly general assumptions on g. The main results of this section are Theorems 3.3.5 and 3.3.9. In Theorem 3.3.5, we show the existence of local weak Hq solutions. In Theorem 3.3.9, we show the local well-posedness of the Cauchy problem in Hq(Q), provided we have the "a priori" information that solutions are unique. The reason we proceed that way is that in order to apply Theorem 3.3.9, we will only need to show uniqueness, and the known techniques for proving uniqueness depend heavily on the the type of nonlinearity and on geometric properties of fl. We make the following assumption on the nonlinearity g: (3.3.5) g — G' for some G £ C1 (H^fl), R). In particular, g £ C(Hq(fl),H~l(Q)). We assume that there exist r,p £ [2, j^^rg) (r, p £ [2, oo] if JV = 1) such that (3.3.6) geCiH^Lr'ifl)) 66 3. the cauchy problem in a general domain and such that for every M < oo there exists C(M) < oo such that (3.3.7) \\g(v) - g{u)\\Lpl < C(M)\\v - u\\Lr for all u,v £ Hq(Q) such that + \\v\\Hi < m. Finally, we assume that, for every u £ #o(ft), (3.3.8) lm(g(u)u) = 0 a.e. in ft. We define the energy E by (3.3.9) E{u) = ^J\Vu\2 dx - G(u) for every u £ H^(Q), a. so that E £ C1{H0l{Q), R) and E'(u) = -Aw - g(u) for every u £ H£(Q). Remark 3.3.3. Assumptions (3.3.5)-(3.3.8) deserve some comments. The energy space being here ^(ft), it is natural to require that g : Hq(Q.) —> H-1(ft), as the Laplacian does. The assumption that g is the gradient of some functional G is stronger. It allows us to define the energy, and the conservation of energy is essential in our proof of local existence. Note that most of the classical examples from theoretical physics possess this property. However, in the case of local nonlinearities in ft = M.N, local existence can be proved without conservation of energy (see Kato [203, 204, 205, 206] and Chapter 4). Assumptions (3.3.6) require that g is slightly better than a mapping -Ho (ft) —* H~1(Q), and assumption (3.3.7) is a type of local Lipschitz condition. Finally, assumption (3.3.8) implies the conservation of charge. It is essential for our proof, but may be replaced by other hypotheses on g with different proofs (see Kato [203, 204, 205, 206], and Cazenave and Weissler [68]). Remark 3.3.4. Note that all the nonlinearities introduced in Section 3.2 satisfy the assumptions (3.3.5)-(3.3.8). We begin with the following result. Theorem 3.3.5. Let g = gi + ■ ■ ■ + gk, where each of the gj's satisfies the assumptions (3.3.5)-(3.3.8) for some exponents rj,pj. Set G = G\ -\- ■ • ■ + Gk and E = Ei + ■ • • + Ek. For every M > 0, there exists T(M) > 0 with the following property: For every if £ Hq(Q) such that \\(i,Hi), \\u'\\LX(ItH-i)}. proof. The result is a consequence of Remark 1.3.11 applied with X = #-1(fi) and p = oo, and of the inequality \\v\\2L2 < \\v\\h-i\\v\\h* (see Remark 1.3.8(iii)). □ Lemma 3.3.7. If g satisfies (3.3.5)-(3.3.8), then, after possibly modifying the function C(M), (3.3.13) \\g(v) - g(u)\\L, < C(M)\\v - u\\aL2, (3.3.14) \G(v) - G(u)\ < C(M)\\v - u\\bL,, for every u,v £ Hq(Q) such that + \\v\Ih1 < M, with a — 1 — — and 6 = 1-JV(I-I). proof. (3.3.13) follows from (3.3.7) and from Gagliardo-Nirenberg's inequality \\M\LrLP ds and the inequality □ proof of Theorem 3.3.5. We give the proof in the case where g satisfies (3.3.5)-(3.3.8). The proof in the case g — g\ + ■ ■ ■ + gk is trivially adapted. The proof proceeds in three steps. We first approximate g by a family of nicer nonlinearities for which we may apply Theorem 3.3.1 in order to construct approximate solutions. Next, we obtain uniform estimates on the approximate solutions by using the conservation laws. Finally, we use these estimates to pass to the limit in the approximate equation. Note that the proof of Theorem 3.3.5 requires at some stage a regularization procedure. Indeed, construction of solutions could be made, under appropriate assumptions on g and in the case ft = RN, by a fixed point argument (see Kato [203, 204, 205, 206], Cazenave and Weissler [70]). However, the energy inequality (3.3.12) is obtained, at least formally, by taking the scalar product of the equation with iut. Note that for a solution with values in Hq(Q), ut is only in H~1(rt), and so one cannot multiply the equation by ut. Hence the necessity of the regularization. Now, in principle, we have the choice on the type of regularization. For a given type of nonlinearity, a natural regularization appears, but which is of a different 68 3. the cauchy problem in a general domain nature according to the nonlinearity. For example, for a local nonlinearity (Example 3.2.4), the most appropriate thing to do would be to truncate / for large values of u. For a linear potential (Example 3.2.1), it would be natural to truncate the potential; and for a Hartree type nonlinearity (Example 3.2.8) it would be natural to use the convolution with a sequence of mollifiers. Since we want a proof that applies to these different nonlinearities, and that works as well when Q = RN or when Q is bounded, we find it convenient to regularize the nonlinearity by applying (7-eA)"1. We obtain estimates on the approximate solutions by using the conservation of energy for the approximate problem. For that purpose, we need g to be the gradient of some functional G (assumption (3.3.5)). As usual, the difficulty is to pass to the limit in the nonlinearity. The crux is that for the limiting problem there is conservation of charge (Lemma 3.3.8). Note that there is necessarily a little bit of technicality at that point. Indeed, we make a local assumption on g (assumption (3.3.8)), we apply a global regularization, and eventually we recover a local property at the limit. This seems rather unnatural, but there does not seem to be any obvious way of avoiding that difficulty. From now on, we consider (p e Hq(Q) and we set M = Step 1. Construction of approximate solutions. Given a positive integer m, let In other words for every / £ H~l{Vt), Jmf e Hq(Q) is the unique solution of the equation u-—Au = f inH-^n). We summarize below the main properties of the self-adjoint operator Jm (see Section 1.5). (3.3.15) \\Jm\\c(H-i,Hl) <™>, (3.3.16) \\Jm\\c(Lv,Lv) < 1 for 1 < p < co. Moreover, if X is any of the spaces Hq(Q),L2(Q,), or H~1(Q), then (3.3.17) \\Jm\\c(x,x) <1, (3.3.18) Jmu —> u in X for all u € X, (3.3.19) if sup < oo, then Jmum - um —^ 0 in X as m —► oo. m We define gm(u) = Jm(g{Jmv)) and Gm(u) = G(Jmu) for every u € Hq(CL) . It is clear from (3.3.15) that the above definitions make sense. It is easy to verify-by using (3.3.15) and (3.3.7) that gm is Lipschitz continuous on bounded sets of L2(0), and by (3.3.15) and (3.3.5) that Gm e C^H^n),^) and G'm = gm. In addition, we deduce easily from (3.3.8) that, for every u e L2(ft), {gm{v),iu)L-2 = (g(Jmu),iJmu)L2 =0. 3.3. local existence in the energy space 69 Therefore, we may apply Theorem 3.3.1. Hence there exists a sequence (um)m^ of functions of C(E, ifo(O)) D C1(R,H~1(Q)) such that (3.3.20) f ium + Aum + gm{um) = 0 \ um(0) = Furthermore, (3.3.21) ||um(t)j|L» = 1Mb and (3.3.22) I J\Vum(t)\2 dx - Gm(um(t)) = | J |V|2 dx - Gm(y>) for all £ € Step 2. Estimates on the sequence um. We denote by C(M) various constants depending only on M. Let (3.3.23) 6m = sup {r > 0 : ||xim(t)||tfi < 2M on (-r,r)} . Note that, by (3.3.17) and (3.3.16), (3.3.24) gm satisfies (3.3.6) and (3.3.7) uniformly in m e N. Therefore, by (3.3.20), (3-3.25) sup \\u?\\Lxi{_em,em),H-i) < C(M). It follows from (3.3.23), (3.3.25), and Lemma 3.3.6 that (3.3.26) \\um{t)-um(s)\\L2 i((-T(M),T(M)),Lp>{n)) such that, for all t € [—T(M),T(M)], (3.3.31) gm(um(t)) /(<) in i/(ft) asm^oo. On the other hand, it follows from (3.3.20) that for every w g Hq(Q) and for every (t)]dt = 0. -t(w) Applying (3.3.30), (3.3.31), and the dominated convergence theorem, we deduce easily that t(m) [~{iu,w)H-\Hl 4>'(t) + (Au + f,w)H-itHi (t)]dt = 0. t(m) This implies that u satisfies f iut + Au + f = 0, (3.3.32) I I «(0) = where the first equation holds for a.a. t g (—T(M),T(M)). Now the crux of the proof is the following result. Lemma 3.3.8. For all t g {-T(M),T(M))) Im(/(t)u(t)) = 0 a.e. on ft. Proof. It suffices to show that for every bounded subset B of ft, (f(t)\B,iu{t)\B)Lp>{B)tLP(B) =0. To see this, we omit the time dependence and we write U^u)lp'{B),Lp(B) = (f ~ Jmg{JrnUm),iu) + (Jmg(JmUm) - g{JmUm),iu) + (g(Jmum),i(u - um)) + (g(Jmum),i(um ~ Jmum)) + {g(Jmum),iJmUm) —> a + b + c-\-d + e. m—>oo Note first that Jmg(Jmum) = gm(um) ->> f in Lp'(tt), hence in Lp'(B). Therefore, a = 0. Next, observe that g(Jmum) is bounded in Lp (ft). It follows from (3.3.19) and (3.3.16) that Jmg(Jmum) - g{Jmum) 0 in #_1(^)> hence in Lp' (B). Therefore, 6 = 0. Since um -± u in H$(Q), we have um -* u in Since g(Jmum) is bounded in Lp (!?), we deduce that c = 0. By (3.3.19), um — Jmwm converges weakly to 0 in H&(Q). It follows that um - Jmum -* 0 in LP(B). Since g{Jmum) 3.3. local existence in the energy space 71 is bounded in Lp' (B), we obtain d = 0. Finally, e = 0 by (3.3.8), and the result follows. □ End of the proof of Theorem 3.3.5. Taking the H"1 ~ Hi duality product of the first equation in (3.3.32) with iu, we find |||ti(*)||£a = 0 for all t e (-T(M),T(M)) and so (3.3.33) IK*)IIl» = IMIl» • It follows from (3.3.21), (3.3.33), and Proposition 1.3.14(ii) that (3.3.34) um -+ u in C([-T(M),T(M)],L2(ft)). Applying (3.3.28), (3.3.34), and Gagliardo-Nirenberg's inequality, we deduce that (3.3.35) um-+u in C([-T(M),T(M)],LP{Q)) for every 2 < p < It follows easily from (3.3.7), (3.3.16), (3.3.18), and (3.3.35) that gm{um(t)) - g{u{t)) = Jm[(g{Jmum{t)) - g(Jmu(t))} + Jm[g(Jmu(t)) - g(u(t))] + Jmg{u(t)) - g(u(t)) —► 0 m—»oo in Lp'(Q) for all t E (-T(M),T(M)). Therefore, / = g(u) and so u satisfies (3.1.1). (3.3.10) follows from (3.3.28) and (3.3.11) from (3.3.33). It remains to prove (3.3.12). This is a consequence of (3.3.22), the weak lower semicontinuity of the /f^norm, and the fact that Gm(um(t)) —>■ G(u(t)) as m —* oo. This completes the proof. □ We now show that the initial-value problem (3.1.1) is locally well posed in Hq(Q), provided we have the a priori information that weak Hq solutions are unique. Theorem 3.3.9. Letg = -----\-gk where each of the gj's satisfies (3.3.5)-(3.3.8) for some exponents rj, pj; and set G = G\ + • ■ ■ + Gk and E = E\ + • • • + Ek. Assume, in addition, that there is uniqueness for the problem (3.1.1). It follows that (3.1.1) is locally well posed in Hq(Q,), and that there is conservation of charge and energy; i.e., \\u(t)h* = IMU2 and E(u(t)) = E{tp) for all t £ (—Tmin,Tmax)? where u is the solution of (3.1.1) with the initial value Proof. The proof proceeds in three steps. We first show that the solution u given by Theorem 3.3.5 belongs to C((-T(M), T(Af)),H^n)) D C\{-T{M), T(M)), H~1(Q)), 72 3. THE CAUCHY PROBLEM IN A GENERAL DOMAIN and that there is conservation of energy. Next, we consider the maximal solutions and show that Tmin and Tmax satisfy the blowup alternative. Finally, we establish continuous dependence. Step 1. Regularity. Let / be an interval and let u e L°°(i,Hi(Q)) n w1,0O(/,Jfir-1(fi)) satisfy iut + Au+g(u) = 0 for a.a. t g /. We claim that u satisfies both conservation of charge and energy, and that u g c(i,h£{Q)) nc1!/^"1^)). To see this, consider M = sup{\\u{t)\\Hi,t€l}, and let us first show that ||w(£)||L2 and E(u(t)) are constant on every interval J c i of length at most T(M), where T(M) is given by Theorem 3.3.5. Indeed, let J be as above and let a,r g J. Let ip = u(a) and let v be the solution of (3.1.1) given by Theorem 3.3.5. v(- — a) is defined on J and, by uniqueness, v(- — a) = u(-) on J. Applying (3.3.11) and (3.3.12), we deduce in particular that (3.3.36) ||u(t)||L2 HK M by Lemma 3.3.7. In view of (3.3.37), this implies that j|u(£)||#i is continuous i —» r. Therefore, w g (7(7,i?o(ft)), and, by the equation, u g C^J, #-1(ft)). Step 2. Maximality. Consider tp g ^(ft) and let Tmax(^) = sup{T > 0 : there exists a solution of (3.1.1) on [0, T]} , Tm\n( 0 : there exists a solution of (3.1.1) on [~T, 0]}. It follows from Step 1 and the uniqueness property that there exists a solution of (3.1.1). Suppose now that Tmax < oo, and assume that there exist M < oo and a sequence tj | Tmax such that ||u(£j)||#i < M. Let k be such that tk + T(M) > Tmax( u in C([-T(M),T(M)],H$(Q)). Since T(M) depends only on M, we may repeat this argument to cover the interval [-Ti,T2]. This completes the proof. □ Remark 3.3.10. By Theorem 3.3.9, if g is a finite sum of terms Oj, where each of the Oj's satisfies the assumptions (3.3.5)-(3.3.8) for some exponents rj,pj, then problem (3.1.1) is well posed in Hq(Q) provided there is uniqueness. Unfortunately, the techniques that are used to prove uniqueness depend on the problem (see the following sections). However, we give below a general sufficient condition for uniqueness. Corollary 3.3.11. Let G e C1{H£(Q),R) and let g - G'. Assume that g{0) <= L2(Q.) and that there exists C(M) for every M such that (3.3.38) \\g{v) - g(u)\\L. < C(M)\\v - u\\L, for allu,v G i?q (S7) such that \\u\\h1 + \\v\\h1 1 and set Hq = (H^Q))1*, H'1 - (iJ_1(fi))M, 74 3. THE CAUCHY PROBLEM IN A GENERAL DOMAIN and Cp = (LPityy. Let (ae)i oo for every interval [S,T] C (-Tmin,Tmax). 3.4. Energy Estimates and Global Existence Given g as in Theorem 3.3.5, there exists a local weak 77o-solution of the problem (3.1.1) for every initial value

0, C(A) > 0, and e G (0,1) such that (3.4.1) G(u)<~\\u\\2m+C(A) for all u G Hq(Q) such that \\u\\Li < A. If tp G Hq(Q) satisfies JMir,* < A then there exists a (global) weak HQ-solution u of (3.1.1) on R. In addition, u G L°°(R,Hq(Q)) and u satisfies the conservation of charge (3.3.11) and the energy inequality (3.3.12) for all t G R. Proof. Let I => 0 be an interval of R. Consider a weak TfQ-solution u °f (3.1.1) on 7. Assume that u satisfies the conservation of charge (3.3.11) and the energy inequality (3.3.12) for all tel. Since \\u(t)\\l1=E(u(t))-2G(u(t)) + \\u(t)\\l1 we deduce from (3.3.11) and (3.3.12) that \\u(t)\\2H1 < \\ 0. We also can argue similarly for t < 0, so that we obtain a weak .íí/q -solution u of (3.1.1) on R which satisfies (3.3.11) and (3.3.12) for all t £ R. Finally, we deduce from (3.4.2) that suPi€» II^WIIh1 < 005 which completes the proof. □ The following corollary is an immediate consequence of Theorem 3.4.1. Corollary 3.4.2. Let g be as in Theorem 3.3.5. Assume further that for every A > 0, there exist C(A) > 0 and e £ (0,1) such that (3.4.1) holds. It follows that for every tp £ Hq(CI), there exists a (global) weak Hq1-solution u of (3.1.1) on R. In addition, u £ L°°(1R, Hq (S7)) and u satisfies the conservation of charge (3.3.11) and the energy inequality (3.3.12) for all tel. Corollary 3.4.2 provides a sufficient condition on the nonlinearity so that for all initial values tp £ Hq(Q), there exists a global weak #0-solution of (3.1.1). We next show that, under a different type of assumption on g, there exists a global weak iřg-solution of (3.1.1) for all sufficiently small initial data

0 and a nonnegative function 6 e C([0, e),R+) with 0(0) = 0 such that (3-4.4) G(u) 0 such that for every

||La)] for all t e J. Note that the right-hand side of (3.4.5) is a continuous function of

0 as above. Since in particular ||<£>||#i < e/2, it follows from Theorem 3.3.5 that there exists a weak iJo-solution u of (3.1.1) on [0, T(e/2)] which satisfies (3.3.11) and (3.3.12) for all t e [0,T(e/2)] and such that, ||w||l^((o,t(£/2));hi) < e- We deduce in particular from (3.4.5) that \\u(T(e/2))\\Hi < e/2. Setting tp = u(T(e/2)), we again apply Theorem 3.3.5. We see that there exists a weak Hq-solution u of (3.1.1) (with the initial value eH?;(n). Proof. It follows from Proposition 3.2.5 that g satisfies (3.3.38), and the result follows from Corollary 3.3.11. □ Corollary 3.5.2. Let g be as in Theorem 3.5.1. If F(x,u) 0 such that, for every

then < K. Therefore, G(u)■-lXx(i)E^~ for p large enough. Therefore, Jo (3.6.4) Let now (t) < Cp ds for all t g /. $p(i) = /" 0(a) JO ^ ds. It follows from (3.6.4) that &p(t) < Cp\$p{t)(p~Wp\ for all t g J. Integrating this inequality yields |$p(i)| < (2C|*|)p/2. Therefore, if 2C|r| < 1, we obtain liminf $P(T) =0, p—>oo which implies that / (p(s)ds = 0. Jo Thus «; = 0 on [—T,T]. This gives the result. □ 80 3. the cauchy problem in a general domain COROLLARY 3.6.2. Letti be an open subset ofR2 and let g be as in Theorem 3.6.1. -f/IMU2 is small enough, then the maximal strong HQ-solution of (3.1.1) given by Theorem 3.6.1 is global and uniformly bounded in H1. If F(x, u) < C(l + \u\5)\u\2 for some 6<2, then the same conclusion holds for every

Lp(fi) for all e > 0, and (3.7.2) sup {||(/-^)_1[U(jlp)lp) : e > 0} < oo, Consider a function g G C(Xa,X\) such that (3.7.3) g = G' for some G G C1(Xa,R) , and assume that there exist r,p e [2, jfr^) (r,p G [2, oo] if N = 1) such that (3.7.4) g€C(XA,IS'(n)) 3.7. COMMENTS 81 and such that for every M > 0, there exists C(M) < oo such that (3.7.5) \\g(v) - g(u)\\LP> < C(M)\\v - u\\Lr for every u,v £ XA such that \\u\\xA + \\v\\xA ^ M- (0r> more generally, assume g = gi + • • • + gk, where each gj satisfies the above assumptions for some rj, pj.) Finally, assume that for every u £ Xa, (3.7.6) lm(g(u)u) = 0 a.e. on Q,, and let E be defined by (3.3.2). We consider the problem {iut + Au + g(u) = 0 w(0) = x for a given x £ Xa- We have the following result. Theorem 3.7.1. Let A and g be as above. Assume, in addition, that there is uniqueness for the problem (3.7.7). It follows that the initial value problem (3.7.7) is locally well posed in Xa- Moreover, there is conservation of charge and energy; i.e., \\u(t)\\L2 = ||a:||L2 and E(u(t)) = E(x) for allt £ (—Tmin, Tmax), where u is the solution of (3.7.7) with the initial value x £ XA- {Here, the notions of uniqueness and local well-posedness are as in Section 3.1). proof. The proof is an adaptation of the proof of Theorem 3.3.9. We only point out the modifications that are not absolutely trivial. Lemma 3.3.6 is easily adapted with the duality inequality \\u\\x < \\u\\xA\\u\\x^- The proof of Lemma 3.3.7 is adapted as follows. Consider 2 < p < q < ^f^. By Holder's inequality and (3.7.1), there exists a £ (0,1) such that NIlp < lltillMMfc0 < HtCNlir. and the rest of the proof is unchanged. To adapt the proof of Theorem 3.3.5, we need inequalities of the type (3.3.15)-(3.3.16). They follow easily from the self-adjointness of A, except for (3.3.16), which follows from (3.7.2). The rest of the proof, including Lemma 3.3.8, is unchanged except that one has to apply Proposition 1.1.2 instead of Proposition 1.3.14. □ Remark 3.7.2. Corollary 3.3.11 is easily adapted to the above situation. Remark 3.7.3. Like Theorem 3.3.9 (see Remark 3.3.12), Theorem 3.7.1 is stated for one equation, but the method applies as well for systems of the same form. More precisely, considering an integer p > 1, one may assume that A is a self-adjoint operator on (L2(£7))^ and replace everywhere LP(Q) by (Lp(fi))^. It follows that the conclusions of Theorem 3.7.1 remain valid. Remark 3.7.4. Using Strichartz-type estimates (see Remark 2.7.3), it is possible to solve the local (or global) initial value problem for certain nonlinear Schrödinger equations in a cube of with periodic boundary conditions. See Bourgain [34, 35, 37, 38] and Kenig, Ponce, and Vega [214]. 82 3. the cauchy problem in a general domain REMARK 3.7.5. For g{u) = -\u\2u in the unit disc of R2, it was shown that the initial value problem is ill-posed in HS(Q.) for s < 1/3. More precisely, given T > 0 and a bounded subset B of HS{Q), the map

0. (4.1.1) then takes the form f iut + Au + X\u\au = 0, (4.1.3) < V ^ 1 u(0) = ip. We observe that if a < 4/(N - 2) (a < oo if N = 1,2), then u € L°°{I,H^{Q)) is a solution of equation (4.1.3) on I if and only if u satisfies the integral equation (4.1.4) u(t) = 7(t)

N/2 (Section 4.10). 83 84 4. the local cauchy problem Section 4.11 is devoted to a nonautonomous Schrodinger equation that is derived from (4,1.1) by the pseudoconformal transformation. We will use that equation in Section 7.5. Finally, we observe that the results of this chapter are stated for one equation, but similar results obviously hold for systems of the same form. See Remark 3.3.12 for an appropriate setting. 4.2. Strichartz's Estimates and Uniqueness As we have seen in Section 3.3, uniqueness is a key property for the local well-posedness of the initial-value problem (4.1.1). In the present case where 0, = RN, Strichartz's estimates are a powerful tool to establish uniqueness. We note that most of the results of this section are due to Kato [206]. We begin with the model case of the pure power nonlinearity; i.e., we consider the problem (4.1.3). We note that if a < 4/(N - 2) (a < oo if AT = 1,2), then g(u) = \\u\au satisfies g G Cf/f^R^),^"1^)) so that we may consider weak ^resolutions of (4.1.3). It turns out that they are unique, as the following result shows. Proposition 4.2.1. Assume A G C and 0 < a < 4/(N - 2) (0 < a < oo if N = 1,2). If tp G H1(RN) and ui,u2 are two weak Hl-solutions of (4.1.3) on some interval I 3 0, then u\ = u2 ■ Proof. We may assume without loss of generality that I is a bounded interval. It follows from (4.1.4) that (4.2.1) (ui-u2){t) = %X f T(t - 8)(\ui\aui - \u2\au2)(s)ds. Jo Since |K|a«l - |«2|a«2| L'-))IIw1 - u2\\l*'(j,l') • It follows from (4.2.1), (4.2.2), and Strichartz's estimate that (4.2.3) Hit! -U2||l*(J,L'-) < C(hl\\l~{J,Lr) + l|w2||L~(JiLr))||ui -u2\\L,>(JtLr) ■ Since H1^1*) ^ Lr(RN) and |/| < oo, (4.2.3) yields ||w1 ~ U2\\L*(J,Lr) < C\\U! - U2\\Lq>{J>Lr) for some constant C independent of J. The result now follows by applying Lemma 4.2.2 below with k = 1, <$>i(t) = ||«i(t) - u2{t)\\L-, ax = q', and b\ = q. □ 4.2. strichartz's estimates and uniqueness 85 Lemma 4.2.2. Let I 3 0 be an interval. Let 1 < aj < bj < oo and 4>j £ Lbj(I), for 1 < j < k. If there exists a constant C > 0 such that k k (4.2.4) Y^Uj\\l'hj)%cTi\\Ml'hj) for every interval J such that 0 € J C I, then \ — • • • = 4>k — 0 a.e. on I. Proof. We first consider the case 7 = [0, T) for some 0 < T < oo. Suppose that We deduce from what precedes that 9 > 0 (starting with r = 0). If 8 < T, then we let r = 9 and we deduce that 0X = • - ■ = <£fc = 0 a.e. on (0, # 4- g) for some £ > 0, which contradicts (4.2.5). Thus 8 = T which shows the desired conclusion. The case I — [-T, 0] is treated similarly (by changing t to —t). In the general case, we apply the above results to all T > 0 such that [0, T] c 7, then to all T > 0 such that [—T, 0] C 7, and we deduce that \ — ■ ■ • = ^ = 0 a.e. on 7. □ Proposition 4.2.1 can be extended to more general nonlinearities. In particular, we have the following result. Proposition 4.2.3. Consider gugk e C(771(RAr)), H~1(RN)) and let 9 = 9i +----1- 9k ■ Assume that each of the gj's satisfies the assumption (3.3.7) for some exponents rj,pj G [2,2N/(N - 2)) (rj,pj 6 [2,oo] if N = 1); i.e., there exists C3 such that (4-2.6) \\g3(u) - fc(t/)||L,; < Cj(M)\\u - v\\Lr3 for allu,v€ H1(RN) such that \\u\\Hi, ||vf|Hi < M. If ip € 771(RAr) andux,u2 are two weak H1-solutions of (4.1.1) on some interval I 3 0, then u\ —u2. Proposition 4.2.3 is a consequence of the following simple lemma. 86 4. the local cauchy problem LEMMA 4.2.4. Let I 3 0 be an interval and k > 1 be an integer. For every 1 < j < k, let (qj,rj) and (7,-,^) be admissible pairs and fj e L7i(/,L^(K^)). Finally, set fc rt (4.2.7) w(t)=iJ2 I 7{t~s)fj{s)ds i=i y° /or all t € I (so that w e 2> (I, Lrj (MN)) /or all 1 < j < k by Strichartz's estimates). If for every I < j < k there exist 1 < aj < qj and a constant C3- such that (4-2-8) H/illLi(JiL^)^^HI^WL'i) /or a// bounded intervals J such that 0 e J c /, then w = 0. Proof. Letting Jo Wj[t) =1 'j(t- s).U(s)ds Jo and applying k times Strichartz's estimate, we see that there exists a constant Kj such that k for all bounded intervals J such that 0 € J C I. It follows from (4.2.7) that there exists a constant C such that k k J2WwWl"hj,l^) 3. Indeed, we would be lead to apply Lemma 4.2.2 with a\ — ř>i, in which case the conclusion of the lemma is clearly false. In fact, we do not know if the conclusion of Proposition 4.2.1 holds in the limiting case a = 4/(N — 2). A slight modification of the method of proof, however, shows uniqueness of strong H1 -solutions. More precisely, we have the following result. Proposition 4.2.5. Assume N > 3. Let X € C and a = 4/(N - 2). // 0, set fM = l{|Ul|+|W2|>M}(|«l|7^5Wl - \u2\^U2) , ÍM = l{|«i|+|t*2|M}(|iti| + |u2|)^|«i - u2\. In order to show that U\ = u2: we use the endpoint Strichartz's estimate. We have (ui - u2)(t) = *A / T(t - s)(fM + fM)ds , Jo so that for every 0 < r < T, (A 2 10) uiw.i. ) O(ll/-ll^((..r,.l., + I/-||£,((0iT)il^I)). Using (4.2.9), we see that (4.2.11) ||/m||li((o,t),l2) < CMt£*\\Ul - u2\\LimrhL2h and that (4.2.12) "*«°.*>.^> Finally, we observe that {u^ + |«2| e C([0,T]:H1(RN)), so that we have |wi| + M £ C([0,T],L^(R7V)). It follows easily by dominated convergence that (4.2.13) ll^l^l+^l^f^l + l^^lLcc^r),^)^0- 88 4, THE LOCAL CAUCHY PROBLEM Applying (4.2.13) we see that, by choosing M large enough, we can absorb the right-hand side of (4.2.12) by the left-hand side of (4.2.10). Therefore, we deduce from (4.2.10)-(4.2.12) that there exists C such that ||«1 - W2||l«((0,t),L2) < - W2||li(0,t),L2) for every 0 < r < T, and the result follows from Lemma 4.2.2. □ Remark 4.2.6. Note that it is precisely for showing (4.2.13) that we use the assumption that u\ and u2 are strong /^-solutions. For weak iJ1-solutions, we only know that + \u2\ g L°°((0,T),i^1(MJV)), which does not imply (4.2.13). Proposition 4.2.5 can be extended to more general nonlinearities. We will not study the general case. Note, however, that an immediate adaptation of the proof of Proposition 4.2.5 yields the following result concerning local nonlinearities. Proposition 4.2.7. Assume N > 3. Let g e C(C,C) with g(0) ~ 0 satisfy \g(Zl) - 9M\ < C(l + \zi\t£* + |22|^)|2l - z2\ for all z\,z2 g C. Iftp g H1(RN) and Ui,u2 are two strong H1 -solutions of (4.1.1) on some interval I 3 0, then u\ = u2. We will construct in the following sections solutions of (4.1.1) that are not H1-solutions, and we now study uniqueness of such solutions. We begin with a lemma about the equivalence of (4.1.1) and (4.1.2) for such solutions. Lemma 4.2.8. Let I 3 0 be an interval, let s,a g R, and let g : HS(M.N) —► ifa(R^) be continuous and bounded on bounded sets. If u g L°°(I,HS(RN)), then both equations (4.1.1) and (4.1.2) make sense in H^(RN) for \i = min{s — 2, a}. Moreover, u satisfies equation (4.1.1) for a. a. t g I if and only if u satisfies the integral equation (4.1.2) for a.a. t g I. Proof. Let u g L°°(I,H'(Rn)). Since A g £(Hs(Rn),Hs-2(Rn)), we see that Au g LX(I,HS~2{RN)). Moreover, g(u) is measurable I ~> Ha{RN) because g g C(Hs\rRN),Ha(RN)), and bounded because g is bounded on bounded sets, and so g{u) g L°°(I, Ha(RN)). Thus we see that both equations make sense in Jf^R-^). Since (T(t))tem is a group of isometries on ^(R-^), the equivalence between the two then follows from the results of Section 1.6. □ According to the above lemma, under appropriate assumptions on g, we can address the question of uniqueness of solutions of (4.1.1) in L°°(I,HS(RN)). We have the following result, which is an easy application of Lemma 4.2.4. Proposition 4.2.9. Consider s > 0. Let (4.2.14) 0i,... ,gk g C(HS(RN)),H(7(RN)) be bounded on bounded sets, for some a g R, and let g — g\ + ■ ■ ■ + gk- Assume that there exist exponents rjtpj g [2,2N/(N - 2)) {rj,pj g [2,oo] if N = 1) and functions Cj g C([0, oo)) such that (4.2.15) \\9j(u) - 9j{v)\\lP,. < CjiM^u - v\\v-,, 4.2. strichartz's estimates and uniqueness 89 for all u,v G ^'(R^) such that \\u\\Hs,\\v\\Hs < M. Let ip G HS(RN) and ui,u2 G L°°(I,HS(RN)) be two solutions of (4.1.1) on some interval I 9 0. If k (4.2.16) ui ~u2 G f)LV(I,Lri{RN)), 3 = 1 where qj is such that (qj,rj) is an admissible pair, then u\ = u2. Remark 4.2.10. The assumptions (4.2.14), (4.2.15), and (4.2.16) deserve some comments. (4.2.14) ensures that equation (4.1.1) makes sense for a function u G L°°(I,H'(RN)). (See Lemma 4.1.8.) Assumption (4.2.15) is a Lipschitz condition for the Oj's on bounded sets of HS(RN). It is rather natural, since a Lipschitz condition of some sort is necessary for a uniqueness property. Finally, (4.2.16) is a regularity assumption on the difference of the solutions u\ and u2. In practice, it is verified by requiring that HS(RN) LV:>(RN) for all j's, so that both ux and u2 belong to the prescribed space and in particular the difference u\ - u2. Note, however, that (4.2.16) is in principle weaker than assuming that both and u2 belong to the prescribed space. For example, for the Navier-Stokes equation, the difference of two solutions has a better regularity in certain spaces than each of the solutions (see, e.g., [225]). However, it seems that no one could use such a property for the Schrodinger equation to take advantage of the fact that (4.2.16) concerns the difference of two solutions (see Furioli and Terraneo [120] for interesting comments on this problem). Proof of Proposition 4.2.9. The proof is identical to the proof of Proposition 4.2.3. □ Remark 4.2.11. Given s > 0, we apply Proposition 4.2.9 to the model case g(u) = AJu|au where a > 0 and A G C. There are three conditions to be checked, namely (4.2.14), (4.2.15), and (4.2.16). We note that, since g is a single power, we do not need to decompose g = g\ + ■ ■ ■ + gfc. We investigate the condition (4.2.14). Suppose first s > N/2, so that HS(RN) ^ ^(R^) for every 2 < p < oo. It follows easily that (4.2.14) is satisfied with a = 0. Suppose now s < N/2, so that HS{RN) ^ Lp(Rn) for every 2 < p < 2N/(N - 2s). We deduce easily that if (N -2s)(a + l) < 2N, then g G C(HS(RN), L ("-a-Ka+D (R*)) . Condition (4.2.14) is then satisfied, for example, with o < -N/2. If, on the other hand, (N — 2s)(a + 1) < 2N, then g{u) (which is a measurable function) need not be locally integrable, so that g does not map HS(RN) into any space of the type Ha{RN). Therefore, we see that (4.2.14) is satisfied if and only if (In particular, there is no condition if a < 1.) We next investigate the condition (4.2.16). As observed in Remark 4.2.10, we require that HS(RN) <-» Lr(R*), where r is as in (4.2.15). This is obviously 90 4. THE LOCAL CAUCHY PROBLEM satisfied if s > N/2. If s < N/2, then we need 27V (4.2.18) 2 < r < N-2s We now turn to the condition (4.2.15) and we first study the case s > N/2 by using the inequality \\g(u) - g(v)\\Lr> < C(\\u\\aL^ + \\v\\^)\\u - v\\Lr. We note that HS{RN) ^ LP(RN) for all 2 < p < oo, so we see that (4.2.15) is satisfied with p = r for all r > 2 sufficiently close to 2. We then study the case s < N/2, so that HS(RN) ^ LP(RN) for every 2 < p < 2N/(N - 2s), and we use the inequality \\9(u) - g{v)\\L, < C(\\u\\*LP + \\v\\lP)\\u - v\\Lr , where 1 < p < oo satisfies a _ _ I _ 1 p p r We begin with the case N = 1. Since the admissible values of p are 2 < p < oo and the admissible values of r are (by (4.2.18)) 2 < r < 2/(1 - 2s), we see that the admissible values of p are 2a/(l + 2s) < p < oo. Of course, we want HS(RN) LP(RN), and this is compatible with the above restriction provided 2/(1 — 2s) > 2a/(l + 2s). We note that this is exactly (since s < 1/2) the condition (4.2.17). We now assume N > 2 and we begin by assuming s > 1. In this case (4.2.18) is not a further restriction on r, so that the admissible values of r and p are 2 < r, p < 2N/(N - 2). Thus the admissible values of p are Na/2 < p < oo. We want HS(RN) LP(RN), and this is compatible with the above restriction provided 2N/(N~2s) > Na/2, i.e., a < 4/(N-2s). If s < 1, then we have the further restriction (4.2.18) on r, so that the admissible values of p are Na/(l + s) < p < oo. We want HS(RN) LP(RN), and this is compatible with the above restriction provided 2iV/(iV - 2s) > Na/(1 + s), i.e., a < (2 + 2s)/(N - 2s). In conclusion, we see that if s > N/2 there is always uniqueness in L°°(I,HS(RN)). If s < N/2 and N = 1, then there is uniqueness as soon as the equation makes sense, i.e., as soon as (4.2.17) holds, that is N + 2s (4.2.19) ^irrs- If s < N/2 and N > 2, then there is uniqueness provided the equation makes sense, i.e., provided (4.2.19) holds, but under the additional assumption min{4,2 + 2s} (4.2.20) Q<_L^_2. Remark 4.2.12. Suppose g G C(C,C) satisfies #(0) = 0 and \9(zi) ~ g{*2)\ < C(l + \Zl\" + \z2\a) \Zl - z2\ for some a > 0. It follows that the conclusions of Remark 4.2.11 hold. More precisely, there is uniqueness in L°°(I, HS(RN)) provided s > N/2, or provided 4.2. strichartz's estimates and uniqueness 91 s < N/2, (4.2.19), and, if N > 2, (4.2.20). To see this, we decompose g = gi + g2, where #i(0) = #2(0) = 0, g\ is globally Lipschitz, and g2 satisfies \92{zi) ~ 92{z2)\ < C(\Zl\a + \z2\a)\Zl - z2\. (See Section 3.2.) We may apply Proposition 4.2.9, since g2 is handled with exactly the argument of Remark 4.2.11 and pi clearly satisfies (4.2.15) with r\ — pi = 2. We now state an analogue of Proposition 4.2.7 in the case of H8 solutions. Proposition 4.2.13. Assume N > 3 and 1 < s < N/2. Let g 3 and s > 1, (4.2.17) is satisfied. This implies that equation (4.1.1) makes sense for a function u e C(I,HS(RN)) (see Remarks 4.2.11 and 4.2.12). The proof is similar to the proof of Proposition 4.2.7. Note that we use the same admissible pairs (00,2) and (2,27V/(JV - 2)), and that we use the property u e C{I, L~^(RN)), so that I|1{m+m>m}(K| + M)\\L^mT);L7m7} 0 as M -*00• □ Remark 4.2.14. The observations of Remarks 4.2.11 and 4.2.12 and Proposition 4.2.13 are part of the work of Kato [206]. In the single power case, we observe that when N = 1 or when s > N/2 there is always uniqueness in L°°(I,HS(RN)) as soon as the equation makes sense. When N > 2 and 0 < s < N/2, there are some cases where the equation makes sense; but uniqueness is not a consequence of Remark 4.2.11, namely when min{4,2 + 2s} ^ N + 2s N - 2s ~ ~ N -2s In fact, we will see in Section 4.9 that when a < 4/(N — 2s), one can construct Hs solutions, while for a > A/(N — 2s) the existence problem is open. Even if one is willing to consider the restriction a < 4/(N - 2s) as essential, there still are cases when uniqueness in L°°(I,HS(RN)) does not follow from Remark 4.2.11, namely when N > 2, 0 < s < 1 and 2 +2s 4 -- < Q < - . TV - 2s - - N-2s This has been an open problem since the work of Kato [206] but there was a recent breakthrough by Furioli and Terraneo [120] who were able to fill part of the gap by using in particular negative order Besov spaces. 92 4. the local cauchy problem 4.3. Local Existence in H1(RJV) Consider gi,...,gk£ C(H1(RN)),H~1(RN)) and let 9 = 9i + --- + 9k- Assume that each of the g/s satisfies the assumptions (3.3.5)-(3.3.8) for some exponents Tj, pj. Let G = G\ + • • • + Gk , and set £(u) = i J \Vu(x)\2dx-G{u) for u £ H1^1*). We will apply the results of Section 3.3 to establish the following result. Theorem 4.3.1. If g is as above, then the initial-value problem (4.1.1) is locally well posed in H1(RN). Furthermore, there is conservation of charge and energy; i. e. = IMU* , E(u(t)) = E( 2. Set (4-4.4) g(u)(x) = f(u(x)) for all measurable u : RN —> C and Theorem 4.4.1. Let f e C(C,C) satisfy (4.4.1)-(4.4.3) and let g be defined by (4.4.4). If f (considered as a function R2 —► R2) is of class C1, then the initial-value problem (4.1.1) is locally well posed in if1(RAr). Remark 4.4.2. Since we assume neither (3.3.5) nor (3.3.8), we cannot expect conservation of charge and energy. If, in addition to the hypotheses of Theorem 4.4.1, we assume (3.3.5) (respectively, (3.3.8)), then there is conservation of energy (respectively, conservation of charge). See [203, 204] and Theorem 4.4.6 below. 94 4. THE LOCAL CAUCHY PROBLEM PROOF OF THEOREM 4.4.1. We consider the case N > 2, the proof in the case N = 1 being easily adapted. Let 0 e CC°°(C,R) be such that 9(z) = 1 for \z\ < 1. Setting /i(u) = 9{u)f{u), /2(«) = (l-0(u))/(ti), one easily verifies that \fi(u) - fi(v)\ < C\u - v\, \h(u) - f2(v)\ < C(\ur + \v\a)\u - v\, where a is given by (4.4.3). Set gt(u)(x) = fe(u(x)) for £ = 1,2 and let r = a + 2. Using (4.4.5), we deduce from Holder's inequality that , IMW) ~ Pi(«)IUa < C\\u - v\\L2 , \\92(U) -9!i(v)\\Lr> 0, to be chosen later, and let q be such that (q, r) is an admissible pair. Consider the set e = {u£r((-r,r),ff1(RA'))nL'((-r,r),^r(Rw)); (4.4.8) IMU~((-7\t),*p) < Af, ||u||l«((-t,t),ivi.*-) < M] equipped with the distance (4.4.9) d(«, V) - \\U - v\\Lu^-T,T),V) + \\U - v||L«({_r,T),La)- We claim that (E, d) is a complete metric space. Indeed, we need only show that E is closed in Lq {{-T ,T), Lr(RN)). Consider (un)n>0 C E such that un -> u in Lq((—T,T),Lr(RN)). In particular, there exists a subsequence, which we still denote by (un)n>0, such that un(t) —> u(t) in Lr(RN) for a.a. t £ (—T,T). Applying Theorem 1.2.5 twice, we deduce that u e L°°((-T, T), H^R1*)) n L9((-T, T), ^1|,'(RJV)) and that ||w||L«(-T,T),Wri.'-) < \\un\\Li(-T,T),Wi L2(KN), it follows that <7i(«) : (-T,T) —> L2(R^) is measurable, and we deduce easily that 0i (u) € I°°((-T,r),i2(RJV)). Similarly, since #2 is continuous J7 (RN) —> l/ (R^), we see that g2(u) € Lg((-T,T), Lr'(RN)). Using inequalities (4.4.6) and (4.4.7) 4.4. KATO'S METHOD 95 and Remark 1.2.2(iii), we deduce the following: gi(u) £ L°°((-T, T),i/1(RJV)), g2(u)eL*{(-T,T),W1S(RN)), \\gi(u)\\L°°((-T,T),m) < Cilw||L-((-r,T),7/i) i \\92{u)\\L and \\9i(u) - 9i(v)\\l°°((-t,t),l*) < C\\u - v||L~((-r,r),L3), ||P2(«) - 92(v)\\LH(-T,T),Lr') ^ ^(llwltL°°((-T,T),Z.r) + N|£«((_7yT),l'-))llu ~ v\\Li{{-T,T),Lr) ■ Using the embedding H1(RN) ^ I/^R^) and Holder's inequality in time, we deduce from the above estimates that (4.4.10) lbiH||LM(-r,T),if1) + ll52(«)||L,'((_T)T),iyi^) < C(T+T^){\+Ma)M and \\9l{u) ~ gi{v)\\m(-T,T),L?) + \\92(U) -p2(v)||L,'((_TiT)jLr') < c(r + r «' )(i + MQ)d(u,t;). Given if £ H1^1*) and u e E, let W(u) be defined by (4.4.12) H(u)(t) = T(t)¥3 + i ( 7{t- s)g(u(s))ds. Jo It follows from (4.4.10) and Strichartz's estimates that (4.4.13) H{u) £C{\-T,T},Hl{RN))fM9{{-T,T),W^r{RN)), and ||W(u)|jl«((-r,r),^) + l!^(M)llL9((-T,r),^1.-) < (4 4.14) C\\ i.e., H(u) £ E and d(W(tt),W(v))o C H1(RN) such that 00; and let un be the maximal solution of (4.1.1) corresponding to the initial value 0 depending on IMIh1 such that un is defined on [—T,T] for n large enough and un —* u in C([—T,T],H1(RN)) as n —► 00. The result follows by iterating this property in order to cover any compact subset of (-Tmin, Tmax). We now prove the claim. Since H^nlln1 < 2||y>||j/i for n sufficiently large, we deduce from Step 1 that there exists T = T(||<£>||jji) such that u and un are defined on [—T, T] for n > no and (4.4.16) f!«||L«((-T,T)tif») + SUP ||«n||L~((-T,T),/fi) < C\\*P\\hi ■ n>no Note that un(t) - u(t) = T(t)(L2) + ||V(Un ~ «)||L9((-T,D,L'-) < C[||„ - + II(/iK) - /{(u))Vu|Ui{(_T,T),L») + ||(/2(Wn) - /2>))Vwlliy ((-t,t),zx)] ■ Therefore, if we show that ||(/l(«n)-fl(«))Vu||Li((_T>r)>La) (4-4-18) + IK/2M -/2(t*))Vti||L,.((.TiT)(iw) — 0, we obtain that (4.4.19) ||V(w„ - u)||l»((-t,T),l=) + IIV(m„ - u)I|l««-t,t)iL'-) n"^o 0, which, combined with (4.4.17), yields the desired convergence. We prove (4.4.18) by contradiction, and we assume that there exist e > 0, and a subsequence, which we still denote by (un)n>o such that 4 20) "(j>n) " •fl^)VUHil((-T,T))L2) + IK/2K) -/2(u))Vu||Lg'((_T,T),L.') >£■ By using (4.4.17) and by possibly extracting a subsequence, we may assume that un u a.e. on (—T,T) x RN and that there exists w e Lq((~T,T), Lr(RN)) such that |tin| < w a.e. on (—X,T) x RN. In particular, (/i(un) — f[(u))Vu and (/2(u„) - /2(w))V« both converge to 0 a.e. on (-T, T) x RN. Since l(/i(«n)-/i(«))V«| 00 as t T ^max if Tmax < 00 and as 11 -Tmm if Tmin < 00. (ii) u depends continuously on ip in the following sense: There exists T > 0 depending on IMI//1 such that if u in C([-T, T],LP(RN)) for all2 0 to be chosen later, we consider the complete metric space (E, d) defined by (4.4.8)-(4.4.9). We now proceed in three steps. Step 1. Existence, uniqueness, regularity, the blowup alternative, and continuous dependence. We first claim that if y> G jEf1(RAr), then the mapping 7i defined by (4.4.12) is a strict contraction on E for appropriate choices of M and T. Given 1 < j < k, we consider qj,~fj such that {qj,Tj) and (7j,p^) are admissible pairs. It follows from Holder's inequality that 2(r-rj) r(rj-2) II II ^ II II rl(r-2) M II ri(r-2) so that 2(t—r3-) rQj-2) IMIl9j(<-T,T),W'1',V) ^ ll^llz/°°((—t,t),//'1) ll^llz,9((—T)r),w'1'r) • In particular, if u G £, then u € L"'((-T, T), W1'ri(RN)) for all 1 < j < fc and (4.4.23) ll«||L«,((_TtT),H^) ^ M^iM^ = M. Next, it follows from (4.4.21)-(4.4.22) that gi is continuous 7J1(RAr) -* Lpj'(Mw). We deduce that if w 6 E, then : (-T,T) —» LP'(RN) is measurable, and it follows easily that ^(u) G L°°((-T,T),L^(RAr)). Applying Remark 1.2.2(iii) and (4.4.22)-(4.4.23), we conclude that gj{u) G L*:((~T, T), W1*"* (RN)) and where Cm depends on M. It follows that (4.4.24) IJffi(«)ll^((_Ttr),^-;) ^ +M)r^f. Applying now Strichartz's inequalities, we deduce from (4.4.24) that if T < 1, then H(u) G Lq((-T, T), W^r(RN)) n C([-r, T], tf1^)) and I|W(u)IIl»((-t,t),iv».-) + HW(«)||L-((-r,D,ffi) ^ *1Mljy* + *^W(1 + M)T* , where a = min 93 ~ f7j > 0. We now choose M,T so that M > 2K\\ 0, we set Is = (I — eA)~2. The reader is referred to Propositions 1.5.2 and 1.5.3 for all the relevant properties of I£. We define 9jAw) - h9j{hw) for 1 < j < k and w e H 1(RN), and we set fc g£ = ^2gj,£ and G£(w) = G(I£w). We observe that the 0 and that (4.4.26) g£ = G'£. We denote by u£ the solutions of (4.1.1) with g replaced by g£. It follows from the estimates of Step 1 that there exists T = T(|| H~l{RN), thus ge(u£) is continuous [-T,T\ -> H2(RN). Since

|[#i), (4.4.29) u£ —* u in C([-r, T], LPfR*)) for all 2 < p < 2N/(N - 2). Indeed for every j we write 9j,e(Ue) ~ 9j («) = K) ~ 9j,e(«) + /e(0. We also deduce from (4.4.21) applied to g^£ and (4.4.27) that (see the estimates of Step 1) \\9jAue) -^Wil^^.^r)^) ^ C^IK " w||z,«((-7yr),L*-). Using the above estimates and Strichartz's inequalities, we conclude that ||«e - u\\l~((~T,T),L2) + IK ~ u||L,((-T,T),L-) < + CTa||u£ ~ «||l,«((-iVr),L'-) with ae —► 0 as e j 0. By choosing T sufficiently small, we deduce that IK - w||l~((-t,t),l2)-* 0 asej.0, and (4.4.29) follows by applying (4.4.27) and Gagliardo-Nirenberg's inequality. Next, we deduce easily from (4.4.21)-(4.4.22) that (see the proof of (3.3.14)) (4.4.30) \G(u) - G(v)\ < C(M)(\\u - v\\L* + \\u - v\\Lr) for all u, v e H1(RN) such that \\v\\hi < M. In particular, (4.4.31) \Ge{ in ff1(MJV) by a sequence (<£n)n>i C H2(RN), and we denote by un the corresponding solutions of (4.1.1). We note that un satisfies (4.4.33). Letting n —■» oo, using continuous dependence (property (ii)) and the argument just above, we deduce that u satisfies (4.4.33). This means that E(u(t)) has a local maximum at t — 0. The same property applied after replacing

0 depending on \\- such that un is defined on [—T, T] for n large enough and un —► u in C([-T, T], i?1(RN)) as n —► oo. The result follows by iterating this property in order to cover any compact subset of (-Tmin, Tmax). We now prove the claim. By Theorem 4.4.6(h), we know that there exists T > 0 depending on H^JJh1 such that un is defined on [—T, T] for n large enough and un -> u in C([-T, T], for all 2 < p < 2N/(N-2). It follows that (see (4.4.27)) G(un) —* G(u) in C([-T, T]). By conservation of energy, ||Vu„||L2 -» ||V«|j£2 in C([-r,T]), so that (see Proposition 1.3.14) Vun -»• Vu in C([-T,r],L2(RJV)). This completes the proof. □ Remark 4.4.8. We may apply Theorem 4.4.6 to the case g(u) = Vu + f(u(-)) + (W*\u\2)u, where V,W e L6(RN) + L°°(WN) for some 6 > 1,5 > N/2, f is as in Theorem 4.4.1 (for example, f(z) = X\z\az with A G C and (N - 2)a < 4), and W G L^R^) + L°°(RN) for some a > 1, a > N/4. This follows easily from the estimates of Section 3.2. Note that in this case, even though the assumptions of Corollary 4.4.7 are possibly not satisfied, the initial-value problem (4.1.1) is, however, locally well-posed in fl'1(RiV). We need only prove the continuous dependence, and this follows from the argument used in Step 3 of the proof of Theorem 4.4.1. The term V[F(un — u) + (W * \un\2)un - (W * |u|2)w] is easily estimated by using the formula V[Vu + (W *\u\2)u\ = Wu + Wu + {W* \u\2)Vu + (W* uVu)u + (W * Vuu)u together with Holder and Young's inequalities. Note, in addition, that there is conservation of charge provided V and W are real valued and lm(f(z)z) — 0 for 4.5. a critical case in H1 (Kw) 103 all z € C. Moreover, there is conservation of energy provided V and W are real valued, W is even, and f(z) = zB(\z\)J\z\ for all z ^ 0 with 6 : (0, oo) -+ R. 4.5. A Critical Case in H1^) In this section we assume N > 3. If we consider the model case g(u) = X\u\au with A E R and a > 0, then it follows from Corollary 4.3.4 that the initial-value problem (4.1.3) is locally well posed inif^R*) if a < A/(N-2). If a > 4/(iV-2), then g does not map HX(RN) —► if-^R^), so we may consider the problem out of the reach of our method. (See Section 9.4 for some partial results in that case.) In the limiting case a = 4/(JV — 2), g G C(Hl(RN),H~1(RN)), the energy is well defined on i/1(R7V), and the various notions of H ^solutions make sense. On the other hand, the methods we presented do not apply at several steps. However, since this is a borderline case, we may think that an appropriate refinement of the method will yield some local well-posedness result. This is indeed the case, and below is such a result. (See Cazenave and Weissler [69].) Theorem 4.5.1. Assume N > 3. Let g(u) = A|u|^u with X € R. For every (p 6 H1(RN), there exists a unique strong H1 -solution u of (4.1.3) defined on the maximal interval (—Tm\n, 2max) with 0 < rmax,rmjn < oo. Moreover, the following properties hold: (i) There is conservation of charge and energy. (ii) u e Lfoc(~Tmin,Tmax), W1'P(RN)) for every admissible pair (q, r). (iii) If Tmax < oo (respectively, Tmin < oo), then ||Vu||Lq((0,Tmax),L-) = +oo (respectively, \\^u\\Lg^-Tmia,(i),Lr) = +oo) for every admissible pair (q,r) with 2 < r < N. (iv) u depends continuously on ip as follows. The functions Tmax,Tmjn are lower semicontinuous H1(RN) —► (0, oo]. Moreover, if in i/1(RAr) and if un is the maximal solution of (4.1.3) with the initial value ipn, then un —► u in Lp((-S,T),H1(RN)) for every p < oo and every interval [S,T] C ( ^min) -^max ) - REMARK 4.5.2. Here are some comments on Theorem 4.5.1. (i) We do not know whether there is uniqueness in the sense of Definition 3.1.4, i.e., uniqueness of weak if1-solutions. In our proof of uniqueness, it is essential that we consider strong if ^solutions. (ii) We do not know whether the usual blowup alternative holds (i.e., the blowup of In particular, we cannot deduce global existence results from the a priori estimates of ||u(£)||#i that follow from the conservation laws when A < 0. (iii) The statement of continuous dependence is weaker than usual, since un —> u in Lp((-S, T), H1 (RN)) for every p < oo, but possibly not for p = oo. In the case A < 0, then there is also convergence for p = oo; see Remark 4.5.4(iii). There are at least two methods for proving the existence part in Theorem 4.5.1. One can use a variation of Kato's method. This provides a simple proof, but it is then delicate to establish the conservation of energy. Instead, one can truncate the nonlinearity g and obtain solutions of the truncated problem for which there is 104 4. THE LOCAL CAUCHY PROBLEM conservation of energy. Next, one uses the Strichartz estimates to pass to the limit. This is the method we follow here. We begin by introducing the truncated problem. Given n € N, let sax -n / \ S 9^ if N n. In particular, gn : C —> C is globally Lipschitz. Set (4.5.2) Gn(s)= f' gn{ 0, and let En e C1(H1(RN), R) be defined by (4.5.3) En(u) = 11 |Vtx|2 - y Gft(u) for all u € #KR")- Given T > 0 and u : [0,T] Hl{RN), define W„(ti) by (4.5.4) Hn(u)(t) = + * / 7{t~ s)gn(u(s))ds Jo for 0 < t < T. H is defined similarly, by replacing gn by 5 in (4.5.4). Finally, let 2iV2 2N (45-5) P=N2-2N + r 1 = —2' so that (7, p) is an admissible pair. We will use the following lemma. LEMMA 4.5.3. If (q,r) is any admissible pair, then \\Hn(u) ~ Wn(^)II.L«((0,T)vL-) < (4.5.6) ) C(||Vw||z,^((o,T),lp) + II Vt;||lt((0,t),lp)) ||« ~ v\\L-r^0tt),lp) , (4.5.7) ||VWn(u)||L,((0fr)fLO < ^II^OV^Hl^^d.l-) + C||Vti||J^0fT)iLP), \\Hn{u) -H{u)\\L^0)T^Lr) < (4-5.8) _ „_2 ^T^n-^||VU||-2(0iT))LP)|iu||L^o,r),^)> /or some constant C independent of n, T, and by Sobolev's inequality, (4.5.6) follows by applying Strichartz's estimate. (4.5.7) is proved similarly, by using the inequality |Vn}u|. 4.5. A CRITICAL CASE IN H1^) 105 We deduce that 4 (4.5.9) \\gn(u) - g{u)\\Ly>{{0!T)!LP'} < C\\Vu\\^{QThLP)\\l{lu\>n}u\\L^({0tT)tLP). Finally, (4.5.10) ||l{|„|>n}«IU, < n-mhv < Cn-^T\\u\\P^ . (4.5.8) follows from (4.5.9), (4.5.10), and Holder's inequality in time. □ Proof of Theorem 4.5.1. We consider only positive times, the problem for t < 0 being treated by the same method. We proceed in six steps. Step 1. Uniqueness. This follows from Proposition 4.2.5. STEP 2. Approximate solutions. Since gn defined by (4.5.1) is globally Lip-schitz C —► C, there exists a unique, global solution un € C([0, oo), Hl(WN)) of the problem (4.5.11) Un(t) = Hn(u)(t) for all t > 0. Moreover, there is conservation of charge and energy, (4.5.12) K(*)||l* = IMU*, En(un(t)) = En( 0. See, for example, Corollary 4.3.3 and Corollary 6.1.2 below. Furthermore, it follows from Remark 4.4.3 or Theorem 4.4.6 that (4.5.13) un € L«((0,T), W^r{rn)) for every admissible pair {q, r) and every T > 0. Consider now any admissible pair (q,r) and any T > 0. We deduce from (4.5.13), (4.5.11), and (4.5.7) that (4.5.14) |jVun|U,{(0,r),L-) < imOV^Uw),^) + Q\\^\0(o,t),l») • Similarly, we deduce from (4.5.6) that 4 (4.5.15) ||«n||L«((o,T),L') < C\\ip\\L2 + C\\Vun\\^f{OThLP)\\un\\L-r({0!T)!LP). Finally, given £ > n, we may write un - ue = [Hniun) - nn(ue)} + [?in(ue) - 7i{ue)} + \H{ue) - Ue{ue)}, and we deduce from (4.5.6) and (4.5.8) that \\un -Ue\\Li{(q,T),L<-) _4 (4.5.16) - G(Wun\\L^((o,T),L») + l|Vu*||i,-ir((o,r),Lp)) N' -2 N*-2N+4 Note that the constant C in (4.5.14), (4.5.15), and (4.5.16) may depend on the admissible pair (q, r), but is independent of n, £, and T. Step 3. Passage to the limit. We will solve the equation (4.1.4) (which is equivalent to (4.1.3)), by letting n —> oo in (4.5.11). Consider K larger than the 106 4. THE LOCAL CAUCHY PROBLEM constant C appearing in (4.5.14), (4.5.15), and (4.5.16) for the particular choice of the admissible pair (q, r) = (7, p). Fix 5 > 0 small enough so that (4.5.17) K(A5)^ |U,((0,r),LO then (4.5.19) supJlVun||LT((0,T),i>) < 25 n>0 and (4.5.20) sup||wn||L<,((0iT)jVi/i,r) < 00 n>0 for every admissible pair (q, r). (Note that, given tp G i?1(EJV), (4.5.18) is satisfied if T > 0 is sufficiently small. Indeed, T(-)V

0 as T J. 0.) Set Bn(t) = ||VtiTXj|L,((o,t),iP). It follows from (4.5.14) that for every 0 < t < T, en{t) < 6 + cen(t)®%. If 6n(t) = 25 for some t € [0,T], then 25 < S + C(2d)^i < 25, by (4.5.17), which is absurd. Since 8n is a continuous function with 6n(0) = 0, we conclude that 0n(t) < 25 for all t G [0,T), which proves (4.5.19). Applying now (4.5.14) for any admissible pair (q,r), we find that (4.5.21) sup ||Vun||L,((o,T),L'-) < 00 . n>0 Applying (4.5.15) with (q, r) = (7,p) and using (4.5.17) and (4.5.19), we obtain \\Un\\L^((0,T),LP) < 0\\ip\\L2 + ^\\Un\\Ly((pfT)tLP) , and so H^nlU^o,!1),!^) ^ 2C7|| n and for all 0 < r < T, t < 00. (Note that we used again (4.5.17).) It follows that (un)n>o is a Cauchy sequence in L7((0, r), LP(RN)). Applying again (4.5.16), but with an arbitrary admissible pair (q, r), we conclude that (Mn)n>o 4.5. A CRITICAL CASE IN /f1(RN) 107 is a Cauchy sequence in Lq((0, r),Lr(RN)). If we denote by u its limit, then for every admissible pair ( u in L«((0,r),Lr(3RJV)) for all 0 < r < T, r < oo. By using Lemma 4.5.3 we may let n —► oo in (4.5.11), and we obtain that u satisfies (4.1.4) for all 0 < t < T, t < co. Since #(u) G L^((0,T), W^'fR*)), we deduce from Strichartz's estimate thatu€ C^rj^fR*)) for every 0 < r < T, r < oo. In particular, u is a strong I^-solution of (4.1.3) by Proposition 3.1.3. Step 4. The conservation laws. We deduce from the conservation of mass for un (see (4.5.12)) and from (4.5.23) applied with (g,r) = (co,2) that ||tz(£)j|L2 = IMIl2- We now show the conservation of energy. Applying (4.5.23) with (q, r) = (oo, 2) and using (4.5.20), we deduce easily that un —► u in L»((0, r),L^{RN)) for every r < oo, r < T, and every q < oo. In particular, there exists a subsequence, which we still denote by (un)n>o, such that un(t) —> u(t) in L w-2 (Rw) for a.a. t G (0,T). It follows that Gn(un) -+ G(w) in L1(RJV) for a.a. t G (0,T). Using the conservation of energy for un and the lower semicontinuity of the gradient term, we deduce that E{u{t)) < E(ip) for a.a. t G (0,T), hence for all t G (0,T) by continuity of u(t) in if1(RJV). Considering the reverse equation, one shows the converse inequality. STEP 5. The blowup alternative. By uniqueness, we may consider the maximal solution, defined on the interval [0, Tmax). We show the blowup alternative by contradiction, so we assume that Tmax < co and u G LQ((Q,Tma,x),W1>r(RN)) for some admissible pair (q,r) with 2 < r < N. Let b G (2, -j2^) be denned by 6 N-2\r NJ' 4 and let o be such that (a,6) is an admissible pair. Since |V#(«)| < C\u\ N~2 |Vwj, one easily verifies by using the Sobolev inequality ||tt|| < C\\ VwjJl^ that (4.5.24) \\9(u)\\L*'i(3,t),wi.o') ^ c\\Vu\\£q(lStt)!Lr)\\u\\L«{{s,thwi,b), with C independent of 0 < s < t < Tmax. Since (4.5.25) u(s + t) = 1(t)u(s) + i f 7(t - a)g(u(s + e))do~, Jo we deduce from Strichartz's estimate that 4 HlL"((«,t),wM) < C\\u(s)\\m + C\\Vu\\^{Stt)Lr)\\u\\La{{Stt^Wi,b), with C independent of 0 < $ < t < Tmax. Fix s close enough to Tmax so that C||V^II^,Tmax),L0 < 1/2- It follows that I|w||l-((m),ivi.>) ^ 2C,II«(s)IIhi for all s < t < Tmax, and so u G La((s,Tmax),W^b(RN)). Therefore, u G La((0,rmax), W1'b{RN)) and, applying again (4.5.24), we conclude that g(u) G 108 4. THE LOCAL CAUCHY PROBLEM l/((0,Tmax), W1-b'(RN)), so that u e L^((0,Tmax), W1'"(rjv)) by Strichartz's estimate. We finally deduce from (4.5.25) and Lemma 4.5.3 that Af + 2 l|^(Ow(*)ll^((0,r„Iax-t),^.") < \\u\\L->((tiTmux),Wi.e) +C\Mw(t,Tm„),W*,r)i where C is independent of £ £ [0, Tmax]. Therefore, we may choose t close enough to Tmax so that \\7(-)u(t)\\L-motTmax^t)jWi,p) < S with 6 given by (4.5.17). Therefore, there exists £ > 0 such that \\^{')u(t)\\L-y^0jTmax+£_t^wi,P) < S. We deduce from Step 3 that the solution u can be extended to the interval [0, Tmax + e], which contradicts the maximality. Step 6. Continuous dependence. Fix 0 < T < Tmax and let 5 > 0 satisfy (4.5.17). Since u € C{[Q,T\,Hl{RN)), Uo<* 0 such that (4.5.26) sup ||T(.)u(t)||L->((o,T)(ttn.p) < - • 0T),L'') ^ 25. Arguing as in Step 4, we deduce that un —> u in L*((Q,t),Lt(Rn)) and that un is in a bounded subset of L9((0,r), VT1'r(rn)) for every admissible pair (g,r). Therefore (see (4.5.6)), (4.5.27) * Un->u in C([0,t], L2(RN)). Choosing r > 2 arbitrarily close to 2, so that q < oo is arbitrarily large, and applying Gagliardo-Nirenberg's inequality, we obtain that un -► u in £«(((), t),!.^^)) for every g < oo. Since E{un{t)) = E{ipn) -> J5( ||Vu||x,2 in L9(0, r) for every g < oo. On the other hand, since un is bounded in C([0,r], ff^r^)), we deduce from (4.5.27) that un(t) w(t) in H1(RN) for every t e [0,t]. Using the convergence of the norm, one concludes easily that un —* u in L^((0, r), if 1(rar)) for every g < oo. In particular, there exists (tn)n>o C [r/2,r] such that ||uM(in) — u(tn)—» 0. Repeating the above argument (using (4.5.26) and (4.5.27)), we deduce that un exists on [0,3r/2j for n large enough and that un -t- u in Lg((0,3r/2), ff^R*)) for every g < oo. We may now iterate the same process to cover the interval [0, T]. □ Remark 4.5.4. Here are some further comments on Theorem 4.5.1. (i) If || V^||jf/2 is small enough, then we may take T = oo in Step 3 of the proof of Theorem 4.5.1. Indeed,'||T(-)Vv?||l-»(r,lp) < C\\V 0 4.6. l2 solutions 109 where uM = u1{|u|>m}- In fact, the limit is not only positive, but bounded from below by a positive number independent of the solution. This indicates a concentration phenomenon in LN-2 (R-^). (iii) In the case A < 0,ithe continuous dependence statement (iv) can be improved. More precisely, un —* u in C{[—S, T], i71(RJV)) for every interval [—S,T] C (—Tmin,Tmax). In other words, there is the usual continuous dependence property. The proof is in fact simpler. We have un(t) —> u(t) strongly in L2{RN), weakly in H1^), and £(un(i)) E(u(t)) for all t G [0, r]. Since A < 0, both terms in the energy are lower semicontinuous so that indeed \\un{t)\\-»• ||w(*)lii7^fj 811(1 II^WnWIlL8 ~* 11 Vu(t)j|^,2. Thus un(t) —► «(£) strongly in iJrl(RiV) and one concludes as above. (iv) In the case A < 0, one would expect that the solution is global for every initial value. This is only known if we assume further that ||Vv?||l2 is small (see (i) above) or if ip is spherically symmetric (see Bourgain [39]). 4.6. L2 Solutions In this section we construct solutions of some nonlinear Schrodinger equations for initial data in L2{RN). Such results were first obtained by Y. Tsutsumi [343] (see also Cazenave and Weissler [69, 70] and Kato [204]). We assume that (4.6.1) g : L2(RN) n Lr(RN) -> Lr'(RN) for some (4.6.2) r € [2, — J (re[2,oo]if/V = l). Furthermore, we assume that there exists a > 0 such that, for every M > 0, there exists K{M) < oo such that (4.6.3) \\g(v) - g(u)\\Lr. < K{M){\\u\\tr + H£r)ll« - «|U- for all w, v e I2(lJV)nIr(lJV) such that ||u||L2, ||v||£a < M. We have the following result. Theorem 4.6.1. Assume (4.6.1)-(4.6.3) and set f so that (q,r) is an admissible*pair. If a + 2 < q, then for every

—> Tmin,Tmax are tower semicontinuous L2{RN) —► (0,oo]. If

(RN) ^ H~2(RN), we see that Au + g(u) £ L{f? ((-Tmin,TmaLX),H-2(RN))."it follows that equation (4.1.1) makes sense in T>'((— Tmjn,rmax), H~2(RN)). In particular, ut £ L{^{(-Tmin,Tmax),H-2(RN)) and (4.1.1) makes sense in H~2(RN) for a.a. t £ ( Tmin j Tmax j. " For the proof of Theorem 4.6.1, we will use the following lemma. Lemma 4.6.3. Let g satisfy (4.6.1)-(4.6.3) and let I be an open interval of R. If u is measurable both as a function I —> L2(RN) and as a function I —> Lr(RN), then g(u) is measurable I —► Lr (RN). Proof. Note that for a.a. t £ 7, u{t) £ L2(RN) n Lr(RN), so that g(u(t)) £ Lr (RN) is well defined. Consider a function

(RN) such that 1 and x £ RN. Using the dominated convergence theorem, we see that tpnu(t) —» u{t) in L2(RN) and in Lr(RN) as n '—► 00 for a.a. tel. In particular, g(tpnu) —» _»(«) in //"'(R-^) as n —» 00 for a.a. t £ I. Therefore, we need only show that for any given n > 1, g(o C C(I,Lr(RN)) such that ttfc(i) —► u(t) in I^R^) as A; —> 00 for a.a. tel. Since ipnuk(t) is supported in a fixed compact subset of Kn c RN and Lr(Kn) ^ L2(Kn), it follows that L2(RN) n Lr(RJV), and so g((pnuk) is continuous 7 -* Lr'(RN). Since v'nWfc ¥>n« a^fc 00 in Lr(RN), hence in L2(RN)nLr(RN) for a.a. t £ I, we have g((pnUk) —» g( Z/'fR^), which completes the proof. □ Proof of Theorem 4.6.1. For the existence part, we use a fixed point argument as in Section 4.4. For the conservation of charge, we need a regularization process. We proceed in five steps. Step 1. Existence. Fix T, M > 0 and set E = {u£ L°°((—T,T),L2(RN)) n L*{{-T,T),IS{RN)yy (4.0.4) IMlL«((-_yr),L3) + \\u\\Li((-T,T),Lr) < M) ■ It follows that E is a complete metric space when equipped with the distance (4.6.5) 6(u,v) = ||u-uj|L»((-T,r),L=) + 11« - v|U«((-t,t),l-)- Consider u £ E. It follows from Lemma 4.6.1 that #(«) : 7 —► LT {RN) is measurable. Moreover, we deduce from (4.6.1) and (4.6.3) that for a.a. t £ (~T,T), < ll5(0)IL^ + K{M)\\u(t)\\ltl. 4.6. L2 SOLUTIONS 111 Therefore, by Holder's inequality in time, q-(Q-t-2) UMhsi-TW) < CT7\\g(0)\\L*> +C^WII«C1+J„'((_r,r),Lr) < CT7\\g(0)\\Lr, + CT2^f±21X(M)||u||^(1(„T>T))Lr); and so (4.6.6) \\9(u)\\L*a-T,T),Lr>) < CT?\\g(0)\\Lr, +CTq-^K(M)Ma+l. Similarly, one shows that for u, v £ E, (4.6.7) ||0(u)-$WIJl«'<(-^^ ^ ^ Applying (4.6.6), (4.6.7), and Strichartz's estimate, we see that g(u)(t) = i f 7{t- s)g{u{s))ds Jo is well defined, that G{u) £ C([-T,T], L2(RN)) n Lf{(-T,T),L<>(RN)) for every admissible pair (7, p), and that (4.6.8) ||0(u)||L,((_riD,x,,) ||£2, we see that if T is sufficiently small (depending on ||v?||l3)) Hiv) £ E for all u £ E. Moreover, we deduce from (4.6.9) and Strichartz's estimate that, by possibly choosing T smaller (but still depending on ||v?||l2)» d(H(u),H(v))<~d(u,v) for all u,v £ E. Thus H has a unique fixed point u £ E. Note that g(u) £ Li'((-T,T),Lr'(RN)) ^ L«'((-r,r), JJ-^R*)). It follows (see Section 1.6) that u e C([—T,T], H~1(RN))*r) W1'1((—T,T), H~3(RN)) and u satisfies (4.1.1) in H~3(RN) for a.a. t £ (-T, T). This proves local existence. Step 2. Uniqueness. We first note that uniqueness is a local property, so that we need only establish it on possibly small intervals, 'ib see this, we argue for positive times, the argument tor negative times being the same. Suppose we know that if u,v £ C([0,T],L2(RN)) n Lq((0,T),Lr(RN)) are any two solutions of (4.1.1), then u = v on (0, r) for 0 < r < T sufficiently small. We may then define 0 < 6 < T by 9 - sup{0 < r < T; u = v on (0, r)} . It follows that u = v on [0,9]. If 9 = T, uniqueness follows, so we assume by contradiction that 6 < T. We see that Ui(-) = u(9 + •) and vi(-) = v(B + •) are two solutions of (4.1.1) with

jt8) + ||v||L-(/|La)) x (||u||l«(I,L*-) + ||u||l«(/tl'))a||w - ^llz/W) . Since Q{u) — Q{v) = u — v, we deduce that if |/j is sufficiently small; i.e., if T is sufficiently small, then 11« - v|U«(/,l»-) < ^ ||« - v\\li(I,L-) , i.e., u = i> on J. Step 3. The blowup alternative and continuous dependence. Arguing as in the proof of Theorem 3.3.9, we define the maximal solution by using the uniqueness property; and since T depends on IMiz,2; we deduce the blowup alternative. Next, using again (4.6.9), we deduce easily that, if tp,ip G L2(RN) and if u,v are the corresponding solutions of (4.1.1), then for some T depending on JMIl2, II^IU2, d(u,v) < C\\

0 and let Je = (I - sA)'1. (The reader is referred to Proposition 1.5.2 and 1.5.3 for all the relevant properties of J£.) We define the nonlinearity g£ by ge(w) = JEg(JEw)) for all w e L2(RN). We observe that J£w G HX(RN) C L2(RN) n Lr(RN) so that g(J£w) G Lr'{RN). Since Lr'(RN) tf-^R"), we have gs{u) G HX(RN). Moreover, since Je is a contraction" in LP(R^) for ull 1 < p < oo, we see that g£ satisfies the assumption (4.6.3) uniformly in e > 0. Moreover, {g£{w),iw)L2 = (g£(w),iw)Lr>Lr = (g(J£w),iJsw)Lr> Lr =0. We now proceed as follows. Consider

0, we deduce from the estimates of Step 1 that u£ is defined on some interval [—T, T] with T independent of e > 0. Since g{JEu£) G Lq'(-T, T),Lr>(RN)) (see the estimates of Step 1), it follows that g£(u£) G L1((-T,T),H1(RN)). This implies that u£ G C([-T,T],H1(RN)), 4.6. l2 solutions 113 Therefore, we may take the H~x - H1 duality product of equation (4.1.1) (with g replaced by g£) by iue, and we obtain 1 d 2 dt"We^"^2 = ~^U^iu^H-l,H^ ~ {9e{Ue),iu£)H-iiHx = 0, so that (4.6.10) IK(t)IU' = Ml* for |i| < T. We claim that, after possibly choosing T smaller, (4.6.11) ue —> u in L°°{(-T,T),L2{RN)). Indeed, we write 9e(u£) - g(u) = g£(u£) - fle(u) + J£(g(J£u) ~ g(u)) + (J£ - I)g(u), and we deduce that IKK) - 9{u)\\Lo'({-t,t),l-') < ll&K) ~ 9s{u)\\l<>' ({—t,t),lr') + \\g(Jeu) - 9(u)\\l^'({-t,t),l-') + l!K ~ ^)5(«)IIl«'((-T,T),L^) • Since g(u) u in Lq((-T,T),Lr(RN)), we deduce from (4.6.7) that \\g{Jsu) -p(M)||L,'((_T,r)tLr') -^0. We also deduce from (4.6.7) (applied to p£) that q-(o + 2) \\g£(ue) - 5e(«)llL9'((-T,r),l'-') ^ ct " IK ~ u||l«((-T,T),I,-) - Using the above estimates and Strichartz's inequalities, we conclude that IK - u\\l^((-T,T),L2) + IK - u\\lo{{-T,T),L-) j)-(q + 2) Q C if1 (R^) and we use the continuous dependence to obtain the conservation of charge. Global existence follows from the blowup alternative. □ Theorem 4.6.4. Let g — g\ + • • • + gk, where each of the gj's satisfies (4.6.1)-(4.6.3) for some exponents rj,aj. Set and letr — max{ri,... , nt} andq = min-j^,..., q^}- If2+aj < qj for j = 1,..., k, then all the conclusions of Theorem 4.6.1 hold. 114 4. the local cauchy problem proof. Fix M > 0 and consider (E, d) defined by (4.6.4)-(4.6.5). We see (cf. the proof of (4.4.23)) that \M\L*n(-T,T),Lrn < 11 ^ 11t-,r) ) 11 ^ I i 14 a-V,t> ) • In particular, \\u\\Lij((-t,r,lrj) < f°r all w g E and ||w - v||l'j((-t,t,li'j) ^ d(w,v) for all u,v £ E. We deduce (see the proof of (4.6.6)-(4.6.7)) that llfc(«)lliVT,r),L'i) ^ ^j|^(0)||^ +CTI^i±^^(M)M^+1, Ito(ti) - P^)!!^.^^ < CT^-^li^M)^ d(u,tO. It follows that £,(«)(*) = « / T(i - s)^-(u(s))ds ./o is well defined, that ^(«) g C([-r,T],L2(RJV)) n L7(-r,T),L^(RJV)) for every admissible pair (7,p), and that ll^(")ll^(-r.T),Lp) < llfli(0)||L.< + CT * ^(M)M^+1, HC7i(lt) " Gj{v)\\Li{r-T,T)tL>) < CT* ^ Kj(M)M^ d(u, «) . Given G £2(R*), set now H{u){t) = T(t)^ + £i(«)'(*) + ■■ • + &(«)(*)• We deduce that for every u g E, ||W(u)||loo(_t]T)>L2) + ||W(u)liL9(-r,r),L»-) < c|M|L9 +C^(r^||^(0)||Lr, +r2^±^irj(M)MQ>+1). 3 = 1 Choosing M = 2C|j 1, 6 > N/2 and W £ L*(RN) + L°°(RN) for some 1, a > N/2. Let / : RN x C —» C be measurable in x g RN and continuous in z g C. Suppose that f(x, 0) = 0 for all x £ RN and that \f(x,Zl) - f(x,z2)\ < C(l + \Zl\ + \z2\Y\zx - z21 for some 0 < 0 < A/N. Set g(u) = Vu + /(., «(•)) + {W * |u|2)u. We have the following result. Corollary 4.6.5. If g is as above, then the conclusions of Theorem 4.6.4 hold. Moreover, if V and W are real valued and if f(x, z)z £ R for all z £ C and x £ RN, then there is conservation of charge and all solutions are global. 4.7. A CRITICAL CASE IN L2(&N) 115 Proof. We let V = Vi + V2 with Vx G L6(RN) and V2 G L°°(RAr), W = + W2 with € Z/(RN) and W2 G £°°(RN). We need only show that each of the terms V\u, V2u, /(•,«(•)), (VFi * \u\2)u and (W2* |u|2)u satisfies the assumptions of Theorem 4.6.4. It is immediate that V\,u (respectively, V2u) satisfies the assumptions with K(M) = C, a = 0, and r = 26/(8 - 1) (respectively, r = 2). Applying Holder's and Young's inequalities, one easily verifies that (Wi*|u|2)u (respectively, (W2*\u\2)u) satisfy the assumptions with K(M) = M2, a = 0, and r — 2a/(a- 1) (respectively, r = 2). Finally, one may write f(x,z) = fi(x,z) + f2(x,z), where fi is Lipschitz continuous in z, uniformly in x, and l/2(a:, Zl) - f2(x, z2)\ < C(\Zlf + \z2f)\Zl - z2\ for a.a. x G RN and all z\,z2 G C. One easily verifies that /i(-,u(-)) (respectively, /2(-,w(-))) satisfies the assumptions with K(M) = C and with a = 0 and r = 2 (respectively, a = (3 and r = (3 + 2). □ 4.7. A Critical Case in l2(rn) If we consider the model case g(u) = A|«|Qu with A G C and ct > 0, it follows from Corollary 4.6.5 that the initial-value problem (4.1.3) is locally well posed (in an appropriate sense) 'mL/^RN) if a < 4jN. In the limiting case a — 4/TV, the method we presented does not apply at several steps. However, an appropriate refinement of this method yields local well-posedness, and below is a typical result. (See Cazenave and Weissler [69].) Theorem 4.7.1. Let g(u) = X\u\au with A G C and a = 4/N. For every

a + 2. (ii) u G £foc((-Tmin,rmax),Lr(RAr)) for every admissible pair (g,r). (hi) //, in addition, if G 771(RJV), then u G C((-Tmin,Tmax),Hl(RN)). (iv) (Conservation of charge.) If A G R, then ||u(t)|jL2 = \\(p\\l* for all t G (—Tmin 1 2m ax ) • (v) (Continuous dependence.) The mappings

t-+ Tmin,TmiiX are lower-semi-continuous L2(RN) —* (0,oo]. If (pn —> (p in L2(RN) and if un denotes the corresponding solutions of (4.1.3), then un —> u in Lq(I,Lr(RN)) for every interval I ^ (~Tm\n,TmSi,x) and every admissible pair (q,r). Remark 4.7.2. Arguing as in Remark 4.6.2, we see that if u is as in Theorem 4.7.1, then equation (4.1.3) makes sense in H~2(RN) for a.a. t G (—Tmjn,Tmax). Proof of Theorem 4.7.1. Consider an interval I C R with 0 G I and let u, v G La+2(I,La+2(RN)). It follows from the estimate \\u\au - \v\av\ <{a + l)(\u\a + \v\a)\u - v\ 116 4. the local cauchy problem and Holder's inequality that \\\uru-\v\av\\ m m < (4.7.1) l^t(i,l^t) (a + 1)(IMIl«+2{jijc«h-2) + NIl«»+»(j,l»+2))!!u ~ ^IUq+2(J,l°+2) • Setting Q(u)(t)= f 7(t - s)\u(s)\Qu{s)ds, Jo it follows from (4.7.1) and Strichartz's estimates that G{u) £ C(7,L2(RN)) n Lq{I, Lr(RN)) for every admissible pair (q, r), and that (4.7.2) \\G(u)\\Lq{JtLr) < C||u||2+i(J|Ln+9) and (4 7 3) S^-^*^) - c(\\u\\l°+hi,l<*+*) + \\v\\l°+hi,l*+*)) \\u ~ v\\l°+Hi,l°+*) for some constant C independent of 7. We now proceed in four steps. Step 1. There exists 6 > 0 such that if

0, to be chosen later, and let ip £ L2(RN) satisfy (4.7.4). Consider the set E = {u £ La+2(I,La+2(RN)); ||u||La+a(JtIl«+a) < 25} , so that (E,d) is a complete metric space with d(u,v) = ||w - uiiL°+2{/,LQ+2)- For u £ E, set H(u){t) = T(t)

Tmax. Since the equation (4.1.3) is invariant by space translations and since the gradient is the limit of the finite differences quotient, we deduce easily from (4.7.5) that liVv|iLoc{(0jTmax)jjL2) < C||Vv?|jL2 , which contradicts the blowup alternative for the H1 solutions. Thus Tmax > b and one shows by the same argument that a > — Tmin. Step 3. For u as in Step 1, we show that there is conservation of charge. Indeed, let (pn -+ (p in L2(rn), with u in C(I, L2(rn)), where un is the solution associated to • oo, we obtain ||ti(£)||^2 = |j<£>||z,3 for all t g I. Step 4. Let tp g L2(rn). Since 7(-)

a + 2. It follows from Holder's inequality that for any T < IMIl«+2((0,T),I/*+2) < llullL«((0)t)ILa)lluilLT({o,t),L'-) ^ IMl£2 IIwlli7(%,t),l-) > With A* = (a+2)(r-h ' LettinS ^ T rmax, we Obtain ||u[|l9((0,Tjmx),L'-) = 00. Finally, we show the continuous dependence. Consider T < Tmax. Since u € C([0,T], L2(RN)), it follows from Strichartz's estimates and an obvious compactness argument that there exists r > 0 such that II^(Ou(OIIl°+2((0,t),L«+3) < £ for all t e [0, T], where 5 is as in (4.7.4). Fix an integer n such that T < nr, let K > 1 be the constant in (4.7.5), and let M be such that ||J'(-)i;|jZ/*+2(KtLo+2) < M||v||L2. Let e > 0 be small enough so that MKn~xe < 5/2. We claim that if JI^-^Hi,2 < e, then Tm&x(tp) > T and |jw-v||C([0)r])L2) + ||w-i;||Iia+2((0>T)iLo+2) < nKn\\(p~t/j\\L2, where v is the solution corresponding to the initial value Indeed, if \\) > T/n and that \W ~ v\\c([o,T/n},Li) + \W ~ 'u||L-+2((0,T/n),L«+2) < K\\

0 and let p € Hl(RN) be such that | • \(p(-) e L2(RN) and E(tp) < 0. It follows from Theorem 6.5.4 below that u blows up in H1 for both t > 0 and t < 0. Therefore, Tmax < 00 and Tmin < co, by Theorem 4.7.1(iii). Remark 4.7.4. We conjecture that if A < 0, then Tmin = Tmax = 00 for all

o C if1(RJV), with >pn —►

in L2(RN) and xipn(x) bounded in L2(RN). The corresponding solutions satisfy un e CCR.fT^R^)) and | ■ |w„ € C(R,L2(RJV)) (see Lemma 6.5.2 below), and 4.8. H2 solutions 119 from the pseudoconformal conservation law (see Theorem 7.2.1 below), we see that j|un(i)||£_"+2 < Ct~2 for all t £ R. By continuous dependence, this implies that \\u(t)\\1t+2 < Ct~2 for a.a. t £ (-Tmin,Xmax). In particular, we see that if Tmax < oo, then u £ La+2((Q,Tmax),La+2(RN)), which contradicts the blowup alternative (note that u £ LQ+2((0, T), La+2(RN)) for all 0 < T < Tmax). We see as well that Tmjn = oo. In addition, it is clear that the above estimate implies that u £ La+2(R, La+2(RN)). The estimate for an arbitrary admissible pair follows easily from Strichartz's estimates. Remark 4.7.5. There exists n > 0 such that if (4.7.7) ||F(-)vllL°+a(R,L°+3) < 77, then Tmin = Tmax = oo. Moreover, u £ LQ(R,Lr(RN)) for every admissible pair (q,r). This follows easily from Step 1 of the proof of Theorem 4.7.1 (see in particular (4.7.4)). However, this conclusion does not hold in general for large data. Indeed, if A > 0, there exist nontrivial solutions (standing waves) of the form u(t,x) = eiu;tcj)(x), with £ H1(RN), 0^0 (see Section 7.2). These solutions obviously do not belong to Lg(R, Lr(RN)) if q < oo. On the other hand, by Strichartz's estimates, (4.7.7) is satisfied if IMIl2 < p for p small enough. 4.8. H2 Solutions In this section we construct H2 solutions by a fixed-point argument, and we follow the proof of Kato [203, 204]. See also Y. Tsutsumi [340]. We note that obtaining H2 estimates by differentiating twice the equation in space would require that the nonlinearity is sufficiently smooth (see Section 4.9 below). Instead, we differentiate the equation once in time, and then deduce H2 estimates by the equation. Let g : H2(RN) -> L2(RN). Assume there exist 0 < s < 2 and 2 < r,p < 2A^/(iV - 2) (2 < r, p < oo if N = 1) such that (4.8.1) g £ C(HS(RN), L2(RN)) is bounded on bounded sets and (4.8.2) \\g(u) - g{v)\\Lpl < L(M)\\u - v\\Lr for all u, v £ H2(RN) such that ||tt||j_., < M. Theorem 4.8.1. Let g = e/i + • • • + gk, where each of the gj's satisfies the conditions (4.8.1)-(4.8.2) for some exponent Sj,rj,pj and some function Lj(M). For every ip £ H2(RN), there exist Tm&x,Tmm > 0 and a unique, maximal solution «eC((-rmin,rmax),7f2(RN))nC1((-rmin,rmax),JL2(EAr)) of (4.1.1). Moreover, the following properties hold: (i) u £ W£c*((-!_„„, Tmax), Lr(RN)) for every admissible pair (q,r). (ii) (Blowup alternative) // Tmax < oo (respectively, Tmln < oo), then IIwWIIh2 —> 00 05 11 Tmax (respectively, as t I — Tmjn). (iii) u depends continuously on ip in the following sense. There exists T > 0 depending on |J

h-2) < || 0 to be chosen later, we set I = (—T, T) and we consider E={ue L°°{It H'(RN)) n Whoo(I, L2(RN)) n W^q(I, Lr(RN)); (4.8.9) i u(0) = tp and ||u||L~ + \\u\\Wi,x{TfL2} + IMIun^L-) {u) °(i,V) )||L» + ™(M) + F(M) Er1"*""*) , 4.8. H2 solutions 123 and ||H(u)j|l,~(J,i/2) < k k (48-19) Cb/lMI** + E \M E. A similar, though simpler, argument shows that, after choosing T possibly smaller, H is a strict contraction on (E, d). Therefore, H has a fixed point u G E, which is a solution of (4.1.1). It remains to show that u G C(I,H2(RN)) n C1(I,L2(RN)) and that u G W1'a(I,Lb(RN)) for every admissible pair (a, b). This follows from Lemmas 4.8.2 and 4.8.5. The only point which is not immediate is that g(u) G C(I, L2(RN)). To see this, we observe that u G C(I,L2(RN)). Moreover, by (4.8.19), u G L°°(J, H2(RN)). Applying the inequality (4.8.20), we deduce that u G C(J, HS(RN)), so that #(u) G C(7,L2(RJV)) by (4.8.1). Step 2. Uniqueness, the blowup alternative, and continuous dependence. Uniqueness follows from Proposition 4.2.9. For the blowup alternative, we proceed as in the proof of Theorem 3.3.9: using uniqueness, we define the maximal solution; and since the solution u of Step 1 is constructed on an interval depending on ||<£>||h2 (as is easily verified), we deduce the blowup alternative. Arguing as in Remark 4.4.5, we see that there is boundedness in L°°((—T,T),H2(RN)) and continuous dependence in L°°((—T,T),L2(RN)) for some T > 0 depending on iMi/f2-Applying (4.8.20), we deduce the continuous dependence in L°°((-T, T), HS(RN)) for every s < 2. Step 3. Property (iv). Since equation (4.1.1) makes sense in L2(RN) for all t G (—Tmin, Tmax), we may multiply it (in the L2 scalar product) by iu, and we obtain (ut,u)L2 = (-Au,iu)L2 + (g(u),iu)L2 ~ 0. Therefore, ^\\u(t)\\h = 2{iH,u)H-itHi =0, and the result follows. 124 4. THE LOCAL CAUCHY PROBLEM Step 4. Property (v). Since equation (4.1.1) makes sense in L2(RN) for all t G (~Tmin,Tmax), we may multiply it (in the L2 scalar product) by ut, and we obtain {iut,ut)L2 = (-Au,ut)L2 + (g(u),ut)L2. Since (iutlut)Ľ2 = 0, we deduce that -j-tE(u(t)) = 0 and the result follows. Note that the identity —G(u) = {g(u),ut)L2 holds in principle for u G C1((-71min,Tmax), H2(RN)). However, it is equivalent to G(u(t)) = G(u(0)) + / (g{u{s)),ut(s))L2 ds for all t G ( JO This last identity is easily established for u as in the statement by an obvious density argument. This completes the proof. □ We now give an application of Theorem 4.8.1 in a model case. Corollary 4.8.6. Let V e LS(RN) + L°°(RN) for some ô > 1, 5 > N/2 and W E L°(RN) + L°°(RN) for some a > 1, a > N/6. Let f G C(C,C) satisfy /(O) = 0 and |/(u)-/(v)| 0 and (N - A)a < 4 if N > 4. Finally, set g{u) = Vu + /(u(.)) + (W ★ \u\2)u. It follows that all the conclusions of Theorem 4.8.1 hold. If, in addition, f G C1 (C,C) (in the real sense), there is continuous dependence in a stronger sense as follows. The mappings (p i—► Tmin,Tmax are lower semicontinuous H2(RN) —+ (0,oo]. If (pn —*

0, and (TV — 4)a < 4. A similar result holds in the H2 "critical" case N > 5 and a = 4/(N — 4); see Cazenave and Weissler [70], theorem 1.4. 4.9. Hs Solutions, s < N/2 In this section we study the existence of solutions of the nonlinear Schrodin-ger equation (4.1.1) in the Sobolev space HS(RN) for s > 0. (We note that the cases s = 0, s = 1, and s — 2 have been studied in the preceding sections.) In principle, a local existence result can be established by a fixed point argument by using Strichartz's estimates in the Sobolev spaces Hs'r(WN) (see Remark 2.3.8) along with estimates of ||<7(w)||#*,r-. This is the program carried out by Kato [206] and it makes use of a delicate estimate of H#(u)ll#3'r (Lemma A3 in [206]). Here, we rather use the Besov space B^.2{RN) as an auxiliary space because estimates of ||#(w)||b*2 are much simpler to obtain (see Cazenave and Weissler [70]). We also note that, regardless of the auxiliary space, the case s > 1 tends to be more complicated as it requires more regularity of the nonlinearity. Thus we restrict ourselves to s G (0,1) and we comment on the case s > 1 at the end of the section. Also, consider the case s < N/2 (we comment on the limiting case s — N/2 at the end of the section). When s > N/2, the embedding HS(RN) ^ L°°(RN) allows a simpler treatment of the equation (provided the nonlinearity is sufficiently smooth, though); see Section 4.10 below. Note that if we consider a nonlinearity of the form g(u) = \\u\au, then the results of this section provide local existence in HS(UN) under the condition a < 4/(N — 2s) (and, also, a regularity assumption). Thus, in principle, any power a > 0 can be handled in the Hs framework with s < N/2. Furthermore, and for the sake of simplicity, we only consider local nonlinearities. The first result of this section is the following. theorem 4.9.1. Let 0 < 5 < min{l,N/2}. Let g G C(C,C), and assume that ^(0) = 0 and that there exists (4.9.1) 0 < a < ^ such that (4.9.2) \g(u) - g(v)\ < C(l + \u\a + \v\a)\u - v\ for all u,veC. Let (7, p) be the admissible pair defined by N(a + 2) 4(a + 2) (4-9-3) p= J, 7- V N + sa a{N-2s) Given if G HS(RN), there exist Tmax,Tmin G (0, 00] and a unique, maximal solution u G C((-Tmin,Tmax),#s(M*))^ of the problem (4.1.1). Moreover, the following properties hold: (i) u G Lqoc((-Tmin,Tmax), B*t2(RN)) for every admissible pair (q, r). (ii) (Blowup alternative) If Tmax < 00 (respectively, if Tmjn < 00^, then \\u(t)\\H* -> 00 as t T rmax (respectively, as 11 -Tmin). (iii) u depends continuously on

u in Lq((—T,T),Lr(RN)) as n —> oo. In particular, un -> u in C([-T,T], HS~£(RN)) for all e>0. Remark 4.9.2. We decompose g = g\ + g2 where ^i(O) = g2(0) = 0, g\ is globally Lipschitz C —► C, and (4.9.4) Mzi) - 92(z2)\ < C{\Zl\a + \z2\a)\zi - z2\ for all zi,zi e C (see Section 3.2). Let now I B 0 be a bounded interval and let u G L00{I1Hs(RN))nL^(I1Bar2(RN)). In particular, u G L°° (I, L2 {RN)) so that 9i(u) € L°°(I,L2{RN)). Next, we note that p > 2 so that Bapfi(RN) ^ HS^(RN). Since 2s < TV, we see sp < TV, and it follows that (4.9.5) BaPt2(RN)-->L*(RN) for all p < p < Np/(N - sp) = TV(a + 2)/(TV - 2s). Using (4.9.4) we easily deduce that g2(u) e L^(I,LP(RN)) for all max(1'^TTP^(Tv-2s)(Q + i)- In particular, we see that g2(u) G £7(T, i/(Rw)). Since Lp'(RN) ^ H-ff(RN) for cr = a(TV-2s)/2(a+2), we deduce that g2(u) G Lt(ItH-ff(RN)). Therefore, g(u) G L7(7, ^"^(R^)), so that equation (4.1.1) makes sense. Moreover, we see that equation (4.1.1) is equivalent to (4.1.2) and that (4.1.2) makes sense in H~ff(RN). Remark 4.9.3. In Theorem 4.9.1, uniqueness is stated in C(I,HS(RN)) n Lt(I,BaPt2(RN)). If, in addition to (4.9.1), we assume a < (TV + 2s)/(TV - 2s), then equation (4.1.1) makes sense for any u G L°°(I,HS(RN)), without assuming that u belongs to the auxiliary space LJ(I, Ba2(RN)). Assuming, in addition, a < (1 + 2s)/(l - 2s) if TV = 1 or a < (2 + 2s)/(N - 2s) if TV > 2, we know that there is uniqueness in L°°(I,HS(RN)) (see Section 4.2, especially Remark 4.2.12 for these properties). In this case, we deduce in particular from Theorem 4.9.1 that if u G L°°(I,H8(RN)) is a solution of equation (4.1.1), then Lq(I, Ba2(RN)) for every admissible pair (q, r). For the proof of Theorem 4.9.1, we will use the following nonlinear estimates in Besov spaces. Proposition 4.9.4. Let g € C(C,C). Assume that g(0) = 0 and that there exists a > 0 such that (4.9.6) \g{u) -g(v)\ < C{\u\a + \v\a)\u- v\ forallu,veC. Let 0llj3r-_, < and (4.9.8) \\9{u)\\b^ < C\\u\\l.||u(- - y) - «(.)||l- • Therefore, the inequality (4.9.7) follows from Remark 1.4.4(iii). Inequality (4.9.8) follows from (4.9.7), (4.9.10), and Remark 1.4.4(ii). □ Proposition 4.9.5. Let g e C(C, C). Assume that g(0) = 0 and thatg is globally Lipschitz continuous. Let 0 < s < I, 1 < r, q < oo. It follows that (4.9.11) HsMlljj. 2(RN) ^ £irai(Mw) by (4.9.5), we deduce that (4.9.14) \\92(u)\\B;, 2 < C\\u\\Bt\ for all u € Bsp2(RN). Next, it is clear that (4.9.15) \\9M-gi{v)\\L, 0 and consider two solutions u,v G L°°(I,HS(RN)) r\Li(I,BsPt2(RN)) of (4.1.2). We deduce from (4.9.15) that (4.9.17) \\9l(u) - 5i(v)lli,i(7,L») < C\\u -Next, it follows from (4.9.16) that \\92(u) - 02(«)||Ly < (4 9 18) c(IIuIIl7(i,b;i3) + \H%(i,b;i2)) 11« " vIUp(/,lo , where 1 _ 4 - a(N - 2s) 1 p 4 7 so that p < 7. Therefore, we deduce from Lemma 4.2.4 that u = v. Step 2. Proof of property (i). Let I = (0,T) with T > 0 and consider a solution u £ L00(/!jYs(RAr))nLT(/,^2(MAr)) of (4.1.2). We deduce from (4.9.13), (4.9.14), and Holder's inequality in time that (4.9.19) ll5i(«)||Li(/,Jir-) and property (i) follows. Step 3. Existence. We apply a fixed point argument in the set ■ g E={ue L°°(I,HS(RN)) n£7(7,Bp2{RN)); \\u\\l~(i,h*) < M and \\u\\Llii 0 are to be chosen later. (E, d) is a complete metric space, where the distance d is defined by d{u,v) = \\u- v||L«(/,La) + \\u-v\\L-,{ItLP). (See Step 1 of the proof of Theorem 4.4.1.) It follows from (4.9.21)-(4.9.22) that there exists a constant Co independent of T such that \\7{-)L2) < CT\\u - w||i,<»,(J,£a) and 4-g(N-2s) IM«) - 92{v)\\LYuotT)tlS) ^ CT 4 Mq||u - ^||l-v((0,T),Lp) for all u, v € jE. Applying Strichartz's estimates, it follows that there exists a constant C\ such that (4.9.26) d(H(u),H{v)) < d(T + T4-Q^'2s}Ma)d(u,v) forallu,ve£. Choosing now T possibly smaller so that (4.9.27) CifT + T4~a("~2s) Ma) < 1, we see that H is a strict contraction on E, and thus has a unique fixed point which is a solution of (4.1.2) on I. Step 4. The maximal solution and the blowup alternative. We proceed as in the proof of Theorem 3.3.9: using uniqueness, we define the maximal solution; and since the local solution is constructed by the fixed point argument on an interval depending on ||v?I!h* (by (4.9.24)-(4.9.27)), we deduce the blowup alternative. Step 5. Continuous dependence. This is an easy consequence of the estimates of Step 3 above. Indeed, let ipn —>

no sufficiently large, we see that m > 2Co||¥Jn||ha f°r n > no- It follows easily that if T = T(\\(p\\h*) satisfies (4.9.25)-(4.9.27), then the solutions un constructed by the argument of Step 3 all belong to the same set E (defined by (4.9.22) with T = T{\\ip\\H-)) for n > n0. Estimate (4.9.26) (together with Strichartz's estimates for the term T(-)ipn) implies that d(«n,u) < C\\ipn - vlU2 + (l/2)d(iin, w), i.e., d(un, u) < 2C\\ipn - v||l2-It follows that un -» u in L°°{I,L2{RN)) n L^(I, LP(RN)) and a further use of Strichartz's estimates shows the convergence in Lq(I, Lr(M.N)) for every admissible 130 4. THE LOCAL CAUCHY PROBLEM pair (q, r). Finally, the convergence in ^(I^H8 €{RN)) follows from the convergence in L°°(I,L2(RN)), the boundedness in L°°(I, Ha(RN)), and the elementary interpolation estimate ||u||jj*~c < \\u\\H% \\u\\l2. □ Remark 4.9.6. Here are some further comments on Theorem 4.9.1 and its proof. (i) The choice of the admissible pair (7,/?) given by (4.9.3) is (partially) arbitrary. It is not difficult to see that other choices are possible. The present choice leads to relatively simple calculations and is also a valid choice for the case s > 1 and for the critical case a = 4/(N - 2s) (see below). (ii) If \g(u) — g(v)\ < C(\u\a + \v\a)\u- v\, one can do the fixed point argument in the set E = {u e L7(7, BsPt2(RN)); ||u||L,(7iB. 2) < M}, with the distance d defined by d(u,v) = \\u - ^Hz,^/,^)- (iii) It is not difficult to show that one can replace the set E defined by (4.9.22) by the following set £'={«€ L°°(I, HS(RN)) n U(I,Bl2{RN)); Wu\\l~{i,h°) ^ m and \\u\\l^i,b^2) ^ m) ■ This requires two modifications in the proof. One needs the Sobolev inequality |[u|| N(a+2) < C\\u\\gS (see [28]). One also needs to show that (E',d) i, n —2s p,2 is complete. This amounts to showing a property of the type if un —> u in LP1 (I, LP(RN)) and |KIIL,(j> a) < M, then u Tmin(

0 and consider two solutions u,v € L°°(I,HS(RN)) DL^(I,B^2(RN)) of (4.1.2). Uniqueness being a local property, we need only show that u = v if T is sufficiently small (see Step 2 of the proof of Theorem 4.6.1). We observe that (4.9.18) becomes (4.9.28) ||02(u)-P2(v)||Lt'(j,lp') < C,(||u||^(/ijB,2) + |Jv||^(/,^2))I!«-v||l^{/,jlp) • Using (4.9.17), (4.9.28), and Strichartz's estimates, we deduce that there exists C independent of T such that IJW - v\\l°°(.I,L?) + \\u - al-f{I,lp) < CT\\u - + C(\\u\\ly{IiB,j + \\v\\lHl!Bs2))\\u - v||L,(JiL,). Since ||wJ1l7(/,b« 2) —> 0 as T J. 0 and similarly for u, we see that if T > 0 is small enough, then ||« - v\\Loo{ItL2) + \\u - u||LTr(/,LP) < ^(||« ~ v\\L~(I,l*) + ||« - v\\li(I,L»)) so that u — v. Step 2. Proof of property (i). It suffices to show that if I 3 0 is a bounded interval and u e Lco(I,Hs(RN)) n L^(ItBaPi2(RN)) is a solution of (4.1.1), then u e L*(I, B*2(RN)) DC(I, HS(RN)) for every admissible pair \q,r). This is proved as in Step 2 of the proof of Theorem 4.9.1. Step 3. Existence. We apply a fixed point argument in the set E defined by E = {u e L°°(J,HS(RN)) n£7(I,Bso2(RN)); (4 9 29) IMU»(/,#<) < Mi and ||tt||^(/,B-ia.) < M2}, where I = (0,T) and Mi,M2,T > 0 are to be chosen later. (25,d) is a complete metric space, where the distance d is defined by (See Step 1 of the proof of Theorem 4.4.1.) The proof of (4.9.21) yields l|W(u)lli;. 0 sufficiently small so that C0M?+1 0 sufficiently small so that C0rMi((o,rm„),b'i2) < 00, then we deduce from Step 2 that u e C([0, ^max]i Hs (RN)). By Step 3, we then can construct a solution v of (4.1.2), with

0. It follows that u defined on [0, Tmax+£] by u(t) = u(t) for 0 < t < Tmax and u(t) = v(t-Tm&x) for Tmax < t < jTmax + e is a solution of (4.1.2) on [0, Tmax + e). This contradicts the definition of Tmax. Thus ||u|U~((o,t„.«),ff») + IMU^o.tw),/^) = 00. Step 5. Continuous dependence. This is done as in the proof of Theorem 4.9.1, Step 5. □ Remark 4.9.8. The observations of Remark 4.9.6 apply as well to Theorem 4.9.7 and its proof. We now focus in particular on the case where \g(u) — g(v)\ < C(|u|a-f Ma)|u - v\. Then one can do the fixed point argument in the set E ~ {u € £7(/^^(R^)); ||«||lt(/jb. ) < M}, with the distance d defined by d(u,v) = \\u — v\\L-r(i,Lf>) (see Remark 4.9.6(h) and (in)). In this case, instead of (4.9.31)-(4.9.32), one obtains and ||W(u) - H(v)\\L,{I,LP) < CiM* d(u, v). Since H^O^IIl-vCR,^" 2) - ^IMIjy^ we obtain by letting I = R HW(«)llL-r(r,^ia) ^ C3\\ 0 small enough so that C0Ma+1 < M/2 and C\Ma < 1/2 and then assume that is sufficiently small so that C3||v?j[#* < M/2, we see that H is a strict contraction on E. In this case, we obtain (under 4.9. Hs SOLUTIONS, s < N/2 133 the assumption that H^H/j* is small) a global solution u of (4.1.1). Moreover, this solution belongs (by construction) to L°°(R, HS(RN)) n L7(R,5*i2(RN)). See [70] for details. We now comment on the case s > 1. The restriction s < 1 in Theorems 4.9.1 and 4.9.7 is motivated only by the nonlinear estimates of Propositions 4.9.4 and 4.9.5. The rest of the proof is not subject to the condition s < 1. It turns out that estimates of the type (4.9.8) for s > 1 are true but require more regularity assumptions on g. The corresponding existence results of Hs solutions therefore hold provided g is sufficiently smooth. See Cazenave and Weissler [70] and Kato [206]. See also Pecher [295], where the author uses time derivatives to obtain Hs solutions with minimal regularity assumptions on the nonlinearity. For completeness, we state below two typical results. Theorem 4.9.9. Assume N > 3 and 1 < s < N/2. Let g(u) = \\u\au with A e C and 4 0 < a < ——— . N - 2s If a is not an even integer, suppose further that (4.9.33) [s] oo as t T Tmax (respectively, as t { -Tmin). (hi) u depends continuously on

3. Let 1 < s < N/2 and g(u) = Aju|7^=27w with A G C. If a is not an even integer, suppose further that (4.9.33) holds. It follows that for every

4, then it can be replaced by the weaker condition s < a + 3. remark 4.9.12. In the limiting case s = N/2, the embedding H^(RN) <-»• U>(RN) for an 2 < p < oo makes it possible to obtain local existence for (sufficiently regular) nonlinearities with arbitrary polynomial growth. See Kato [206]. In particular, there is local existence in the model case g(u) = A|u|Qu for any a > 0 such that [s] < a. Using Trudinger's inequality, one can also consider nonlinearities of exponential growth. See Nakamura and Ozawa [256]. 4.10. H™ Solutions, m > N/2 In this section we study the local existence of "smooth" solutions in Hm(RN) for m > N/2. In principle, one can consider arbitrary real m (see Kato [206]), but we will only consider integers. The estimates are then simpler. The main point in considering m > N/2 is that Hm(RN) ^ L°°(RN). The consequence is that we need regularity of the nonlinearity but we do not need any control on its growth. We follow the method of Ginibre and Velo [135] and for simplicity, we only consider local nonlinearities. The main result is the following. Theorem 4.10.1. Let m > N/2 be an integer and let g G Cm(C,C) (in the real sense) satisfy g(0) = 0. For every ip G Hm(RN), there exist Tma,x,Tmm > 0 and a unique, maximal solution u G C((— Tmin,Tm^),Hm(RN)) of (4.1.1). Moreover, the following properties hold: (i) (Blowup alternative) If TmgbX < oo (respectively, Tmin < oo), then \\u(t)\\H™ —► oo as t t Xmax (respectively, as t I -Tmin). Moreover, limsup ||«(i)||£,oo = oo as 11 rmax (respectively, as 11 -Tmin). (ii) u depends continuously on

in the following sense. The functions Tmax and Tmin are lower semicontinuous Hm(RN) —► (0, oo]. Moreover, if ip in Hm(RN) and if un is the maximal solution of (4.1.1) with the initial value (pn, then un —* u in L°°((—S, T),Hm(RN)) for every p < oo and every interval [—S, T] C ( (iii) // (g(w),iw)i2 = 0 for all w G Hm(RN), then there is conservation of charge; i.e., ||u(t)||i,a = IMU2 for al1 * € (-Xmin,Tmax). (iv) If there exists G G Cl(Hm(RN),R) such that g = G', then there is conservation of energy; i.e., E(u(t)) = E((p) for all t G (-Tmin,Tmax), where E is defined by (3.3.9). The proof of Theorem 4.10.1 relies on the following technical lemma. Lemma 4.10.2. Let m > N/2 be an integer and let g G Cm(C,C) satisfy g(0) = 0. It follows that the mapping u i-> g(u) is continuous and bounded Hm(RN) —> 4.10. Hm SOLUTIONS, m > N/2 135 Hm(WN). More precisely, given any M > 0, there exists C(M) such that (4.10.1) \\g{u)\\H™ 0 and let (4.10.4) K{M)= sup \g'(u)\ + --- + \gW{u)\ 1. Let p3- = 2m/1/?,-[, so that It follows from Holder's inequality that On the other hand, it follows from Gagliardo-Nirenberg's inequality that 'u < C\\u\\H™\\u\\kL-J WDbuWw 0 c C£°(RN) satisfy un~+um Hm(RN). Since Hm{RN) <-> L°°(RN), we see that < 2M and tl«n||i7m < 21|itHi-f^ for n large. In particular, we deduce from (4.10.1), which we already established for un, that (4.10.8) \\g(un)\\H™ < 2C{2M)\\u\\Hm for n large. In particular, (g{un))n>o is bounded in Hm. Since g(un) -+ p(u) in L2(RN) by (4.10.5), it follows that g(un) g(u) in Hm(RN). Applying (4.10.8), we deduce (4.10.1) (with C(M) replaced by 2C(2M)). Inequality (4.10.2) is an immediate consequence of (4.10.5). We finally prove (4.10.3). First note that, given u € Hm(RN) and (w„)n>0 C C£°(RN) as above, we may assume (after possibly extracting a subsequence) that Daun —* Dau a.e. for all |oi| < m. Thus we see that formula (4.10.7) holds a.e. for every u e Hm(RN). Let now M ~> 0 and u,v (E Hm(R^) with ||u||/fm, JW". Given a multi-index a with jetj = m, we deduce that Da[g(u) — g(v)] is a sum of terms of the form k k gik){u) Y[ u - g{lc\v) Y[D^v, 3 = 1 3 = 1 where k and the /?/s are as in (4.10.7). Each of the above terms can itself be decomposed as a sum of terms where the first one is fc (4.10.9) [g(k\n) -g{k)(v)} [Jd&w. The other ones have the form fc gW^HD^Wj, 3=1 where all the w/s are equal to u or v, except one which is equal to u — v. Let now pj = 2m/\(3j\. We see that fc fc g^(v)J{D^wj < || L°°(RN) and Hm(RN) «-► LK(RN). If k < m- 1, then gW is Lipschitz on bounded sets, so that the terms in (4.10.9) are estimated as above by C(M)\\u — u||j7">. It remains to estimate the terms (4.10.9) when k = m. This last term is estimated as above by (4.10.10) \\9{mHu)-g(m\v)\\L~\\u\\%m. Since g(m') is continuous, hence uniformly continuous on bounded sets, and Nlff~, NIh™ < M, we see that \\g^{u) ~ <7(u)IU~ < Mil* - v||L-), with $m($) —* 0 as s | 0. Since \\u — < C\\u — \\u — v\\L2m , we may replace o~m(\\u — v\\l°°) by Sm(\\u - ), so that (4.10.3) now follows from (4.10.10). □ proof of Theorem 4.10.1. We first note that by Lemmas 4.2.8 and 4.10.2, problems (4.1.1) and (4.1.2) are equivalent. We then proceed in four steps. 4.10. Hm solutions, m > N/2 137 Step 1. Existence. We construct solutions by a fixed-point argument. Given M, T > 0 to be chosen later, we set / = (—T, T) and we consider E = {u£ L°°(I,Hm{RN)) : \\u\\Loo{ItHm) < M) . It follows that (E, d) is a complete metric space, where the distance d is defined by d{u,v) = \\u-v\\L{I)Hm) < \\ip\\H™+TC(M)M. Furthermore, it follows from (4.10.2) that, if u, v G E, then \\H(u)(t) - H(v){t)\\L2 < TC{M)\\u - v||Loc(JiL2). Therefore, we see that if (4.10.11) M = 2\\o C Hm(RN) such that

0 138 4. the local cauchy problem depending on jjy?||i*m such that un is defined on [—T,T] for n large enough and un -> u in C([-T,T],Hm{RN)) as n -> oo. The result follows by iterating this property in order to cover any compact subset of (—Tmin,Tmax). We now prove the claim. Since |S no and (4.10.12) \\u\\L^((-T,T),Hm) + SUP \\un\\L™({-T,T),H™) < 4|M|#- . n>no Note that un(t) - u(t) = 7(t){ 2. The conservation laws (properties (iii) and (iv)) then follow by multiplying the equation by u and ut, respectively. See Steps 3 and 4 of the proof of Theorem 4.8.1. □ Remark 4.10.3. Let g(u) = X\u\au with a > 0 and A € C. If a is an even integer, then g e C°°(C,C). Therefore, we may apply Theorem 4.10.1 for any m > N/2. If a is an odd integer, then g <= Cm(C, C) only for m < [a], so that we may apply Theorem 4.10.1 only in the case [a] > N/2 and for N/2 < m < [a]. If a is not an integer, then g g Cm(C, C) only for m < [a] +1, so that we may apply Theorem 4.10.1 only in the case [a] + 1 > N/2 and for N/2 < m < [a] + 1. 4.11. Cauchy Problem for a Nonautonomous Schrodinger Equation In this section we study the Cauchy problem for equation (7.5.5) below, starting from any point t g [0,1]. In fact, we consider the more general Cauchy problem (see [72]) f ivt + Av + h(t)\v\av = 0 (4.11.1) < V ; I t/(0) = V, where h g 1^(1$, R). We study the equation (4.11.1) in the equivalent form (4.11.2) v(t) = 7(t)ip + i [ 7{t-s)h(s)\v(s)\av{s)ds. Jo We have the following existence and uniqueness result. 4.11. nonautonomous schrodinger equation 139 Theorem 4.11.1. Assume 0 < a < 4/(N - 2) (0 < a < oo if N = 1). Let 9 = 4/[4 - a(N - 2)] (9 = 1 if N = 1, 9 > 1 and (2 - a)B < 1 i/ AT = 2), one? consider a real-valued function h g Lfoc(R, R). It follows that for every tp g H1(RN), there exist Tmax,Tmin > 0 and a unique, maximal solution v g C(KminJmax)^H^))nW/1lf((-Tmin)Tmax)^--1(lJV)) of equation (4.11.2). The solution v is maximal in the sense that if Tmax < oo (respectively, Tmm < oo), then ||u(£)||#i —> oo as £ t ^max (respectively, t J. -Tmin). in addition, the solution v has the following properties: (i) //Tmax < oo, then ^mit^tlTam{Ht)\\%4h\\Le{ttTnia)} > 0. (ii) 7/Tmin < oo, then liminfti_Tmln{||v(i)||£ 1 \\h\\Le{.Tminit)} > 0. (iii) u e Lqoc((-Tmin, Tmax), W1'r(RN)) for every admissible pair (q, r). (iv) There exists 5 > 0, depending only on TV, a, and 9 such that if wis J Jh(s)\9ds<6, then [-t,t] C (-Tmin,Tmax) and [jt>||L,((_TiT)jVKi,r) < iiT||0||jtfi for every admissible pair (q, r), where K depends only on N, a, 9, and q. In addition, if ip' is another initial value satisfying the above condition and if v' is the corresponding solution of (4.11.2), then \\v—v'\\Loo^_r^T)jL2^ < K\\ip—ip'\\L2. (v) If\-\ipe L2(RN), then \-\ve C((-Tmin,Tmax),L2(RN)). Proof. We apply the method of Section 4.4. We suppose first that TV > 3, then we indicate the modifications needed to handle the cases TV = 2 and TV = 1. Let 2* = 27V/(TV - 2) and define r by 2 a (4.H.3) X-? = F- Since (TV - 2)ct < 4, we have 2 < r < 2*. Therefore, there exists q such that (q, r) is an admissible pair. A simple calculation shows that 1 1 1 4.11.4 - = - + -.. q 8 q By Strichartz's estimates, there exists K such that ||^(-)^IIl*(h,hi) + ||3'('Mli,«(R,w1.'-) < K\\lp\\Hi for every V e H1(RN). Given M > 0 and 0 0, let E = {v E Loa((-TllT2),H1(RN))nLq((-T1,T2),Wl'r(RN)), IMl^U-TuTi),!!1) + Wv\\ll, which proves property (i). Property (ii) is proved by the same argument. Finally, since v satisfies equation (4.11.1) in Lfoc((—Tmin,Tmax),H~1(RN)) and h is real 4.11. NONAUTONOMOUS schródinger EQUATION 141 valued, property (v) is proved by standard arguments. For example, multiply the above equation by (xpe-^1' v, take the imaginary part and integrate over RN, then let e I 0 (see Lemma 6.5.2 below). If N = 2, the proof is the same as in the case N > 3, except that we set r = 29 and use the embedding H!(R2) <-+ L*>(R2) with p = a0/(9 - 1). If N = 1, the argument is slightly simpler. We let E = {v e ^((-T^T^i^R)) : ||v||Lo0((_Tlira)iHi) < 2M} equipped with the metric d(u,v) — \\v - u\\l^((-Ti,t2),l2), and use the embedding H^R) <-> L°°(R). ' ' □ We now study the continuous dependence of the solutions on the initial value. The result is the following. Theorem 4.11.2. Under the assumptions of Theorem 4.11.1, suppose there exists 01 > 9 such that h € L^.(R). The solution v of (4.11.2) given by Theorem 4.11.1 depends continuously on ip in the following way. (i) The mappings ip t—> Tmax and ip i—► Tmin are lower semicontinuous H^R") -> (0,ooj. (ii) If ipn —► V7 in íT1(RJV) and i/ t>n denotes the solution of (4.11.2) with initial value ipn, then vn —► v in C([—T\, T2],ii1(]RiV)) /or any interval [-Ti,T2] C (~rmin,Tmax). //, in addition, \ ■ |-0n j - \rp in L2(RN), then | ■ |v„-» | ■ |w in C([Ti,T2],L2(R^)). Proof. We apply the method of Section 4.4. We proceed in two steps. Step 1. We show that for every M > 0, there exists r > 0 such that if ip e HX{RN) satisfies \\ip\\m < M, then [-r,r] C (-Tmin,Tmax), and v has the following continuity properties. (a) If IJV'IIh1 < JW? V'n —* "0 in íř1(RJV), and if vn denotes the solution n—»00 of (4.11.2) with initial value i/jn, then vn -* v in C([-t,t], Hl(RN)). (b) If, in addition, | • \ipn —► \ ■ \ip in L2(RN) as n —> 00, then | • \vn —► | • |t> in C([-t,t},L2(Rn)) as n -* 00. We only prove the result in the case N > 3 (see the proof of Theorem 4.11.1 for the necessary modifications in the cases N = 1,2). Given M > 0, we choose r so that the inequality in property (iv) of Theorem 4.11.1 is met whenever llV'llff1 < M. In particular, if < M, then [—r, r] C (~Tmm, Tmax). Next, observe that 1/9 > (4 - q7V)/4, and so we may assume without loss of generality that l/0i > (4 — aN)/4. Therefore, if we define a by a N \ dj'-then 2 < oto < 2N/(N - 2). Let now p be defined by 142 4. THE LOCAL CAUCHY PROBLEM Since a > N/2, we see that 2 < p < 2N/(N - 2). Finally, let 7 be such that (7, p) is an admissible pair. It follows easily from Holder's inequality that for every —00 < a < b < 00, (4.11.9) \\hwz\\L^({atb)M) < [J \h(s)\e> \\w(s)fL^ ' \\z\\L,{{a^LP). Consider now tp such that H^Htf1 < M, and let be as in (a). Let v,vn be the corresponding solutions of (4.11.2). We deduce from Strichartz's estimates that there exists C, depending only on 7 such that (4.11.10) CM ~ tpnWm + \\h(\v\av - \Vn\aVn)\\Li- {{-t,t),w^p') ■ On the other hand, a straightforward calculation shows that (4.11.11) \V{\v\av - \vn\°vn)\< C\vn\a\Vv - Wvn\ + cj>(v,vn)\Vv\, where C depends on a, and the function (x,y) —► 0. Therefore, applying (4.11.9), (4.11.10), and (4.11.11), we get \\V - Vn\\L''((-r,T),W1'p) + \\v _ ^^^((-T.r),^1) < C\\lp - 1>n}\Hi (4.11.12) + \\h\\Lei(-T,T)\\vn\\L~((-T,T),L°°)\\v ~ V„||iT,((_TlT))Wi,p) / rb \ l/Oi _. / 1 ..... a ,. . . ..a. \ C{J \h(s)\^U(v,vn)\\{\J ||«||Ln(-r,T)>H,i,P) Note that by property (iv) of Theorem 4.11.1, vn is bounded in i71(RAr), hence in LatT(RN), with the bound for t G [—r,r], depending only on HVViIIhi, hence (for large values of n) only on M. Also, the bound on ||v||lt((-t,t),iv1'P) depends only on M. Therefore, it follows from (4.11.12) that \\v - Vn\\L-t({-T,T).Wi-1>n\\h*+c[J \h(S)\^U(v,Vn)\\eL\ Therefore, property (a) follows, provided we show that (/Voor ii^oii^^o. By the dominated convergence theorem, it suffices to verify that \\{v,vn)\\L0 for all t G [— t,rj. n—*oo 4.12. COMMENTS 143 To see this, we argue by contradiction. We assume that there exist t and a subsequence, which we still denote by vn(t) such that \\(v(t),vn(t))\\i > 0. Note that vn{t) -» v(t) in L2(RN) and vn(t) is bounded in ^(R^) by property (iv) of Theorem 4.11.1. Therefore, by Sobolev's and Holder's inequalities, vn(t) —> v(t) in LacT(RN). It follows that there exist a subsequence, which we still denote by vn(t), and a function / e Laa(RN) such that vn(t) —> v(r-) a.e. in RN and |^n(*)] < / a.e. in R^. Applying the dominated convergence theorem, we deduce that \\ 0, which is a contradiction. Hence property (a) is proven. Property (b) follows from property (a). (See Corollary 6.5.3 below.) Step 2. Let ip E if1(RJV), let v be the maximal solution of (4.11.2) given by Theorem 4.11.1, and let [-TUT2] C (-Tmin,Tmax). Set M = 2 sup \\v(t)\\m, -Ti 0 given by Step 1. By applying Step 1 m times, where (m - l)r < T1+T2 < mr, we see that if W^-^Wh1 is small enough, then the solution of (4.11.2) with initial value tp exists on [—T\,T2\. Property (i) follows. Property (ii) follows easily from the same argument. □ 4.12. Comments The results of this chapter are mostly based on the Strichartz estimates. Thus we may expect that the results of the previous sections have a counterpart for the abstract equation iut + An + g(u) = 0 u(0) — x, whenever T(t) = ettA satisfies Strichartz-type estimates. Theorem 2.7.1 gives a sufficient condition for such estimates, which we recall below. Let ft be a domain of R^ and let X = L2(Q). Let A be a C-linear, self-adjoint < 0 operator on X with domain D(A). Let XA be the completion of D(A) for the norm ij^ll^ = \\x\\2x - (Ax,x)x, X*A = (XA)*, and A be the extension of A to (D(A))*. Finally, let 7(t) be the group of isometries generated on (D(A))*, X*A, X, XA, or D(A) by the skew-adjoint operator iA. If, in addition, (4.12.2) \\7(t) 0, there exist s(A) > 0 and K(A) < oo such that G(u) Y <—> X such that for every e > 0 there exists a constant CE with (4.12.3) ||u||y < elluH^a) + for all u G D(A). Taking Y = D(Ai) is a possible choice. Depending on the applications, good choices may also be an LP space or even an Hs space. Following the proof of Theorem 4.8.1, it is not difficult to establish the following result. Theorem 4.12.3. Let A be as in the statement of Theorem 3.7.1, and assume that T(t) — eltA satisfies estimate (4.12.2). Assume there exists a Banach space D(A) <—> Y <-»• X such that for every £ > 0 there exists a constant C£ for which (4.12.3) holds. Let g = g\ + ■ ■ ■ + gk with gj : D(A) —> X, and assume there exist exponents 2 < rj,pj < 2N/(N - 2) (2 < rjtpj < oo */ N = 1) such that gj G C(Y,X) is bounded on bounded sets and \\9j(u) - fc-OOU^j < L(Af)||u - t;||Lr, for all u, v G D(A) such that \\u\\Y, \\v\\y < M. For every x G D(A), there exist Jmax?2min > 0 and a unique, maximal solution u G C((—Tmin,Tmax), D(A)) n Cl((—Tm-m,Tma,x),X) of (4.12.1). Moreover, there is the blowup alternative; i.e., if Tmax < oo (respectively, Tmin < oo), then \\u(t)\\o(A) ~* °o ast | Tmax (respectively, ast i -Tmin). 4.12. COMMENTS 145 Remark 4.12.4. Note that one can also show, as in Theorem 4.8.1, a form of continuous dependence as well as the conservation laws (whenever the relevant conditions on g are satisfied). remark 4.12.5. Note that the results of Section 4.6, i.e., the existence of L2 solutions, have an obvious counterpart in the setting of Theorems 4.12.1 and 4.12.3. This is not the case for those of Sections 4.4 (Kato's method), 4.9 (Hs solutions, s < N/2), and 4.10 (Hm solutions, m > N/2). Indeed, those results are obtained by differentiating the equation, in one way or another, with respect to space. It is not clear, in general, what a good analogue of the space differentiation would be. Remark 4.12.6. We established in the previous sections local existence results for (4.1.1) in spaces of the type ^(1^) with s > 0. One may wonder if there is any local well-posedness result in Hs spaces with s < 0, as is the case for KdV, for example. This is a delicate question. For nonlinearities of the type g(u) — \\u\au, the answer is no; see Birnir et al. [32], Christ, Colliander, and Tao [80], and Kenig, Ponce, and Vega [216]. On the other hand, the answer is yes if, for example, g(u) = Aw2. See Kenig, Ponce, and Vega [214]. More generally, one can investigate the minimal value of s for which the initial value problem (4.1.1) is (locally or globally) well-posed in Hs. This question has been (and is still being) studied by many authors. See, for example, Bourgain [38], Kenig, Ponce, and Vega [214], Staffilani [318], and Tao [335]. Note also that the nonlinear Schrodin-ger equation (4.1.1) can be solved in certain spaces that are not based on L2, like Lorenz spaces Lp,°° (see Cazenave, Vega, and Vilela [67]) or Besov spaces -62,00 (see Planchon [298, 299]). CHAPTER 5 Regularity and the Smoothing Effect In this chapter we consider the nonlinear Schrddinger equation (4.1.1) in RN, We address the problems of regularity of solutions and the C°° smoothing effect. The problem of regularity of solutions can be formulated as follows. Suppose

0 and suppose the nonlinearity g is such that there is a local existence theory of Hs solutions (see Chapter 4). It follows that there is a maximal solution u e C((—Tm[n,Tm&x),Hs(RN)) of (4.1.1). Suppose now that tp is smoother than just HS(RN), say

s. The question is then: does u belong to C((~Tmin,Tm&x),HSl(RN))? In fact, one can simplify the problem by assuming that there is also a local existence theory of H31 solutions. It follows that there is a maximal solution m 6 C((-T^in, r^ax), HSi (RN)) of (4.1.1). Of course, an HSl solution is in particular an Hs solution. By uniqueness of H9 solutions, we deduce that u = U\ on the larger interval where the two solutions are defined. We then see that (~Tmin,Tm&x) c (-Tm[n,Tma,x) because, again, an HSl solution is an Hs solution. The question then becomes: does Tmin = Tmin and Tmax = 2^ax? In other words, can u blow up in HSl(RN) before it (possibly) blows up in HS(RN)? At this level of generality, there is no complete answer to this question. (It seems that there is no counterexample either.) We give, however, partial answers to this question in Sections 5.1-5.5. The problem of the C°° smoothing effect is the following: Let and g is the solution u in C°°((I \ {0}) x R^)? In other words, when are the properties of Section 2.5 preserved for the nonlinear equation? We study this question in Section 5.6. Finally, we observe that the results of this chapter are stated for one equation, but similar results obviously hold for systems of the same form. See Remark 3.3.12 for an appropriate setting. 5.1. H8 Regularity, 0 < s < min{l,iV/2} In this section we consider local nonlinearities, so that we may apply the Hs theory of Section 4.9. In this case, there is regularity at the HSl level for any s < Si < min{l, N/2}, as the following result shows. Theorem 5.1.1. Let 0 < s < min{l,JV/2}. Let g e C(C,C) satisfy g(0) = 0 and \g(u)~g{v)\ < C(l + \u\a + \v\a)\u-v\ forallu,v e C with 148 5. REGULARITY AND THE SMOOTHING EFFECT Let (7, p) be the admissible pair defined by (4.9.3). Let ip G HS(RN) and let u G C((-rmin,rmax),i/s(MN))nL/oc((-Tmin,Tmax),^2(RiV)) be the maximal Hs solution of (4.1.1) given by Theorem 4.9.1 (case s > 0) or Theorem 4.6.1 (case s = 0). // tp G HSl (RN) for some s < si < min{l, N/2}, then for every admissible pair (g,r), u G C((-Tmin,Tmax),i^(RN)) n Lfoc((-Tmin!Tmax),B^2(R^)). Proof. We consider t > 0, the argument for t < 0 being the same. We know that u is an HSl solution (in the sense of Theorem 4.9.1) on some maximal interval [0, T) with T < Tmax, and we need to show that T = TmSiX (see the discussion at the beginning of this chapter). We argue by contradiction, and we suppose T < Tmax. In particular, T < 00 so that (5.1.1) \Wt)\\H'i ^00. Moreover, since T < Tmax, (5.1.2) ||w|!l^((o,t),b=2) + sup \\u(t)\\Hs < 00. p' 0 L N~2* (R™). It then follows from (5.1.4), Holder's inequality and (5.1.2) that for any interval /C(0,T), (5.1.5) ||«|lL-r{/,B^a) ^ c\\u\\li(i,b;j)\\u\\l^i,b%2) ^ c\\u\\lv(i,b^2) » where 1 < p < 7 is given by l_l 4 - a(N - 2s) p 7 4 We now apply Strichartz's inequalities in Besov spaces and we deduce from equation (4.1.2) and from (5.1.3) and (5.1.5) that (see the proof of Theorem 4.9.1) (5.1.6) ||u||i/«=(j,/f.i) + ||w||L,(/>B^2) {IiH.t) + £*~a{"~*S)C\\u\\L,{I}B;]2), where the various constants are independent of r < T. We therefore may fix e small enough so that the last two terms in the right-hand side are absorbed by the left-hand side. It follows that \u \\l°°(I,H>i) + \W\\l^(i,b'\) < C) where C is independent of r < T. Letting r | T, we obtain a contradiction with (5.1.1). □ 5.2. H1 Regularity In this section we establish H1 regularity. This can be done for local non-linearities, by starting from the Hs solutions of Section 4.9. For more general nonlinearities, this can be done starting from the L2 solutions of Section 4.6. We begin with the first case. Theorem 5.2.1. Let 0 < s < min{l, JV/2}. Let g & C(C,C) satisfy g(0) = 0 and \g{u)-g{v)\ 0) or Theorem 4.6.1 (case 8=0)). IfifGH1^), thenueCa-T^T^H1^)). Proof. The proof is very similar to the proof of Theorem 5.2.1, except that we use the Sobolev spaces H1^) and W/1'r(RJV) instead of H3l(RN) and Bf.\l(RN), and the inequalities llffi(u)||j_i < C||u|r ||u||Wi,p < Cljullf,. ||ti||wi,p instead of (5.1.3) and (5.1.4). □ We now consider more general nonlinearities and study the Hl regularity of the L2 solutions of Section 4.6. Since there are two slightly different results for the local existence in Hl (Sections 4.3 and 4.4), there are two possible regularity results, depending on what set of assumptions on g we choose. For simplicity, we only establish one such result. We recall the assumptions of Theorem 4.6.4. Let 27V (5.2.1) 2-r 0 such that for every M > 0, there exist K(M) < oo such that (5.2.2) \\g(v) - g(u)\\Lr, < K(M)(\\u\\%r+ \\v\\lr)\\v - u\\^ for all u,v € L2(E^) n 7/^) such that ||u||La, ||v||L2 < M. Set so that (g,r) is an admissible pair and assume (5.2.3) a + 2 0, the argument for t < 0 being the same. We know that u is an Hl solution (in the sense of Theorem 4.4.6) on some maximal interval [0, T) with T < Tmax, and we need to show that T = Tmax (see the discussion at the beginning of this chapter). We argue by contradiction, and we suppose T < Tma,x. In particular, T < oo so that (5.2.5) ||u(t)||Hi—roo. 11 i Moreover, since T < Tmax, (5.2.6) sup |K*)||x,2 + sup ||u||L*J((o,r),i,rJ) < oo-o 0. It then follows from (5.2.8) and the above estimate that (5.2.9) ll^(w)ll^(w1^) ^c^ + £l~^~cWu\ym,w^)^ where the constants are independent of r < T. We now apply Strichartz's inequalities and we deduce from equation (4.1.2) and from (5.2.9) that \\u\\L~(i,Hi) + 2JllwH^(/1w1'^) 1, S > N/2, and W e Lff(RN) + L°°(RN) for some a > 1, a > JV/2. We have /(0) = 0 and \f(zi) - f{z2)\ < C(l + N + N^ki - z2\ for some 0 < j3 < ~ . The fact that the assumptions are satisfied is easily verified; see Corollary 4.6.5 and Remark 4.4.8 for the details. 152 5. regularity and the smoothing effect 5.3. H2 Regularity In this section we study the H2 regularity of solutions. Since we already established the H1 regularity in the preceding section, one possibility is to start from an H1 solution. However, we then need the assumptions of either Theorem 4.3.1 or Theorem 4.4.6. This imposes either the Hamiltonian structure or W1,p regularity of g. On the other hand, if we start from an L2 solution, then we do not need such assumptions on g (see Theorem 4.6.4). Since the assumptions on g for local existence in H2 require neither the Hamiltonian structure nor the W1,p regularity (see Theorem 4.8.1), it may be more economical for L2 solutions to jump directly to the H2 regularity. For this reason, we present two regularity results, one for Hl solutions and the other for L2 solutions. Theorem 5.3.1. Let g = g\ + • • • + gt satisfy the assumptions of either Theorem 4.3.1 or Theorem 4.4.6. Assume further that there exists 0 < s < 2 such that, for all 1 < j < k, (5.3.1) \\9j(u)\\L2 < C(M)(1 + \\u\\H.) for all u £ HS(RN) such that \\u\\Hi < M. Let

2 being treated as in the proof of Theorem 5.2.1 above. We first observe that by (5.3.1), g(tp) £ L2(RN). Next, we recall that (by (3.3.7) or (4.4.21)) there exist 2 1, 6 > N/2. W is an even, real-valued potential, W e L°(RN) + L°°(RN) for some a > 1, a > N/4. The function / : x R -> R is measurable in x e R^ and continuous in u 6 R and satisfies (3.2.7), (3.2.8), and (3.2.17). We extend / to RN x C by (3.2.10). It follows that g satisfies the assumptions of Theorem 4.3.1 (see Example 3.2.11), and similar estimates show that (5.3.1) holds (for $ sufficiently close to 2). In particular, we may let f(x,u) = X\u\au with A G R and 0 < a < 4/(/V - 2) (0 < a < 00 if TV = 1,2). Consider next g(u) = Vu + />(■)) + (W*\u\2)u, 154 5. REGULARITY AND THE SMOOTHING EFFECT where V, VV G L5(RN) + Lco(RN) for some S > 1, 6 > N/2, W e Lff(RN) + L°°(RN) for some a > 1, a > N/4, and / is as in Theorem 4.4.1 (for example, f(z) = X\z\az with A £ C and (N - 2)a < 4). It follows that g satisfies the assumptions of Theorem 4.4.6 (see Remark 4.4.8) and that (5.3.1) holds (for s sufficiently close to 2). We now study the H2 regularity of L2 solutions. theorem 5.3.4. Let g = g\ + • • • + gk be as in Theorem 4.6.4; i.e., each of the gj's satisfies (5.2.1)-(5.2.3) for some rj,aj,qj. Set r — max{ri,... ,rk} and q — min{gi,... ,qk}. Assume, in addition, that there exists 0 < s < 2 such that (5.3.9) \\g(u)\\L2 < K(M)(1 + \\u\\Hs) for all u G H2(RN) such that ||ujjL2 < M. Let

1, <5 > N/2, and W G La(RN) + L°°{RN) for some a > 1, a > N/2. We have f(x, 0) = 0 for all x£RN and \f{x,zx) - f{x,z2)\ < C(l + \zi\ + \z2\f\zi - z2\ for some 0 < 0 < ~ . The fact that the assumptions are satisfied is easily verified, see Corollaries 4.6.5 and 4.8.6 for the details. 5.4. Hm Regularity, m > iV/2 In this section we consider a nonlinearity g that satisfies the assumptions of Theorem 4.10.1 for some m > N/2 and we study the Hmi regularity of the solutions for mi > m. This is a particularly simple case as the following result shows. Theorem 5.4.1. Let m > N/2 be an integer and let g G Cm(C,C) (in the real sense) with g(0) = 0. Let

m and if g G CTOl(C,C), thenue Cm^((-Tmm,TmgiX), Hm^ (RN)). Proof. We consider t > 0, the argument for t < 0 being the same. We know that u is an Hmi solution on some maximal interval [0, T) with T < Tmax, and we need to show that T = Tm&x (see the discussion at the beginning of this chapter). Consider r < Tmax. It follows that sup \\u(t)\\H™i < oo 0<* r. Thus T = Tmax. □ 5.5. Arbitrary Regularity So far, we have established regularity up to the level H2(RN) for Hs solutions, 0 < s < 1, and regularity of arbitrary level for Hm solutions with m > TV/2. In higher dimensions, there is of course a gap between H2 and Hm with m > N/2. It seems that there is no general result concerning regularity at higher order. See Ginibre and Velo [135] and Kato [206, Corollary 4.3] for some partial results in that direction. Here is a result concerning a very particular nonlinearity. Let g(u) = A|w|aw with A € C and a an even integer. In particular, g G C°°(C,C). By Theorems 4.6.1 (case s = 0), 4.9.1 (case 0 < 5 < min{l,7V/2}), 4.4.1 (case s = 1) or 4.9.9 (case 1 < s < N/2), there is local existence in HS(RN) for (4.1.1) when 0 < s < N/2 and s > N/2 — 2/a. Moreover, for every admissible pair (q,r), u G Lqoc((—Tmin,Tmax),Hs>r(RN)) (see the above-mentioned theorems and Remark 4.4.3 for the case s = 1). We have the following regularity result. Theorem 5.5.1. Let g(u) = X\u\au with A G C and a an even integer. Let 0 < s < N/2 satisfy N 2 (5.5.1) s > — - -. v ' 2 a Finally, let (f G HS(RN) and let u be the corresponding maximal Hs solution of (4.1.1) , u G C((-rmin,Tm_x),ffa^ for every admissible pair (g,r) (see above). If tp G Hm(M.N) for some m > N/2, then u G C^-T/min, T^ax), Hm(RN)). proof. Suppose first s < 1. If s < 1 and m = 1, then regularity follows from Theorem 5.2.1 and if m = 2, regularity follows from Theorem 5.3.1. If m > 3, then in particular u is an H2 solution by Theorem 5.3.1. If < 2, then regularity follows from Theorem 5.4.1, and if iV > 3 we are reduced to the case s > 1 and N > 3, which we study below. We define r0 > 2 by (5.5.2) We observe that a > 2 so that (5-5-3) 2 < r0 < ^ _ 2 1 _ 1 2 r0~ 2 Na 2N (with equality if a = 2). Moreover, it follows from (5.5.1) and (5.5.2) that (5.5.4) rQs>N. We deduce from (5.5.3)-(5.5.4) that there exists r such that 2N 156 5. REGULARITY AND THE SMOOTHING EFFECT In particular, there exists q such that (q, r) is an admissible pair. We also note that by (5.5.2), 1 1 2 r 2 ~ Na so that q > a. It follows that u G L?oc((-Tmin,Tm&x),Hs>r{RN)). Since H"'r(RN) «-> L°°(RN) because rs > N, we obtain (5.5.5) u G I£c((-rmiD,rinax),L«,(RJV)). We now observe that, due to the particular structure of g(u), the proof of Lemma 4.10.2 yields the estimate (5.5.6) \\9(v)\\h<» < C\\v\\tx\\v\\Hm for all v G Hm(RN). We consider t > 0, the argument for t < 0 being the same. We know that u is an Hm solution on some maximal interval [0, T) with T < Xmax, and we need to show that T = Tmax (see the discussion at the beginning of this chapter). We use equation (4.1.1), the property that 7(t) is an isometry in Hm(RN), and estimate (5.5.6) to obtain \\u(t)\\Hm < \\ ||w(<)||/fm < oo, a contradiction with the blowup alternative in Hm(RN). □ 5.6. The C°° Smoothing Effect In this section we present a result of Hayashi, Nakamitsu, and Tsutsumi [177, 178, 179] describing a C°° smoothing effect similar to the one observed for the linear equation (see Section 2.5). More precisely, under suitable assumptions on the nonlinearity, if the initial value ip decays fast enough as \x\ —> oo, then the corresponding solution of (4.1.1) is smooth in both t and x for t ^ 0, even if

0, let ip e i?1(R), and let u be a strong H1 -solution of {iut + uxx + \u\2u = 0 M(o)=; °"'°-r]- If (p has compact support, then u £ C°°((0,T) x R). 5.6. THE C°° SMOOTHING EFFECT 157 PROOF. Let us first do a formal calculation, in order to make clear the idea, which is quite simple. We use the operators Pa defined in Section 2.5. More precisely for any positive integer £, let (5.6.2) ue(t, x) = (x + 2itdx)eu(t, x). We deduce from formula (2.5.4) that (5.6.3) u%x) = (2it)eei^d£x(e-i^u(t,x)) . It follows from (5.6.2), (2.5.5), (5.6.1), and (5.6.3) that (5.6.4) iuet+uix + (2it)eet^dex(\e~i^u\2e-i^u) =0. Note that \v\2v — vvv, and so dex\v\2v= Therefore, setting -■si (5.6.5) v(t,x) = e %**u(t,x), we deduce from (5.6.4) that n+j+k=£ Since u*(0) = xeip, we see that u'(t') = 7(t)tf(s)\\L% \\%v(8)\\lv \\dZv(8)\\Lv n+j+k=£ Furthermore, it follows from Gagliardo-Nirenberg's inequality that ||a^(S)|| u ((ott),iř1)-' In particular, given 0 < e < T, v G L°°((e,T),He(R)) for every positive integer Í. Since the mapping v i-+ \v^v is continuous #£ —► Ji€ (see above), it follows from (5.6.4) that u{ G L°°((e,r),L2oc(R)), and so vt G L°°((e,T),fřfoc(R)) for every positive integer £. In particular, t> € C([e,T), Hfoc(R)), for every positive integer £. Applying again (5.6.4), we deduce that v g Cl([e,T),Hfoc(R)) for every positive integer £. Differentiating the equation k times with respect to t, we obtain eventually, with the same argument, that v G Ck([e,T), Hfoc(R)) for all positive integers £ and k. Therefore, v G C°°{[e,T\ x R), which means that u G C°°([£,T\ x R). The result follows, since e > 0 is arbitrary. Now, we want to make that argument rigorous. In order to do that, we need the following result. Lemma 5.6.2. Suppose

0, let us show that it is true for p + 1. Set uk(t) = dxu(t). Given a positive integer £, (5.6.8) iu\ + ulxx+ J2 unvJuk = °- n+j+k-i Taking the L2 scalar product of (5.6.8) with ze~2ea:2x2p+2u£, where e G (0,1), we obtain ~||c-«2a^V(t)||ia =Im J uxxe-2^x2^ (5-6-9) +Im f (e~2ex2x2p+27Š Yl ^n^k) J ^ n+j+k=C ' = a + p. 5.6. the C°° smoothing effect 159 We integrate the term a by parts, and we note that Im u^u^ =0. It follows that a=-Im / (((2p+2) -4ex2)e-ex2xpue+1e-£x2xp+i:i7) (5.6.10) J < C(p)\\xpui+1\\L2 \\e-^2xp+1ue\\L2 < C(pJ)\\e-£x2xp+1ue\\L2 by the induction assumption. On the other hand, P<\\e-ex\p+1ue\\L2 ]T \\e~£x2xp+1unuluk\\L2 n+j+k=£ (5.6.11) t < C(£)\\e-£*2xp+1u%2 Y |je—V+V||L2 , fc=0 since uj is bounded in L°° for every j, by Step 1. It follows from (5.6.9), (5.6.10), and (5.6.11) that \jt\\e-£X\p+1u\t)\\l2 < t C(p,£)\\e-£x2xp+1ue\\L2 +C(£)\\e-£x2xp+1ue\\L2 ]T \\e~ex2 xp+luk\\L2 k=0 for every nonnegative integer £. Therefore, i í C(pJ) Y \\e-£x2xp+1uk\\L2 + C{£) Y j|e~£XV+1ufc||22 . fc=0 fc=0 Hence the result follows by integrating the above differential inequality and letting s i 0. Step 3. Applying Step 2 and equation (5.6.8), we see that, for all nonnegative integers £ and p, xpu\ e L°°((0,T),L2); in particular, xpue € C([0,T],L2). Considering again (5.6.8), we deduce that xpul € C1([0,T],L2) for all £ and p. Iterating that argument, we obtain that xpu£ € C°°([0,T], Ľ2) for all nonnegative integers i and p. This completes the proof. □ End of the proof of Theorem 5.6.1. Consider a sequence tpk g S(R) such that ipk —> (p in H1(R) as fc -> oo, and such that ||a;pt^fc||L2 < 2||xpv?||l2 for all positive integers k and p. Let uk be the solution of (5.6.1) with initial value 1. One concludes as in the formal argument that we described before. □ Remark 5.6.3. Note that we have shown in fact that the function v defined by (5.6.5) belongs to C°°((0,T), Hm(RN)) for all m > 0, whenever (1 + x2)^^ g L2(R) for every positive integer m. Evidently, there are also partial results if we only assume that (1 + x2)~^

0, there exists a constant Ce for which < e|(tt||JD(J4) + ||w||x for aM u £ T)(A). Let ip € XA, and consider the maximal solu- tion u £ C((—rmin, Tmax)j XA) of the problem (3.7.7) given by Theorem 4.12.1. If ip e D(A), it follows that u € C(( 7min) Tmax proof. Local existence in D(A) follows from Theorem 4.12.3. The proof of Theorem 5.3.1 is then easily adapted (with the estimates from the proof of Theo- Remark 5.7.2. Let be a smooth, open subset of R2, and let g satisfy the assumptions of Theorem 3.6.1. Consider

£ Hq(Q), and let u be the maximal solution of (3.1.1) given by Theorem 3.6.1. If

0, A € R, let

0, C(A) > 0, and s e (0,1) such that (6.1.1) G{u) < ±-~\\ufH1 + C{A) for all u € H\RN) 163 164 6. GLOBAL EXISTENCE AND FINITE-TIME BLOWUP such that \\u\\L2 < A. Consider

1, 5 > N/2; W is an even, real-valued potential, W G Lcr(RN) + L°°(RN) for some o > 1, a > N/4, and f : RN x R —► R is measurable in x G RN and continuous in u G R and satisfies (3.2.7), (3.2.8), and (3.2.17). The function f is extended to RN x C by (3.2.10). Assume further that there exist A>0 and 0 < v < 4/N such that (6.1.2) F{x,u) 1, 6 > N/2 (and 9 > 1 if N = 2). It follows that for every if G H1(RN), the maximal strong ^-solution u of (4.1.1) given by Corollary 4.3.3 is global and s\xp{\\u(t)\\H^ ■ t G R} < oo. Proof. We claim that, with the notation of Corollary 4.3.3, (6.1.4) G(u) <^\\ufHl+C(\\u\\L2) for all u G H1(RN). The result then follows from Theorem 6.1.1. To prove the claim, let V = V\ + V2, where Vx G L°° and V2 G L6, and let W+ = W± + W2, where W1 G L°° and W2e Le. We have 4G(u) < 2||Vi|U-|M|£a + 2||V2||L*H|2_^ + 4A\\u\\\2 + 4A\\ufL% + l|Wij|l-Nli» + l|w2||LHMI^ < C(l + HtilltO + C\\u\\2l^ + C\\u\\lt2+2 + C\\ufL^ . On the other hand, it follows from Gagliardo-Nirenberg's inequality that N 26-N 2 ^ /~MI„,IIT 5~ in;^ 0 and a nonnegative function n e C([0, e),R+), with tj(0) = 0, such that (6.1.6) G(u) < ^Ml2^ + r?(NL2) forallueH^R") such that \\u\\hi < e. It follows that there exists a > 0 such that, for every

1, 5 > N/2; W is an even, real-valued potential, W £ L°(RN) + L°°(RN) for some a > 1, a > N/4; and f : RN x R —* R is measurable in x £ RN and continuous in u £ R and satisfies (3.2.7), (3.2.8), and (3.2.17). The function f is extended to RN x C by (3.2.10). It follows that there exists a > 0 such that, for every tp £ H1(RN) with |j 0 such that if ip e 771(MiV) satisfies Hv^lltf1 < £o, then the corresponding maximal H1 solution u of (4.1.1) given by Theorem 4.4.6 is global, i-e., Tmin = Tmax = oo. Moreover, u e Lq(R, Wl)\u - v\ for all u,v e C. We consider the admissible pairs (ij,Pj), j = 1,2, such that pj = aj + 2; in particular, 71 = pi = 2 + 4/TV. Given 0 < t < Tmax, we set /(*) = IMU»((o,t),jn) + ||uj|__Tr1((o,t),w1."x) + \\u\\Li2((Q,t),w^n) ■ Since 7j < 00 (and u is continuous with values in i/1(RAr)), we see that (6.2.4) fit) I Mm as U0. On the other hand, it follows from Strichartz's estimates that there exists C independent of t such that (6.2.5) f(t) < CUvll^ + C|l^i(«)ll^_((0.„).wx.^) +CHite(«)||J_^ao_0pWrx.^) - Since \gj(u)\ + |V_«j(u)| < C|u|°y(|u| + |Vu|) by (6.2.3) and Remark 1.3.1(vii), it follows from Holder's inequality in space and time that there exists C independent of T such that 2 6) llffl(u)l^i((o,t),ivl,',i) - CH'UllL'i((O)t),L^)llWll^((0,t),^.Pi) LP2 and W1'?2 <-* LP2, we see that \\u\\Lt,^t)jL(>2) < f{t), and so (6-2-7) ll52(«)ll^((o1t),^^)^C/W0a+1-It now follows from (6.2.5)-(6.2.7) that f{t) < CIMI/ji + C/(i)ai+1 + Cf(tr*+1 for all 0 < t < Tmax. Applying (6.2.4), we deduce easily that if < £0 where s0 > 0 is sufficiently small so that (2C£0)ai+1 + (2CsQ)a2+1 <1, 6.3. global existence for oscillating data 167 then f(t) < 2C\\(p\\Hi for all 0 < t < Tmax. Letting 11 Tmax? we deduce in particular that ||u||Loc((o,T,max),.H':l) < 00» so that Tmax = oo by the blowup alternative. Thus /(£) is bounded as t —+ oo, so that NU~((0,oo),jn) + NUm(0,oo),W1.'>i) + IMIl."'3({Q,oo),W1.»a) < 00 • By Strichartz's estimates and the previous estimates of gj(u), this implies that u E Lq((0, oo), W1,r(RN)) for every admissible pair (q,r). The estimate for t < 0 is obtained by the same argument. □ Remark 6.2.2. Theorem 6.2.2 says that if IMIiJ1 is small, then the corresponding H1 solution u is global and decays as t —* oo in the sense that u E LQ(U, W1,r(RN)) for all admissible pairs (q, r). Note that we already mentioned results of the above type. See in particular Remarks 4.5.4, 4.7.5, and 4.9.8 where it is assumed that ||Vy||L2> IMU2, and IMI^p, respectively, are small. These results, however, apply to (essentially) homogeneous nonlinearities. This is a major difference with Theorem 6.2.2 which applies, for example, to the case g(u) = a|ujaiu + b\u\a2u. Remark 6.2.3. The smallness condition on \\ip\\m can be improved, depending on the assumptions on g. Also, instead of considering H1 solutions, one can consider more generally Hs solutions. See Section 5 of Kato [206] and Pecher [295]. Note that global existence results for small data hold under various assumptions on the nonlinearity and for smallness of the initial data in various spaces. See, for example, Hayashi and Naumkin [183], and Nakamura and Ozawa [257]. 6.3. Global Existence for Oscillating Data In this section we consider the model nonlinearity (6.3.1) g(u) ■= \\u\au, where 4 (6.3.2) A E C, 0 < a < - (0 < a < oo if AT = 1). We show that the solutions of (4.1.1) are "positively global"; i.e., Tmax — oo if the initial value

0 satisfies A^a:2 + (N - 2)a ~ 4 > 0 if and only if a > ao- We also note that 2 4 4 4 t7 < T7-7: < »0 < t7 < ~t7—- ii N > 2, N ~ N + 2 u N N -2 - ' | ao, where Qo is defined by (6.3.3), and let 2a(a + 2) (6-3.4) a = --v. _ _ ' . v ' 4 - a(N - 2) There exists e > 0 such that ifip e H1(miv) and || J'(*) q/2 if and only if a > ao, with ao defined by (6.3.3). For such values of a and a, and for 0 < T < oo, we have the following estimates for A defined by Af)(t) = [ 7(t-s)f(s)ds forOT),Lr) < C\\u\\li{(QtT)tLr) for every u£ La{(0,T),Lr(RN)). (ii) Ifu € i°((0, T), Lr(RN))nLQ((Q, T), WljT(RN)) and if (7,p) is any admissible pair, then A(\u\au) £ I7((0,T), W^fR*)). Furthermore, there exists C independent of T such that (6.3.6) ||-4(Mau)|JL-r((o,t),h".p) < C\\u\\t«([0,T),Lr)\W\\L«({0,T),W^-) for every u £ L°((0, T), Lr(RN)) n L«((0, T), W^r(RN)). Proof. The first part of the lemma is a simple calculation, which we omit. For assertions (i) and (ii) consider a defined by (2.4.2). Since (a+l)r' = r, (a+l)a' = a, and 11a - = - + -, q' q a we see that IHwrwllLa'((0,T),L-') = WU\\t«((0,T),Lr) and (applying Holder's inequality twice) that \\\u\au\\Li'((0,T),W^') ^ C\\u\\L"((0,T),Lr)\\u\\L-i((0,T),W^-) ■ The results now follow from (2.4.3) and Strichartz's estimates, respectively. □ Proof of Theorem 6.3.2. We use the notation of Lemma 6.3.3. Let e > 0, let tp £ H1(RJV) be such that I|9TVIIl«((o,oo),l-) < £, 6.3. global existence for oscillating data 169 and let u be the maximal solution of (4.1.1) defined on [0,Tmax) with 0 < Tmax < oo. It follows from equation (4.1.2) and from (6.3.5)-(6.3.6) that there exists K independent of T and ip such that (6-3.7) IMIl-<(o,t),l-) < e + #11^+^^> and (6.3.8) I|w||l«((o,D,wi.'-) < #IMI/f * + K\\u\\l"((o,T),Ln\\u\\li((0,T),W^) for every T < Tmax. (The term i^|| a0, where ao is defined by (6.3.3), and let a be defined by (6.3.4). Let

bo, then TmSiX((pt,) = 00. Moreover, ub € La((0,oo),La+2(RN)), and ub e L^((0,00), W1'"(R7V)) for every admissible pair (7, p). Proof. Let r = a + 2 and let (g, r) be the corresponding admissible pair. A direct calculation, based on the explicit kernel of the Schrodinger group (see Lemma 2.2.4), shows that H*l2 [7(t) 0, is defined by Dpw(x) — P^w(px). It easily follows that II^(-)^IIl-«oioo)ilo = /Vb(l - brf-^WnrMh-dT. Jo 170 6. GLOBAL EXISTENCE AND FINITE-TIME BLOWUP Since \\7{T)||Hi < C\\tp\\Hi and since Q by Lemma 6.3.3, we see that Jim ||T(-)Vb!U«((o,oo),l-) =0. The result now follows from Theorem 6.3.2. □ Remark 6.3.5. We note that |<#,(x)| = \ bo means that (pb is sufficiently "oscillating" as \x\ —► oo. Remark 6.3.6. It is the condition | • \tp(-) e L2(RN) that ensures 4/N, then q < a, where a is given by (6.3.4), and that if a < 4/N, then q > a. Next, if u e Lt((0,oo), W^R*)) for every admissible pair (7,p), we have in particular u £ LQ((0,00),La+2(RN)), and also u £ L°°((Q,oo),La+2(RN)) by Sobolev's embedding theorem. Therefore, u £ ^((O.oo),!0-'-2^)) if a > 4/JV. On the other hand, if a < 4/N, then the property u £ La((0,00), LQ+2(RAr)) expresses a better decay at infinity. (ii) If A < 0, then all H1 solutions of (4.1.1) are global (see Section 6.1). Therefore, Theorem 6.3.4 means that all the solutions ub have a certain decay as t —> 00 for b large enough. (iii) If A > 0 and a < 4/N, then all H1 solutions of (4.1.1) are global (see Section 6.1). Therefore, Theorem 6.3.4 means that ub has a certain decay as t —» 00 if b is large enough. Note that certain solutions do not decay, in particular the standing waves, i.e., solutions of the form elwt 0 and a > 4/N, then (4.1.1) posesses solutions that blow up in finite time (see Section 6.5 below). Theorem 6.3.4 means that for any (p £ H1(RN) with j ■ \ ao, where ao is defined by (6.3.3), and let a be defined by (6.3.4). Let y? £ J(RN) satisfy | • \tp(-) £ L2(RN). Given s £ R, let us be the maximal Hl solution of (4.1.1) with the initial value tps = 7(s) s0, Tmax(ips) = 00. Moreover, us £ La((0,00), La+2(RN)), and us £ £7((0,00), W1,P(RN)) for every admissible pair (7,p). Indeed, since \\7(-)ip\\La((o,oo),Lr) is finite (See Corollary 2.5.4), we see that \\7(-)tps\\L-((o,oo),Ln = II^(-)vIIl*((*,oo),lo —► °» 6.4. GLOBAL EXISTENCE 171 and the result follows from Theorem 6.3.2. 6.4. Global Existence for Asymptotically Homogeneous Initial Data In this section we consider the model nonlinearity (6.4.1) g(u)=X\u\au, where 4 (6.4.2) AgC, aQ-» T(£)y? is continuous R —> S'(RN), so that the definition (6.4.8) makes sense. We also observe that Xt and Wt are Banach spaces when equipped with the norms (6.4.9) \\u\\xT = esssupt/3||u(£)||icr+2 , 0o is a Cauchy sequence. It follows that (wn)n>o defined by un(t) = 7(t)o such that itnfe(£o) —► w(*o) in La+2(RN), hence in 5'(RN). Therefore, . We begin with the following existence result. Theorem 6.4.1. Assume (6.4.1)-(6.4.2) and consider the spaces XT and WT defined by (6.4.3)-(6.4.10). There exists 6q > 0 such that the following properties hold: (i) Let 0 < T < oo and

o 2(a + I) and if e is sufficiently small (depending on \x), then (6.4.12) suptM||u(t) - v(t)\\La+2 < 2A. t>o In particular, t^\\u(t) — u(t)||_,Q+2 —* 0 as t —> oo. (iii) Let

o and if e is sufficiently small (depending on ji), then suptM||n(t)||LQ+2 < 2A and t^\\u(t) - 7(t) 0 as t -> oo. t>o Remark 6.4.2. We note that the equation (4.1.2) makes sense for (p £ Wt and u £ Xt- Indeed, 7(t)

La+2(RN). Note also that t i-* 7(t)ip is continuous [0,oo) -> S'(RN) and bounded [S,oo) -> La+2(RN) for every S > 0. It follows that T(i)v is weakly continuous (0, oo) —» LQ+2(RJV). Since 0 satisfies 2K£% < 1, then for any ip £ Wt with |M|wy < e < so and any 0 0. 0 0 so that a(t) is well defined. We deduce from equation (4.1.2) that t»\\u(t)-v(t)\\La+2 ft 0, where the last identity follows from (6.4.4). Since (6.4.16) a0 + fi 0. If e > 0 is sufficiently small so that C(2e)a < 1/2, we deduce that a(t) < 2A for all t > 0 and (6.4.12) follows by letting t T oo. Step 3. Proof of (iii). The first part of the statement follows from (ii) applied with ip = 0. It remains to show that — 7(t) oo. We observe that \\U(t) - nm\L°+* < |A| f\t - 8)-T#te\\u(8)\\li}a ds Jo 174 6. GLOBAL EXISTENCE AND FINITE-TIME BLOWUP and that ||u(£)||La+2 < Ct~v for every 0 < v < p,. Assuming (6.4.17) a(3 + n < (a + l)v < 1, which is possible by (6.4.6) and (6.4.16), we obtain \\u(t) -7(t)s-(Q+l^ds Jo = Ci1_^)-(Q+1)l/. Applying now (6.4.4), we deduce that t»\\u(t) -7{t) where the last property follows from (6.4.17). □ The relationship between the solutions in Xt constructed in Theorem 6.4.1 and finite energy solutions is given by the following lemma. Lemma 6.4.3. Assume (6.4.1)-(6.4.2) and consider the spaces X? and Wt defined by (6.4.3)-(6.4.10). Let £0 > 0 be given by Theorem 6.4.1. Let 0 < T < oo and if € Wt satisfy \\ o,nd let u € Xt be the unique solution of equation (4.1.2) such that \\u\\xT < 2£0, given by Theorem 6.4.1. If if e H1^1*), then u e C([0,T],F1(MAr)). PROOF. Let u1 e C([0,rmax),iJ1(RJV)) be the maximal strong H1 solution of (4.1.1) given by Theorem 4.4.1. We first observe that, since H1{RN) <-> La+2(RN), \\u^xr{{Ti0)iWn,r) with C independent of r < 9 < Tmax. Since q-2 q-2 IMIl?'((t,0),w1''-) ^ (e ~ t) " \\u\\li{{t,6),w^) < (Tmax - r) i \\u\\L<,{{T>e)tWi,r), we see that if we fix r sufficiently close to Tmax5 L 2 C\\u\\L*'((T,e),w^r) < ■x\\u\\L<1((Tto),w^n , and it follows from (6.4.19) that \W\L°°«t,e),Hi) + ||wj|L,((r>e))VKi,r) < 2C\\u(t)\\hi . Letting 9 ] Tmax, we obtain u £ jC°°((t,Tmax),H1(RN)), which contradicts the blowup alternative of Theorem 4.4.1. □ Remark 6.4.4. Lemma 6.4.3 is a regularity result. Under the same assumptions, one can show that if ip £ HS(RN) for some Na f AH < s < min < 1, — >, 2(a + 2) - [ ' 2 then u £ C([Q,T],HS(RN)) and u coincides with the Hs solution given by Theorem 4.9.1. The proof is similar (see the proof of Theorem 4.9.1). The assumption s > Na/2{a + 2) implies that HS(RN) ^ La+2(RN). Thus ||u||xt — 0 as T { 0 whenever u is an Hs solution. This is an essential step in the proof of Lemma 6.4.3. Note that the assumption 5 > Na/2(a+2) also implies the condition a < 4/(N—2s) of Theorem 4.9.1. Corollary 6.4.5. Assume (6.4.1)-(6.4.2) and consider the spaces Xt and Wt defined by (6.4.3)-(6.4.10). Then there exists £o > 0 with the following property. Let

Woo. Indeed, ||T(t)^||LQ+2 < C\\7(t)0 t»\\7{t)ip\\La+2 < oo for all fx < Na/2(a + 2). (ii) Let p £ C with Rep = 2/a. It follows from Theorem 2.6.1 that ip(x) = \x\~p satisfies £/3||T(£)^>|(l°+2 = c > 0. In particular, ip e Wo©. Note that the assumption (6.4.2) is essential. (hi) Assume a < A/N (in addition to (6.4.2)). Let p £ C with Rep = 2/a and set ip(x) = \x\~p. We see that ^{Ix^i} € /f1({|a;| > 1}) and that Q _j_2 ^l{|i| < \x < 4(a + 2) J r 2(a + 2) ' let p £ C satisfy N (6.4.21) Rep =2fx + a + 2 ' and set tp(x) = |xj-p. It follows from Theorem 2.6.1 that tp\\7(t)ip\\La+2 = c > 0. Moreover, it follows from (6.4.20) that ^|{|x|>i} € -^{{M > *}) and that ^!{|a;| € L^(RN), it follows from Corollary 2.6.7 that ip £ and that \\T(t)(p\\La+2 < C(t~^^ + t~^). Moreover, tIJ'\\7{t) 00. 6.4. global existence 177 Remark 6.4.7. Let ip(x) = 5\x\~p with p £ C such that Rep = 2/ct and 5 £ C. It follows from Remark 6.4.6(h) above that there exists a constant K such that JiV'llu'oc ^ 7sT|(5j. In particular, if \S\ < e0/K, it follows from Theorem 6.4.1 that there exists a unique solution v £ X^ of (4.1.2) (with the initial data ip instead of if) such that (Mlxoc < 2£o- Such a solution is of particular interest since it is self-similar. We recall that if u is any solution of (4.1.1) (or (4.1.2)) on (0, oo) x RN and if 7 > 0, then u7 defined by u~,(t,x) = 7pu(72£, 72) is also a solution. (The assumption Rep = 2/a is essential.) A solution u is called self-similar if it is invariant under the transformation u 1-+ u7, i.e., if u — u7 for all 7 > 0. We claim that v is self-similar. Indeed, it is easily verified that t>7 satisfies (4.1.2) with the initial value i^-y(x) = 7^(72;). Evidently, since ip is homogeneous, = 0-Moreover, it follows from a direct calculation that H^Hx^ = IMIxoc- Therefore, by the uniqueness property of Theorem 6.4.1, vy — v for all 7 > 0. We observe that v is weakly continuous (0,00) —> La+2(RN) by Remark 6.4.2, so that / = it(l) € La+2(RN) is well defined. Applying the identity v(t,x) = jpv(j2t,jx) with 7 = t~ 2, we see that v(t,x) = t~%f i.e., the self-similar solution v is expressed in terms of its profile f. Note that self-similar solutions are not H1 solutions in general; see [73, 75]. For a more detailed study of self-similar solutions, see Cazenave and Weissler [73, 74, 75], Furioli [119], Kavian and Weissler [209], Planchon [298], Ribaud and Youssfi [302], and Weissler [363]. Remark 6.4.8. Here are some more applications of Theorem 6.4.1 and Corollary 6.4.5. (i) Assume a < 4/7V and fix satisfying (6.4-11). Let p £ c with Rep = 2/q and let 6 £ C. Set i^(x) = 8\x\~p and let y £ Hl(RN) be such that ip — (p £ L*+*{RN). If \S\ and ||

0 is as in part (ii) of Theorem 6.4.1 (see Remark 6.4.6(h) and (iii)). If we denote by u and v the corresponding solutions of (4.1.2), then u is an H1 solution by Lemma 6.4.3 and v is self-similar by Remark 6.4.7. Moreover, it follows from Theorem 6.4.1(h) and Remark 6.4.6(iii) that £Mjjw(£) — v(t)||/jQ+2 is bounded uniformly in t > 0. Since t@\\v(£)j|L«+2 = c > 0, we deduce that t/3||u(t)||£Q+2 —> c as t —> 00. In particular, we know the exact rate of decay of ||w(£)|| 2,0+2 as t —► 00. Note also that u(t) is asymptotic to the self-similar solution v as t —+ 00 (in the sense that t^\\u{t) - v(t)\\La+2 —► 0 as t —> 00). (ii) Let /i satisfy (6.4.20) and let p £ C satisfy (6.4.21). Set tp(x) = 5\x\~p and let if £ Hl(RN) be such that ip -

0 is as in part (iii) of Theorem 6.4.1 (see Remark 6.4.6(iv)). If we denote by u the corresponding solution of (4.1.2), then u is an H1 solution by Lemma 6.4.3. Moreover, it follows from Theorem 6.4.1 (iii) and Remark 6.4.6(iv) that £m||w(£)IIl°+2 ~^ c > 0 as t —> 00. In particular, we know the exact rate of decay of |ju(t) as t —► 00. Note also that by Theorem 6.4.1 (iii) and Remark 6.4.6(iv), t**||w(£) - ty(t)tp^ia+2 —> 0 as t —» 00. This means that u(t) is asymptotic 178 6. GLOBAL EXISTENCE AND FINITE-TIME BLOWUP to T(t)^j as t —> oo. Note that is a self-similar solution of the linear Schrodinger equation; see Remark 2.6.6(iii). REMARK 6.4.9. Here are some comments on the decay rates of ||u(t)||i,«+2 that are achieved by H1 solutions of (4.1.1). (See also Remark 7.3 in [75].) (i) Suppose a < 4/N. It follows from Remark 6.4.8(h) above that if /3 < (jl < Na/2(a+2), then there exist Hl solutions of (4.1.1) for which ||u(i)||Le,+2 « t~v as t —► oo (in the sense that tp\\u(i)\\La+2 —> c > 0). By Remark 6.4.8(i), \x = [3 is also achieved if a < and it follows from the results of Chapter 7 below that // = Na/2(a+2) can also be achieved. Moreover, p = Na/2(a + 2) is the fastest possible decay in general (see Begout [20] and Hayashi and Ozawa [187]). On the other hand, it is not known whether some solutions can have a slower decay than t~@. (ii) Suppose a > 4/N. It follows from Remark 6.4.8(h) above that if Na/4(a + 2) < n < Na/2(a + 2), there there exist H1 solutions of (4.1.1) for which ||u(t)||Lc,+2 £ ^ as £ —> oo. A decay like t 2 is also possible and is the fastest possible (see (i) above). Note that the lower bound n > Na/4(a + 2) is also optimal. Indeed, Hue Xoo is a solution of (4.1.1), then u € L^5T((0,oo),La+2(]RJV)) (see Remark 3.12 in [75]). If A in (6.4.1) is a negative real number, the same property holds for any H1 solution with initial value in H1(RN) n L2(RN,\x\2 dx); see Chapter 7 below. In both cases, it follows that liminft_>00i4 1} p > N/2. The function / : RN x R -► R is measurable in x e RN and continuous in it e R and satisfies (3.2.7), (3.2.8), and (3.2.17). Extend / to R^ x C by (3.2.10). The potential W is even and real valued; W £ Lq(RN) + L°°(RN) for some q > 1, q > N/4. In particular, g is the gradient of the potential G defined by 6.5. Finite-Time Blowup g(U) = VU + /(•, «(■)) + (W * \U\2)U , G(u) = J |^V(:r)|W(a:)|2+^ RN 6.5. finite-time blowup 179 and we set E(u) = ^J\Vu(x)\2 dx - G(u) for all u G H1{RN). n We recall (see Corollary 4.3.3 and Remark 5.3.3) that the initial-value problem for (4.1.1) is locally well posed in H1(RN), that there is conservation of charge and energy, and that there is the H2(RN) regularity if the initial value is in H2(RN). Our blowup result is based on the following identities, which will also be essential in the next chapter to establish the pseudoconformal conservation law. Proposition 6.5.1. Let g g(u) = Vu + f(;u(-))+(W+\u\2)u be as in Example 3.2.11. Assume, in addition, that (6.5.1) x-VV(x) e L°(RN)+L°°{RN) for some o~ > l,a > ^ , (6.5.2) f(x,u) is independent ofx, (6.5.3) x ■ VW(x) G LS(RN) + L°°(RN) for some 5>1,6>^. Consider

| • \u(t, •) belongs to C((-Tmin,Tmax),L2(RJV)). Moreover, the function (6.5.4) t^f(t) = J \x\2\u{t,x)\2dx is in C2( Tm\n, Tmax); (6.5.5) f'(t) = 41m Jux-Vudx, and f"(t) = l6E(p) + J(8{N + 2)F(u) - 4N Re(f(u)u))dx (6.5.6) -f-°I(V+12 + 8 / ( V + ^x- W )\u\2dx + aJ (^(w + ^x- vw) * \u\2yu\2 dx for all t G (-Tmm, Tmax). Before proceeding to the proof, we establish the following lemma. Lemma 6.5.2. Let g G C(H1(RN),H-1(RN)). Assume that g(w) G Lloc(RN) and that Im g(w)w = 0 a.e. in RN for allw G Hl(RN). Let I 3 0 be an interval of R, let

0, and take the H"1—!!1 duality product of equation (4.1.1) with ie~2e^2 \x\2u(t, x) G Hl(RN). Setting fS) = \\e-e^2\x\u{t)\\2L>, we deduce easily that f'e(t) = 2Im J{Vu- V{e-2£W2\x\2u)-e-2eW2\x\2g{u)u}dx. Since Im(Vu • Vw) = Im(g(u)u) = 0 a.e., we obtain that f£(t) = 2lm J uVu-V{e-2eW\x\2)dx rn = 4Im J{e-eW\l-2e\x\2)}e-eW2ux- Vudx, and so (6.5.7) fe{t) = fe(0) + 4J Im J{e~£^2(I - 2e\x\2)}e'e^ux-Vudxdt. Note that e~e|x!''(1 - 2e\x\2) is bounded in x and e and that ||e_elx|2|a:MlL2 < ||a; | • •) is weakly continuous I —> L2(MJV) (see Section 1.1). Moreover, we may let £ j 0 in (6.5.7) and we obtain (6.5.9) = IMli2 + 4 / Im / ux • Vudxdt. Note that the right-hand side is a continuous function of t and so the function t | ■ •) is continuous I —»■ L2(RAr). It follows that the right-hand side 6.5. finite-time blowup 181 of (6.5.9) is a W1'00 function and the identity (6.5.5) holds a.e. If u is a strong H1-solution, then the right-hand side of (6.5.9) is a C1 function, so the identity (6.5.5) holds for all tel. □ Corollary 6.5.3. Let g be as in Lemma 6.5.2. Assume that g : H1(RN) —» H~1(RJV) is bounded on bounded sets and let I 9 0 be a closed, bounded interval of R. Let ip € if 1(RJV) be such that | • | 0, let um be a weak H1 -solution of (4.1.1) corresponding to the initial value 0 and that xtpm —> x

oo, i/ien xum -* m C(J\ L2^)) as m —> oo. Proof. Assume by contradiction that there exist (im)m>o C / and e > 0 such that ||a:wm(tm) — xu{tm)\\l2 > e. Without loss of generality, we may assume that there exists r G I such that tm —> r as m —► oo. We deduce from (6.5.9) that for every t £ I, (6.5.10) ||xum(t)||i2 = llx^Hls+4 J Im J u^x-Vumdxdt. It follows from (6.5.10) that ||| ■ |um(£)||£2 is bounded uniformly in t G I and m > 0. Since (tim)m>o is bounded in /^(/^(R^)), (g(um))m>0 is bounded in L^i^H'1^)) so that «)m>o is bounded in L°°{I,if-^R^)), by (4.1.1). Therefore, (um)m>0 is bounded in C0>? (/, L2(R^)). We deduce that wm(tm) -» tt(r) in i/^R*) as m oo and, since | >m(£m) is bounded in L2(RN), \ -\um(tm) u(t) in L2(RN) as m -» oo. Since wm ^ u in L°°(/, if^R^)), it then follows from (6.5.10) and (6.5.9) that ||xu7n(tm)||i2 —* ||xti(r)||£,2 as m —> oo, and so xum(tm) —»• xu(r) in Z/^R^), which yields a contradiction. This completes the proof. □ Proof of Proposition 6.5.1. The first part of the statement follows from Lemma 6.5.2. It remains to show that the function / defined by (6.5.4) belongs to C2(—Xmin,Tmax) and that the identity (6.5.6) holds. Formally, the result would follow by calculating the time derivative of the right-hand side of (6.5.5). This corresponds to multiplying equation (4.1.1) by i(2rdru + Nu), which is not allowed since the equation only make sense in H~1(RN). The proof we give below is based on two regulaxizations. Therefore, we proceed in two steps. step 1. The case

0, consider 6£{x) = e~e|xl2 and let (6.5.11) he(t) = Im J 6£ux ■ Vudx for every t G (-Tm iiii -^~max)- rn We claim that he is C1 and that (6.5.12) h'e{t) = - Im J ut{2eerdru + (N0£ + rdr6e)u}dx. rn 182 6. GLOBAL EXISTENCE AND FINITE-TIME BLOWUP Indeed, (6.5.12) is equivalent to (6.5.13) he(t) = MO) - Im jT J ut{29£rdru + (N9£ + rdr9£)u}dxds The identity (6.5.13) holds in fact for every function u which is continuous in Jf^R*) and C1 in L2(RN). Indeed, by density we need only establish (6.5.13) for u which is C1 in iJ1(RJV). In this case, we deduce from (6.5.11) that h'e(t) = - Im J 6£utx ■ Vudx - Im J 6eux • Vut dx, RAT RJV and (6.5.13) follows by integration by parts, since 9eux • Vul = V • {x9£uul) - N9euui — 9eulx • Vu - rdr9£uul. This proves the claim. Using now equation (4.1.1), we see that (6.5.14) h'e(t) = - Re J(Au + g(u)){29£rdru + {N9£ + rdr9£)u}dx. Next, an elementary calculation based on the identity Re(29£Vu • V(rdru)) = — ((TV - 2)9£ + rdr9£)\Vu\2 + V • (xOe\Vu\2) shows that for every u € H2(RN)7 Re J Au{29£rdru + (N9£ + rdr9£)u}dx (6.5.15) =-2/ ^\Vu\2dx - J{2rdr9e\dru\2 + ((TV + l)dr9£ + rd%) Re(udru)}dx. We now calculate the various terms corresponding to g(u). Since Re[Vu{29£rdru + (N9£ + rdr9e)u}] = V • (xV9e\u\2) - 9£{x - VV)\u\2 , we obtain (6.5.16) Re J Vu{29erdru + {N9S + rdr9£)u}dx = - J 9£(x ■ VV)\u\2 dx, Next, so that (6.5.17) Re[f(u)(29£rdrü)\ = V • (2x0EF(u)) - 2(N9£ + rdr9£)F{u), Re j f{u){29£rdrü+(N0£+rdrO£)ü}dx = J(N9£ + rdr9£)(f{u)ü - 2F{u))dx. RN 6.5. FINITE-TIME BLOWUP 183 Finally, using the identity Re[u{29£rdru + (N9£ + rdr9£)u}\ = V • (xd£\u\2), we obtain Re J{w* \u\2)u{29£rdru + (N9£ + rdr6£)u}dx = - J 9£\u\2x ■ (VW* \u\2)dx. ffi™ rn On the other hand, W is even, so that VW is odd. Therefore, J $£\u\2x-(VW*\u\2)dx = \j 9£\u\2\{x-VW)*\u\2}dx rn + i J J(9£(x)-9£(y))\u(x)\2\u(y)\2x-VW(x-y)dydx, rn and so Re J (W * \u\2)u{29£rdru + (N9£ 4- rdr9£)u}dx rn (6.5.18) eM\* ■ VW) * \u\2]dx rn -\j J(e£(x)-9£(y))\u(x)\2\u(y)\2x-VW(x-y)dydx. rn Applying (6.5.15), (6.5.16), (6.5.17), and (6.5.18), we deduce from (6.5.14) that ■ h'e(t) = 2 J 0s\Vu\2dx+ J 9£(x- VV)\u\2dx 4- J N9£(2F(u) ~ f(u)u)dx+^ J 9e\u\2[(x-VW)-k\u\2)dx rn rn (6.5.19) + rn 4- ({N + l)dr9s + rd29£) Re{udru)}dx i J J(9£(x) -9£(y))\u(x)\2\u(y)\2x-VW(x~-y)dydx. rn rn Note that 9£,rdr9£, and r2d29£ are bounded with respect to both x and £. Furthermore, #£ T 1j dr^e —> 0, r6V#e —> 0, and r926*£ —»• 0 as e J, 0. On the other hand, for every t g (-Tmin,Tmax), we have u(t) g tf^R*) and \x\u(t) g L2(R*) by Lemma 6.5.2, and so we may use the dominated convergence theorem to pass 184 6. global existence and finite-time blowup to the limit in the right-hand side of (6.5.19) as e [ 0, except in the last one. We obtain (6.5.20) lim^(i) = 2 J \Vu\2dx + N J (2F(u) - Re{f{u)ü))dx + J \u\2x- VVdx + i j \u\2((x- VW)*\u\2)dx + L, r" ew where L is the limit as £ | 0 of the last term in (6.5.19). We claim that (6.5.21) L = 0. Since also lim he(t) — Im J ü~x ■ Vu dx = h(t), we see that h is of class C1 and that h'(t) = 2 J \Vu\2dx + N J(2F(u) - Re(f(u)ü))dx + J \u\2x-VVdx + ^ J H2((x- VW)*\u\2)dx. Equation (6.5.6) now follows from the above identity, (6.5.5), and conservation of energy. We finally prove the claim (6.5.21). Note that (6.5.22) J J(ee(x)-9£(y))\u(x)\2\u(y)\2x-VW(x-y)dydx j j^\0e(x)-ee(y) < \x - y\ \u(x)\"\u(y)\'\(x - y) ■ VW\x - y)\dy dx Also, it follows from assumption (6.5.3) and from Young's and Sobolev's inequalities that J J \u(x)\2\u(y)\2\(x - y) ■ VW(x - y)\dydx < C\\u\\i* + C||ti||^ < C\\u\\4H1 • Since \ee(x)-ee{y)\ \ee{x)-e£{y)\ sup kc-;-1-< oo ana \x\-:-;- x^yl \x-y\ \x~y\ eio * 0 a.e. in RN x EN, we may use the dominated convergence theorem to pass to the limit in (6.5.22) as e I 0, and we obtain (6.5.21). Step 2. Conclusion. Let (y?TO)m€n C H2(RN) be such that (pm —► ip in H1(RN) and xtpm —> x

oo, and let um be the corresponding maximal solutions of (4.1.1). Let <$>(£) denote the right-hand side of (6.5.6) and let 6.5. FINITE-TIME BLOWUP 185 $m(i) denote the right-hand side of (6.5.6) corresponding to the solution um. It follows from Step 1 that (6.5.23) j|a;wm(t)|||,2 = ||^m|||2 + 4tlm ftp^x-S7 oo in (6.5.23) and we obtain IMOIIl* = \\X(P\\h + 4tlm flpx ■ V 0, (6.5.25) V + ix-W<0 a.e., (6.5.26) W + ~x-VW<0 a.e. Let ip e Hl(RN) be such that | ■ \ 0 and t < 0. Proof. It follows from (6.5.24), (6.5.25), (6.5.26), and Proposition 6.5.1 that for every t £ ( Jmin i -^max)) (6.5.27) \\Mt)\\h t2E(ip). Observe that d(t) is a second-degree polynomial and that the coefficient of t2 is negative; therefore 6(t) < 0 for |t| large enough. Since ||xu(t)|||2 > 0, we deduce from (6.5.27) that both Tmin and Tmax are finite. □ Remark 6.5.5. Note that the proof of Theorem 6.5.4 does not show that ||cci((£)||l2 —> 0 as t f Tmax or t I — Tmax. (See Ball [16, 15] for an interesting discussion of related phenomena.) This is sometimes the case (see Remark 6.7.3), but not always. Consider the model case g(u) = X\u\au with A > 0 and a = 4/N. First, observe that by the invariance of the equation under space translation, one constructs easily a solution such that ||artt(i)||^2 /» 0 as 11 Tmax. Indeed, it follows from the conservation of momentum (3.1.5) that, given x0 £ RN, J \x-x0\2\u\2 = J\x\2\u\2 + \xQ\2 j M2 + 2 J x-XoM2 + 4tIm J 0. To see this, we follow an argument of Merle [248]. Consider a real-valued, spherically symmetric function ip G H1^) such that xo be a sequence of real-valued, spherically symmetric functions such that xipn(x) G L2(RN) for all n > 0 and ipn —► ip in H 1(RJV) as n —► oo, and let un be the corresponding solutions of (4.1.1). In particular, E(ipn) —> E((p) and n—*oo liVnllz,2 —► IMIl2- Therefore, it follows from the proof of Theorem 6.5.10 below (see in particular formula (6.5.42)) that there exists a function \P G W4,0°( ^ > 0, such that ^2 / #K|2 < 2E(n is real valued, one easily verifies that d_ dt J *\un\ = 0, t=o so that J %un\2<2 J *\\\(x- x0)ipn\\2L2 + WE(ip)t2 for 0 < t < Tmax{ oo, it now follows from (6.5.28) that inf {||(x-xoK(0||l2 :0 2E( 0 then Tmin < oo, if E(ip) > 0 and Im J Tpx • Vtpdx < -y/2E( 0 and Im J Tpx ■ V y/2E( 0 and 4/JV < a < 4/(JV - 2). We claim that for any Eq > 0, there exists tp such that E( 0, to be chosen later, set ip(x) = atp(fjix). It follows from (6.5.30) that Im J 2E(A-—B Fix now and let /x be given by C 0 < e < min tA, — >, „2 _ B „<* ß - —e° In particular, (6.5.34) is satisfied and (6.5.33) reduces to 2-N (B \ 2 4-(N-2)c which is achieved for a suitably chosen. Remark 6.5.9. Theorem 6.5.4 shows the existence of solutions for which both Tmax < oo and Tmjn < oo. As a matter of fact, there exist solutions for which Tmax = oo and Tmm < oo and solutions for which Tmax < oo and Tmin = oo. Indeed, let g(u) = X\u\au with A > 0 and a > 4/N. Let

0 and t < 0 (see Theorem 6.5.4). Theorem 6.3.4 implies that if b is large enough, then the maximal solution Ub of (4.1.1) with initial value given by (6.3.12) is positively global and decays as t —► oo. Of course, E(ifb) > 0 for such 6's, and one may wonder if still blows up at a finite negative time. The answer is yes, as the following argument shows. Changing iff, to (which changes Ub(t) to U6(-£)), it suffices to show that if E(ip) < 0, then for all b > 0 the solution v of (4.1.1) with initial value fj)(x) — (p(x)e t~^L blows up at a positive finite time. Let Tmax(^) be the maximal existence time of v, and let f(t) = ||| ■ |v(t, Ollia- It follows from formulae (6.5.5) and (6.5.6) that f(t) = /(0) + tf'(0) + 8E(ip)t2 - x4{Na~ 4) f f f \v\a+2dxdads a + z Jo Jo J mN for all 0 < t < Tmax(ip), and so f(t) < /(0) + tf'(0) + 8E&)t2 for all 0 < t < Tmax(^). Setting P(t) = f(0) + tf'(0) + 8E(u{Q))t2 for all t > 0, a straightforward calculation shows that P(t) = Mia + 4«^) - ^IMIis) +8t2^) + |W|Il2 - h-F{^ 6.5. FINITE-TIME BLOWUP 189 with In particular, F((p) = Im J xipVtpdx. and we deduce easily from (6.3.13) that Tma,x(ip) < 1/b (see the proof of Theorem 6.5.4). Hence the result follows. The condition for finite-time blowup in Theorem 6.5.4 is E( 0. Assume N > 2 and ± the solution u of (4.1.1) blows up in finite time for both t>0 and t < 0. The proof is in some way an adaptation of the proof of Theorem 6.5.4. Instead of calculating ||xu(t,a;)[||2, we calculate \\M{x)u{t,x)^2L2, where M : RN —* R is a function such that M(x) = \x\ for \x\ < R and M is constant for \x\ large. Next, we use the decay properties of the spherically symmetric functions of H1(RN) to estimate certain integrals for \x\ large that appear in the calculation of ||M(a?)u(£,a;)||^2. Note that, as opposed to the case x

1 and let k _ C1([0, oo)) be a nonnegative function such that r-(N~Vk(r) G L°°(0,oo) and r-(N-V(k'(r))- G L°°(0,oo). There exists a constant C such that C||^ll|-(K-)(llr-tAr-1^^ij|2(RW) + ilr-^-^^O-lltoolklll,^)) for all spherically symmetric functions u G H1(RN). proof. By density, we may assume that u G V(RN). For s > 0, /OO i — (k(a)\u(a)\2)da /oo poo k'{(j)\u{a)\2da — 2 J k{s)Re (u(a)ur(a))da POO pOO < (k'{o))-\u(a)\2da+ 2 fc((r)K<7)||t/r(a)|dcr. JS J s 190 6. GLOBAL EXISTENCE AND FINITE-TIME BLOWUP Therefore, k{s)\u(s)\2 < C||r-^-V)-|mi*(r") +C'ilu||1,2(MN)||r-(JV-1)fcur||L2(^), and the result follows. □ Proof of Theorem 6.5.10. By scaling, we may assume A = 1. Let u be as in the statement of the theorem. Consider a function W e W4'°°(RN), and set V(t) = i J V(x)\u(t,x)\2dx for all t E (-Tmi„,Tmax). We claim that ^V{t) = 2 I[H{m)Vu,Vu)dx--~ [ AV\u\a+2dx dtz J a + 2 J (6.5.35) - i J A2$\u\2dx for all t g (-rmin, rmax), where the Hessian matrix if(^) is given by H($>) = (djdktyiKj^KN- Assuming o c H2(RN) and using the continuous dependence. Next, we rewrite (6.5.35): ^V(t) = 2NaE(u(t)) - 2 J j^|Vu|2 - (ff(tt)Vu, Vu)}<*r (6.5.36) +o7T2 f (2N ~ A^)\u\a+2 dx - I J &2V\u\2dx. Let now p g V(R) be such that p(x) = p(4 - x), p > 0, fRp = 1, supp(p) c (1,3), and > 0 on (—oo, 2). We define the function 9 by 9(r) = r — I (r — a)p(a)da for r > 0. Jo We consider e g (0,1), to be specified later, and we set and 7(x) = 7(r) = 1 - 0'(£r2) ~ 2£r20"(£r2) = / p(s)ds + 2er2p{er2) Jo for x G and r — \x\. Elementary calculations show that (6 37) I (i/(*)v"'w) = 2(1 " ^r))ki2' ['' } \ AV = 2N{1-j{r))+4(1-N)er29"(er2), 6.5. finite-time blowup 191 and that AH = e(4N(N + 2)9"{er2) + 16(iV + 2)er26'"(er2) + 16(er2)2e""(er2)) . In particular, there exists a constant a such that (6.5.38) I|A2#J|l~ < 2ae. It now follows from (6.5.36), (6.5.37), and (6.5.38) that (6.5.39) ~V(t) < 2NaE{ 1 and 9" < 0. We claim that there exist b and c such that 4 Nc (6.5.40) ^ /7(r)|«|«« < te"^MT (/7l Indeed, we first observe that 7(r) < 1 + 2sups>0 sp(s), so that Jl(r)\ur+2 1/2 for er2 > 2. Furthermore, 7'(r) = 6erp(£r2) + 4e2r3p'(er2), so that 7'(r) > 0 for er2 < 2 and 7'(r) > -4e2r3\pf(er2)\ > -4e? ||s V(*)IIl~(o,oo) • Thus Wr-^-^y'Hll'hoc < Ce% and (6.5.40) follows. Using now (6.5.39), (6.5.40), and conservation of charge, we see that ~V(t) < 2NaE() < 0 and a < 4, it follows immediately from (6.5.41) that one can choose e > 0 depending only on 0, (6.5.42) implies that Tmin < oo and Tmax < oo. □ Remark 6.5.12. There are two limitations in the above proof. The first one is a < 4. If a > 4, powers of ||72urj|i,2 larger than 2 appear with positive coefficients in (6.5.41). This is due to the homogeneity in Lemma 6.5.11. The other limitation is TV > 2, since if TV = 1 the power of e in the second and third terms of the right-hand side of (6.5.41) vanishes. This is due to the fact that the radially symmetric functions in dimension 1 do not have any decay property. However, in the critical case N = 1, a — 4, Ogawa and Tsutsumi [276] have proved that all negative energy H1 solutions blow up in finite time without any symmetry assumption. Their method is a more sophisticated version of the above argument. (See also Martel [240] for certain extensions.) We now give a lower estimate for blowing up solutions (see [70]). Theorem 6.5.13. Suppose g(u) = X\u\au with A > 0 and 4 4/4 \ N N -2 \N ~ J If if e H1(fi&N) is such that Tmax < oo, then there exists 6 > 0 such that (6.5.43) ||Vu(t)||L2 >--TTZI^I f°r0 ^ 1 < (Tmax — t)<* * A similar estimate holds near — Tmjn if Tm[n < oo. Proof. Generally speaking, every time one proves local existence by a fixed point argument, the proof also gives a lower estimate of the blowup. Here, we do not go through the entire local existence argument, but instead we give a direct proof. Set r = a + 2 and let q be such that (q, r) is an admissible pair. Let tp be as above, and let u be the corresponding solution of (4.1.1). It follows from Remark 1.3.1(v) that (6.5.44) \\V(\u\au)\\Lr, < C||u||M|V«||£r. By conservation of energy, X\\u\\lr = -rE^) + r-\\Vu\\2L2. Therefore A||u||2r < C(l + HVulHO* < C(l + HVuM* . From (6.5.44) and the above inequality, we deduce that for any 0 < t < r < Tmax> \\^(W\ay)\\l0'((t,r),L-') ^ C(l + llVuilL-((f,r),L2))a^||Vw||L9'(((!T)iL,.) < C(r - t)^"(l + ||V«||Loc((t)T),La))^ \\Vu\\L,i(tiT)tLr). Set now ft{r) = 1 + \\Vu\\Lec((tfT)iL2) + ||Vu||x,,((t>t))lt-), so that, by the above inequality, (6.5.45) \m\^)h^t,r),Lr') < C(T - t)^ft(T)1^ . 6.5. FINITE-TIME BLOWUP 193 On the other hand, it follows from Strichartz's estimates that l|V«||Loc((ttT)iL9) + ||VulU,^),^, < C||Vu(*)||L9 + C7||V(|u|au)j|I,,.{{t)T)>Lf.) for 0 < t < r < Tmax. By (6.5.45), this implies that (6.5.46) ft(r) < C(l + \\Vu(t)\\L2)+C(T-t)*ff ft(r)1+^ for 0 < t < r < Tmax. Consider now t G (0,Tmax). Note that if Tmax < oo, it follows from the blowup alternative that ft(r) —> oo as r t Tmax. Note also that ft is continuous and nondecreasing on (£,Tmax) and that /t(r)—^l + liVtiCt)^. Tit Therefore, there exists r0 € (i,Tmax) such that ft(r0) = (C + 1)(1 + |jVu(t)||L2), where C is the constant in (6.5.46). Choosing r = to in (6.5.46) yields 1 + ||V«(*)||ta < C(l + C)1+" (r0 - t)^{l + ||VW(£)||L2)1+" < (1 + C7)2+^(Tmax _.t)^(l + ||V«(t)||L»)1+^ , and so 1 + \\Vu{t)\\L. >-—-^ . Hence the result follows, since t € [0,Tmax) is arbitrary and -^-j^ = ~ - ^f^- □ Before proceeding further, we establish an immediate consequence of the above result concerning the blowing up of certain LP norms of the solution. Corollary 6.5.14. Suppose g(u) = A|u|Q« with A > 0 and 4 4/4 \ N~ N -2 \N~ J J If oo for all p > ^. Moreover, 5 Na (6.5.47) \\u(t)\\Lr >-1 jv~ forO-4-lN-ya/i ^ forOa + 2. (Tmax-t) -2 ("?} A similar estimate holds near — Tmm if Tmjn < oo. Remark 6.5.15. Note that if N > 3 and p > or if N = 2 and p = oo, then it may happen that J|w(i)||^P = oo for some (or all) t G (—Tmin,Tmax). Clearly, this does not contradict the above estimates. Note, however, that u G -^k)c((~^min, 2~max), W1'r(RN)) for every admissible pair (g, r), so that by Sobolev's embedding theorem, ||«(£)||lp < oo for a.a. t £ (—Tmin,TmeiyL) provided N < 3 or N > 4 and p < 194 6. global existence and finite-time blowup proof of Corollary 6.5.14. Suppose first ^ < p < a + 2. By Gagliardo-Nirenberg's inequality, Nisri < c\\vu\\ir\\u\rLr with p = Jpj{lN*2)v • By conservation of energy and the above inequality, we obtain 2A \\Vu(t)fL2 < 2E( oo, it follows that ax ||Vu(t)||Ja a + 2. It follows from Holder's inequality that ay 2(p-(a + 2)) HI^+22-a for all 0 < t < Tmax' The lower bound (6.5.48) now follows from Theorem 6.5.13 and the above inequality. □ REMARK 6.5.16. Theorem 6.5.13 and Corollary 6.5.14 give lower estimates of ||Vw||l2 and \\u\\lp near the blowup. They do not give any upper estimate. It is interesting to compare these results with the corresponding ones for the heat equation. If one considers the equation ut — Au = |u|p-1u with a Dirichlet boundary condition, then a simple argument (even simpler than the proof of Theorem 6.5.13) gives the lower estimate ||u|[l°c > (Tmax — £)". If a < then it is known that this is the actual blowup rate of the solutions (see [361, 126, 195, 352]). However for larger ct's, some solutions blow up faster (see [196]). A lower estimate is obtained as well for ||u||lp, p > Na/2. In some cases it is known that \\u\\lp also blows up for p = Na/2 (see [362]) and that \\u\\LP remains bounded for p < Na/2 (see [113]). Remark 6.5.17. In the case a > 4/iV, one does not know the exact blowup rate of any blowing up solution. In addition, one does not know whether \\u\\lp blows up for 2 < p < Na/2. On the other hand, there is an upper estimate of integral form (see Merle [243]). More precisely, if ip e H1^) and x 0. 6.5. FINITE-TIME BLOWUP 195 Since ||xu^)!!2^ > 0, this implies that I / ||Vu(s)||£3 dsdt < oo Jo Jo Since / / ||W(s)H22 dsdt = / (Tmax-t)\\Vu(t)\\2L2dt, Jo Jo Jo it follows immediately from Holder's inequality that / IIVu(t)||£2 dt < oo forO is spherically symmetric, then one obtains the same estimate. Indeed, by using the fact that a > 4/N, one can improve (6.5.42) to ^V(t) < NaE{ip) - (Na - 4)|| Vu|||8 , and the conclusion is the same. Remark 6.5.18. In the case a = 4/jV, then (6.5.43) becomes (6-5.49) l|Vtt(0IU» > 6 (^raax *)2 and (6.5.47) and (6.5.48) become (6.5.50) ||«(t)||iP>--"¥7i rr forp>2. (Tmax-t)^(2-^ In particular, ||w|Up blows up for p > 2. Since J|u|]l2 is constant, estimate (6.5.50) is optimal with respect to p. On the other hand, it is known that the blowup rate given by (6.5.49) and (6.5.50) is not always optimal, since some solutions blow up twice as fast (see Remark 6.7.3 and Bourgain and Wang [41]). Moreover, in space dimension TV = 1, Perelman [297] has constructed a family of blowing up solutions for which /•iogliog(rmax-t)iv^, ^„ I-7p-—;- I h{t)\\L~ —+ c>0, \ max 1 / 111 max which is very close to but different from the lower estimate (6.5.50). Merle and Raphael [250] recently obtained the upper estimate ||Vu(t)||i,0, «=^- We recall that the Hl solutions of (4.1.1) are global and bounded in H1(RN) provided y? G H1(RN) satisfies \\f\\L2 < $ for some 6 > 0 (see Remark 6.1.3). In fact, one can determine the optimal <5. Let R be the (unique) spherically symmetric, positive ground state of the elliptic equation (6.6.3) -AR + R= \R\4rR in RN (see, for example, Definition 8.1.13 and Theorems 8.1.4, 8.1.5, and 8.1.6). Note that any ground state of (6.6.3) is of the form et0R(x — y) for some 6 G R and y G RN. We have the following result of M. Weinstein [356]. Theorem 6.6.1. Assume (6.6.1)-(6.6.2) and let R be the spherically symmetric, positive ground state of (6.6.3). If ip G H1(RN) is such that \$y\\L*<\\R\\Li, then the maximal H1 solution u of (4.1.1) is global and supteR ||u(£)||/fi < oo. Remark 6.6.2. The condition \\ A~« ||-R||l2 (m fact, even for p = A~q j|JR||_,2, see Remark 6.7.3 below) there exists ip G if^M^) such that |M|l2 = p and such that u blows up in finite time for both t < 0 and t > 0. Indeed, let ip(x) = R(V\x), so that ||^||_2 = A~« ||i£||j_2 and ip is a solution of -A^ + A^ = \\ip\aip. It follows that E(ip) = 0 (see formula (8.1.21)). Let p > \~*\\R\\L2, set 7 = p/\\R\\l2 > 1, and consider ipp = yip. It follows that HwIIl2 ~ P and E{ 4/7V; see Begout [19]. Remark 6.6.3. In space dimension N = 1, an elementary calculation shows that 3* (6.6.4) R(x) - - , ycosh(2x) in particular, \\R\\h=*Vs. Therefore, it follows from Theorem 6.6.1 that if if g Jff1(RJV) is such that ^"H^IIl2 < S^y/n, then the solution u of (4.1.1) is global and supieR ||u(£)||#i < co. Proof of Theorem 6.6.1. By conservation of charge and energy and by Lemma 8.4.2 below, we have for all t € (—Tmin,Tmax), |||V«(*)||ia < E{v) + -^||u(*)||SSa < JE7(<^) ^ _f^_|fX7^(*)lJi2[[^(i)j[-2 <^) + |||||il|VW(i)||i2, IL2 and so Hence the result follows, by using the blowup alternative. □ Remark 6.6.4. Assume A > 0 and let d be the supremum of the //'s such that ||o C H1(RN) with ipn —> tp in L2(RN) and xipn(x) bounded in L2(RN). The corresponding solutions un satisfy | • \un(-) g C(R, L2(RN)) (see Lemma 6.5.2), and from the pseudoconformal conservation law we see that (see formula (7.2.8)) 8t2E(vn(t)) = \\xtpn\\2l2 forallteR, where vn(t) = e~ * un(t). In particular, ||vn(t)llL2 = \\un(t)\\l2 - JI^JIl2, so that there exists e > 0 such that ||un(£)||x,2 < A~~° ||-RJ|l2 — e for n large enough. It follows that there exists C such that ||Vvn(£)||£2 < CE(vn(t)) for all i g R (see the proof of Theorem 6.6.1). By Lemma 8.4.2, this implies that \\vn(t)\\2H2 < CE{vn{t))\\ipn\\% < ^ for all t g R. 198 6. global existence and finite-time blowup We conclude as in Remark 4.7.4 above. Theorem 4.7.1 has an immediate application to the study of blowup solutions. Theorem 6.6.5. Assume (6.6.1)-(6.6.2). Let

o is any sequence such that tn | Tmax, then u(tn) does not have any strong limit in L2(RN). A similar statement holds for Tmm. Proof. Assume that u(tn) —* w in L2(RN). By continuous dependence (see Theorem 4.7.1), 1 Tm&x(u(tn)) > -Tmax(w) > 0 for n large enough. This implies that TmB.x(tp) >tn + ~Tmax(w) for n large. This is absurd, since tn —> Tmax. □ In fact, one can prove a stronger result which implies the above theorem (see [71]). Theorem 6.6.6. Assume (6.6.1)-(6.6.2). There exists p > 0 with the following property. Let € L2(RN) and let u be the corresponding maximal L2 solution of (4.1.1) given by Theorem 4.7.1. 7/Tmax < oo and if C is the set of weak L2 limit points of u(t) as t | Tmax, then [jtwJl^ < IMIi2 ~~ P2 for a^ w £ C. A similar statement holds for Tmin • proof. It follows from Step 1 of the proof of Theorem 4.7.1 (see in particular (4.7.4)) that there exists 5 > 0 such that if ll^(-)0llL°+2((O,r),LQ+2) < 6, then Tmax(0) > r. Letting (ft ~ u(t), we deduce that ||T(-)w(i)||L«+2((0>Tmax_t)!jLQ+2) > 5 for all t € [0 Therefore, given any ip e L2(RN) and any t € [0,Tmax), $ < \\J(-)(u(t) - ^)j|La+2((0,Tmax-t),L-+2) + IITO^IU«+2((0,rmax-t),L"+2) - • ax L 6.6. the critical case 199 Therefore, if u(tn) —tp for some sequence tn | Tmax, then — < liminf \\u(tn) - ipfL2 = liminf(u(tn) -ip,u(tn) -ip)Li = Immf (\\u(tn)\\2L2 + ||t/,||22 -2{u{tn),i>)L.) = m\h - ml*, and the result follows. □ For H1 spherically symmetric blowup solutions in dimension N > 2, there is a minimal amount of concentration of the L2 norm at the origin, as the following result shows (see Merle and Tsutsumi [251], Y. Tsutsumi [344], and M. Wein-stein [360]). Theorem 6.6.7. Assume (6.6.1)-(6.6.2). Suppose further that N > 2 and let R be the spherically symmetric, positive ground state of equation (6.6.3). Let 7 : (0,oo) —► (0,oo) be any function such that j(s) —► 00 and 527(5) —> 0. Finally, s|0 sj.0 let

0, u(tn) -± w in £2({|x| > £}), and so 5&inf IK*)IU2(nt) > A ° \\R\\L2, where Qt = {x e RN : \x\ < |rmax -forTmin. *|27(Tmax~ t)}. A similar statement holds ™\\2LH{\x\>e}) < Ijnmf ||«(*™>||i3({|a!,>cJ) • On the other hand, Htn)\\h(Uxl>£}) = \Htn)\\2L2 - \\u(tn)\\2LH{]xl<£}) = \\\\h - ll«(*n)||ia(nf)> 200 6. global existence and finite-time blowup and the result follows from Theorem 6.6.7. □ Proof of Theorem 6.6.7. Set w(t) = ||Vu(i)||~2 so that w(t) —► 0. We ax claim that (6.6.5) l™!,imC llu(*)llL2({|a;|o is a bounded sequence in Hl(RN), so that there exist a subsequence, which we still denote by (fn)n>o, and w G Hl(RN) such that vn w weakly in H 1(RN) as n —» 00. Since the t;n's are spherically symmetric, we deduce that vn —► w in La+2(RN) (see Proposition 1.7.1). In particular, E(w) < 0, and n—»00 by (6.6.8), / 0. By applying Lemma 8.4.2 below, we obtain (6.6.9) A-|HL3 > ||i?||L2. Given M > 0, |MU2({|x| \\w\\L* > \-»\\R\\L2 , n—>oo which contradicts (6.6.6). This completes the proof. □ In fact, Corollary 6.6.8 can be generalized to nonradial solutions (and also to the space dimension N = 1). More precisely, we have the following result. 6.6. the critical case 201 Theorem 6.6.10. Assume (6.6.1)-(6.6.2). Let R be the spherically symmetric, positive ground state of equation (6.6.3). Finally, let ip g H1(RN) and let u be the maximal H1 solution of (4.1.1). 7/Tmax < oo and if C is the set of weak L2 limit points of u(t) as 11 Tmax, then \H\h < \\0 c Hl(RN) \ {0} and u e L2(RN) be such that un-± u in L2(RN) as n —► oo. If furthermore ||Vu7l|j^2 —► oo and n—+oo hmsuplivu ii2 -0, then \\u\\12 ^liminf^-.oo jjwn|||2 - A~« ||i?jj£2. We will use the following lemma. Lemma 6.6.12. If (un)n>0 c H1(RN) is such that (i) \\un\\2L2=a>0, (ii) 0 < infn>0 ||Vwn|lL2 < supn>0 ||Vun||L2 < oo, (iii) limsupn_0O£(u„) < 0, then p, > \~°\\R\\2L2) where p, = //((un)n>o) is defined by (1.7.6). Proof of Proposition 6.6.11. (Assuming Lemma 6.6.12.) Let (wn)n>o be as in the statement of Proposition 6.6.11. Set a = liminfn_+00 H^nll2^- By considering a subsequence, we may assume that \\un\\h -> a- n—+00 i Set ujn = j|V«n|j£2 and define vn(x) — UnUn(u;nx). It follows that IJVnllia = Kll£a, l|Vv„||£a = l, and lim sup B(vn) = limsup ^^^1 < 0. n—*oo |jVwn||^2 We first show that a > 0. Indeed, by Lemma 8.4.2, which implies that A||vn||£2 > II^II^j and so a > A~« |jjR||^2- We now set Wn = 71-H-vn \\vn\\L* 202 6. global existence and finite-time blowup so that (wn)n>o satisfies the assumptions of Lemma 6.6.12. Therefore, MK)«>o)>A--||i2||ia. We apply Lemma 1.7.5 to the sequence (wn)n>o. Given e > 0, it follows from the above inequality that there exists T such that p(wn,T) > A~°" ||i?|j22 — £ for n large enough. Therefore, setting yn = y(wn,T) with y(-, •) defined by Lemma 1.7.4(h), \wn(x)\2 dx > A~° H-^llla — e for n large, / {\x-yn\ \-°\\R\\2L2 -e with zn = ujnyn and tn = a>nT. Note that tn —► 0. By possibly extracting a subsequence, we may assume that either \zn\ —> oo or zn —> z for some n—>oo n—foo z £ RN. In the first case, consider M > 0. Since wn —* u in Z,2({|x| < M}), we deduce that \\u\\lH{\x\M})} = a-limsup||un|||2({|a;|>M}) < o - A"»|j/2|||2 +£■ The result follows by letting M f oo, then e J, 0. In the second case, consider 8 > 0. Since wn u in L2({|:r| > <5}), we see that hfm{\x-z\>6}) < limmf IK||£2({|x_z|>5}) = a-limsup||un|||2({|x_z|<5}) < a- A--||i?j||2 +e. The result follows by letting 8 I 0, then £ j 0. This completes the proof. □ Proof of Lemma 6.6.12. We claim that there exists 8 > 0, depending only on N and A with the following property. If (un)n>0 c Hl(RN) is such that (6.6.10) |KH|2 = a>0, (6.6.11) 0< inf ||Vwn||L2 < sup ||Vun[|22 o) < \-°\\Rfl2, 6.6. THE CRITICAL CASE 203 then a > ^((wn)n>o), and there exists a sequence (wn)n>o C H1(RN) satisfying (6.6.11), (6.6.12), and (6.6.13), and such that ||u„||£2 = a - 0 for some 0 > 8. The result follows, since if (wn)n>o C i71(MN) is as in the statement of the lemma, and if /i((un)n>o) < ^"ll-fr|||2, then we may apply the claim k times to obtain a — kS > /i((un)n>o)» which is absurd for k large. Therefore, we need only prove the claim, and we consider («n)n>o C H1{RN) satisfying (6.6.10)-(6.6.13). The property a > fJ,((un)n>o) follows from (6.6.11), (6.6.12), (6.6.13), and Lemma 8.4.2. Next, we apply Lemma 1.7.5, then Proposition 1.7.6(iii) to the sequence (un)n>o, and we consider the corresponding sequences (vk)k>o and (wk)k>o- We set (6.6.14) 4= + where the constant K is given by (1.7.17). We first show that (6.6.15) n := /x((itn)„>o) > 6, where 5 is defined by (6.6.14). Indeed, it follows from (1.7.17) and (6.6.14) that EM > i(i - /|VU„,P - j^^.W . Assuming by contradiction fi < 8, we obtain by letting k —» oo and applying Lemma 1.7.5(h) and (6.6.11), limsupEK) >\(l~ (jY^htiJ [Vwn|2 > 0, which is absurd. Next, since \wk\ < \unk \ by (1.7.12), it is not difficult to deduce from Lemma 1.7.5(h) and (6.6.13) that (6.6.16) M(«>fc)*>.o) < A* < \\R\\h • Also, it follows from Lemma 8.4.2, (6.6.13), and (1.7.14), that there exists a > 0 such that (6.6.17) E(vk) > a\\Vvk\\2L2 for k large. On the other hand, it follows from (1.7.15) and (1.7.16) that (6.6.18) limmf{£(unfc) - E(vk) - E{wk)} > 0. k—>oo Inequalities (6.6.12), (6.6.17), and (6.6.18) imply that (6.6.19) limsup£(u;fc) < 0. k—> 0. k—too To prove this, we argue by contradiction and we assume that there exists a subsequence, which we still denote by (wk)k>o, such that HVtUfcjjj^ —> 0 as k —»• oo. It 204 6. global existence and finite-time blowup follows that E{wk) —*• 0 as A; —» oo, so that (6.6.12), (6.6.18), and (6.6.17) now imply that UVujtjJL2 0 as fc —> oo. Using (1.7.16), we deduce that |[wnfc||_.°+2 —+ 0, so that, by (6.6.11), I limsup£(un) > - liminf ||Vun|||2 >0, n—too 2 n-»oo which contradicts (6.6.12). Hence we have proved (6.6.21). Setting yja - n we see that 0 < ||«fcl|22 = a — po satisfies estimates (6.6.11), (6.6.12), and (6.6.13). This completes the proof. □ Remark 6.6.13. For more information on the blowup in the critical case, see the series of papers of Nawa [264, 265, 266, 267, 268, 269, 270]. 6.7- The Pseudoconformal Transformation and Applications In this section we consider the model nonlinearity (6.7.1) g(u) = X\u\au, where (6.7.2) AgR, a = jj- In this case, the pseudoconformal conservation law, introduced by Ginibre and Velo [133], becomes an exact conservation law. This conservation law is associated to a group of transformations which leaves invariant the set of solutions of (4.1.1) (see Ginibre and Velo [139]). We describe below this group of transformations (the pseudoconformal transformation). It will be convenient to use the Hilbert space (6.7.3) E = H\RN) fl L2(RN, \x\2dx) = {u € H X(RN) : \ • \u(-) € L2(RN)} equipped with the norm (6.7.4) HI = HuH^ + \\xu\\2L2 . Let now b 6 R. Given (t,x) elx R^, we define the conjugate variables (s,y) e R x R^ by t x . s y s =-— , y =--— , or equivalently t =-— , x — 1 + bt' y 1 + bt ' M J 1 - bs ' 1 - bs Given u defined on (-Si, S2) x RN with 0 < Si, S2 < oo, we set oo if bSi < -1 r oo if bS2 > 1 (6'7-5' Tl = T+bB7 if »Si > -1. T2 = I rffe K < I- We define Ub on (-Ti,T2) by ub{t,x) = (l+bty%ettt^u 1 + bt' l + bt 6.7. pseudoconformal transformation and applications 205 or equivalently (6.7.6) ub(t,x) = (1 - bs)J^ei^^u(s,y). Note that (6.7.7) IM*)||l» = IK*)IIl2, and, more generally, (6.7.8) \\ub(t)\\L0+2 = (1 - &s)^+3) \\u(s)\\L0+2 if j3 > 0. In particular, (6.7.9) ||ub(t)||La+2 = (1 - bs)^ ||u(a)||La+2 so that if bsi > — 1 and bs2 < 1, then (6-7.10) ||w6||L«+2((-t1,t2)LQ+2) = ||«llL-*+a(<-*i,sa)La+2) with £i = Y+fe~ an(^ *2 = 1_f^S2• Next, if u £ C((—Si, S2),S), then it is clear that u6 £ C((—Ti,T2),S). In addition, (6.7.11) ||aTW6(0llL> = (1 - 65)-1||yw(s)||L2, (6.7.12) ||Vtifr(t)||L* = + 2t(l - 6a)V)u(a)||L2, (6.7.13) l|Vu(s)||L3 =. i||(fcc + 2t(l + fc)Vju6(i)||L2 -The interest of the above transformation lies in the following result. Theorem 6.7.1. Suppose u e C((-Si,S2),L2(RN))nL?+2{(-Si,S2),La+2(RN)) is a solution of (4.1.1) (see Theorem 4.7.1). Let b £ R, let T\,T2 be defined by (6.7.5), and let ub be defined by (6.7.6). It follows that uh £ C((-Ti,T2),L2(RN)) nL?+2{(-Ti,T2),La+2(RN)) is also a solution of (4.1.1). //, in addition, u £ C((—Si, S2), £), then ub £ C((-T!,T2),S). proof. It is clear that ub £ C((-T1,T2),L2(RN)). In addition, it follows from (6.7.10) that ub £ L?+2((-Ti,T2),La+2(RN)). Furthermore, one shows that if 0 < Si, S2 < 00 and if bSi > —1 and bS2 < 1, then the mapping u t-» ub is continu-ous C([-S1,S2],L2(RN))nLa+2((-S1,S2),La+2(RN)) ^ C([-Ti,T2],L2(RN))n La+2((-Ti,T2), La+2(RN)). Let now u £ C([~S1,S2],L2(RN)) n La+2((-Si, S2), La+2(R^)) be a solution of (4.1.1), with Si and 52 as above. Let ip = u(0). We have in particular Tm\n(tp) > Si and TmaxM > 52. Consider (o C if2(MiV) such that ipn ^ tp in L2(RN), By continuous dependence (Theorem 4.7.1(v)), Tmin((pn) > Si and Tmax( S2 for n large enough. We denote by un the corresponding solutions of (4.1.1). We first observe that un £ C({-S1,S2), tf^R*)) by Theorem 4.7.1(iii). Applying then Remark 5.3.3, we deduce that un £ C((-Si, S2), H2(RN)); i.e., u is an H2 solution. It follows that u satisfies equation (4.1.1) a.e. on (-Si,S2) x R^. A tedious, but straightforward, calculation shows that (un)b satisfies (4.1.1) a.e. on (—Ti, T2) xRN. 206 6. GLOBAL EXISTENCE AND FINITE-TIME BLOWUP The conclusion follows from continuous dependence (Theorem 4.7.1(v)) and from the continuity property mentioned above. □ Remark 6.7.2. Note that the pseudoconformal transformation preserves both the space L2(RN) and the space E. On the other hand, it does not preserve the space H1! Remark 6.7.3. The pseudoconformal transformation has a simple application (see M. Weinstein [359]), which yields interesting information on the blowup. Assume for simplicity that A = 1 and let ip be a nontrivial solution of (6.6.3). (Note that tp(x) has exponential decay as |a:| —> oo; see, for example, Theorem 8.1.1.) It follows that u(t,x) = elt\jj(x) is the solution of (4.1.1) with

—* 0 if 1 < p < 2, so that f\ 1 (6.7.17) v{t) -> 0 in L2(RN) as 11 1. In particular, the loss of L2 norm at the blowup is equal to ||-R||l2 if ip is a ground state of (6.6.3), but it is larger if ip is an excited state. (Note that excited states exist if Af > 2; see [25].) Therefore, the loss of L2 norm given by Theorem 6.6.10 is not always optimal. Note also that by (6.7.11), (6.7.18) = (1 - t)\\x1>\\L2 —> 0 (cf. Remark 6.5.5.). In particular, v(t) —► 0 in £2({|a;| > e}) for any e > 0. t"\ 1 Furthermore, one easily verifies that v(t) —* 0 in i71({|a;| > e}) and in L°°({jrcj > tf l e}) (this last point because ip has exponential decay). Therefore, v(t) blows up only at x — 0. Furthermore, it follows from an easy calculation that |v(£)|2 —> \\tp\\2L25 tT l in V(RN), where 5 is the Dirac measure at x = 0. 6.7. PSEUDOCONFORMAL TRANSFORMATION AND APPLICATIONS 207 Finally, we observe that formula (6.7.14) also makes sense for £ > 1, and that v given by (6.7.14) is also a solution of (4.1.1) for t > 1. As a matter of fact, the properties of v as £ T 1 and as £ J. 1 are similar. Formula (6.7.14) gives (formally) an extension of the solution v beyond the blowup time Tmax = 1. We know that v satisfies (4.1.1) on (—oo, 1) and on (l,oo), and we investigate in what sense v may be a solution near t = 1. Note first that v £ C((—oo, 1) U (1, oo), L2(RN)) and that |jv(£)||x,2 = \\iP\\l2 f°r £ ^ 1, so that by property (6.7.17) v is discontinuous in L2(RN) at t = 1. On the other hand, v £ L°°{R,L2(RN)), so that A?; £ L°°{R,H-2(RN)). Furthermore, it follows from (6.7.15) that j|K£)|Qu(£)||Li = c|l - £1^- Therefore, if we assume N > 3, then \v\av £ Lloc(R,Ll(RN)). If m is an integer such that L^M^) <—*■ H~m(RN) (so that in particular m > 2), then we deduce that \v\av £ Lloc(R, H~m(RN)). Therefore, ut £ Lj^R, H~m(RN)), so that u € C(R, H~m(RN)). Thus v(t) ^ 0 in H~m(RN) as £ -> 1. This implies easily that v satisfies (4.1.1) in V(R, H~m(RN)). Therefore, we see that v can be extended in a reasonable sense beyond the blowup time Tmax = 1. However, the meaning of this extension is not quite clear. Indeed, if we define (v(t) if£ 1, then the above argument shows that v is also an extension of v beyond Tmax = 1, which satisfies (4.1.1) in T>'(R, H~m(RN)). As a matter of fact, one can define many such extensions. For example, since equation (4.1.1) is invariant by space translation and by multiplication by a constant of modulus 1, we see easily that for any y £ RN and a> € R, v(t) -{ v(t) if £ < 1 eiuv{t, ■ - y) if £ > 1 satisfies (4.1.1) in V(R, H m(RJV)) and is also an extension of v beyond Tmax = 1. About this problem, see Merle [245]. Remark 6.7.4. In space dimension N — 1, the solutions considered in the above remark are completely explicit. Indeed, it follows from formula (6.6.4) that is a solution of the Schrodinger equation iut + uxx + \u\4u = 0 that blows up at £ = 1. Remark 6.7.5. Let v(t) be as in Remark 6.7.3. Given y £ R , set vy(t) — ■y(£, • - y), so that vy is a solution of (4.1.1) for which Tmax = 1, and that blows up at the point y £ RN. Given (yt)i<£ 0, and y € RN, v(t,x) = e^e^^R^x - y)) is a solution of (4.1.1). For any b < 0 and x± E RN, u(t,x) = Vb{t,x - xi) is therefore also a solution. An easy calculation shows that (6.7.19) u(t,x) = V^max *-/ / iij ix ~ t)xQ) with Tmax = -1/6, u = /uTmax, x0 = y/Tmax, and 0 = 7- p?Tmax. Let now if E H1^1*) be such that ||^||l2 = ||-R||l2 and such that Tmax < 00. It follows from Merle [246] that there exist 9 E R, uj > 0, x0,xi E RN such that u is given by (6.7.19). Similarly, if Tmin < 00, then there exist 9 E R, u > 0, x0,Xi E RN such that u is given by jv u(t,x) = (rrr^—i) 2 ^+'^^"i^i?f?r^((x-x1)-(rmin+^^^ . In other words, the only solutions that blow up on the critical sphere are those obtained from the ground state by the pseudoconformal transformation. Note in particular that if u is a solution on the critical L? sphere, then Tmax and Tmjn cannot both be finite. 6.8. Comments We begin with some examples of applications of the results of the preceding sections. Remark 6.8.1. Let g(u) = A|ti|au with A e RandO < a < 4/(N^-2) (0 < a < 00 if iV = 1). (i) If A < 0, then all solutions of (4.1.1) are global. (ii) If A > 0 and a < 4/N, then all solutions of (4.1.1) are global. (iii) If A > 0 and a > 4/N, then the solution of (4.1.1) is global if JMIh1 is small enough. On the other hand, given ip E i/1(RJ'v), tp 7^ 0, the solution of (4.1.1) with (p = ktp blows up in finite time, provided \k\ is large enough. Statements (i) and (ii) follow from Corollary 6.1.2. The first part of (iii) follows from Corollary 6.1.5. Finally, the last part of (iii) follows from Theorem 6.5.4. Indeed, it is clear that (6.5.24) is satisfied, and that E(kip) < 0 for large enough. Remark 6.8.2. Let g(u) = A(|x[~"*|w|2)w, where A e R, and 0 < u < min{AT, 4}. (i) If A < 0, then all solutions of (4.1.1) are global. (ii) If A > 0 and 0 < v < 2, then all solutions of (4.1.1) are global. (iii) If A > 0 and v > 2, then the solution of (4.1.1) is global if W^Wh1 is small enough. On the other hand, given ip E HX{RN), ip ^ 0, the solution of (4.1.1) with ip = ktp blows up in finite time, provided \k\ is large enough. 6.8. COMMENTS 209 Statements (i) and (ii) follow from Corollary 6.1.2. The first part of (iii) follows from Corollary 6.1.5. Finally, the last part of (iii) follows from Theorem 6.5.4. Indeed, it is clear that (6.5.26) is satisfied, and that E(kip) < 0 for \k\ large enough. Remark 6.8.3. Let g{u) = X\u\au+P(\x\-U*\u\2)u, A,/? e K, 0 < a < 4/(7V-2), and 0 < v < min{iV, 4}. (i) The solution of (4.1.1) is global if \\ 0, and 0 < v < 2, then all solutions of (4.1.1) are global. (iv) If A > 0, P < 0, and a < 4/iV, then all solutions of (4.1.1) are global. (v) If X,P > 0, a < 4/N, and 0 < v < 2, then all solutions of (4.1.1) are global. (vi) If A,/3 > 0, a > 4/N, and v > 2, then given ip g H1(RN), ip ^ 0, the solution of (4.1.1) with

0, p < 0, a > 4/N, a > 2, and v < 2, then given ip g i71(EJV), ip 7^ 0, the solution of (4.1.1) with

0, a < 4/N, a < 2, and v > 2, then given ip € H1^**), tp ^ 0, the solution of (4.1.1) with ip = kip blows up in finite time, provided |fc| is large enough. Statement (i) follows from Corollary 6.1.5. Claims (ii), (iii), (iv), and (v) follow from Corollary 6.1.2. Finally, (vi), (vii), and (viii) follow from Theorem 6.5.4. Remark 6.8.4. There are some finite-time blowup results in strict subdomains Q C RN. For example, assume Q c RN is smooth, bounded, and star-shaped about the origin, and g(u) = \u\au for some 4/N 0. Suppose first that a < 4/N. It follows from Theorem 4.6.1 that for every ip g L2(RN), the corresponding L2 solution of (4.1.1) is global. By Hs regularity (Theorem 5.1.1), we deduce that if 0 < s < min{iV/2,1}, then for every ip g HS(RN), the corresponding Hs solution of (4.1.1) given by Theorem 4.9.1 is global. Fix now 0 < s < min{iV/2,1} and assume 4/N < a < 4/(N - 2s) and A < 0. It follows (see Remark 6.8.1 (i) above) that for every ip e H1^1*), the corresponding H1 solution of (4.1.1) is global. On the other hand, it follows from Theorem 4.9.1 that for every ip e HS(RN), there exists a local Hs solution u of (4.1.1). One may expect that the Hs solution is global, but there is no equivalent of the conservation of energy at the Hs level. It is possible to show, however, global existence for all ip g HS(RN) in some cases; see Bourgain [38] and Colliander et al. [88, 89, 90, 91]. See also Vargas and Vega [350] for a related result of global existence for all initial values in a space strictly larger than L2(RN) for the cubic one-dimensional Schrodinger equation. CHAPTER 7 Asymptotic Behavior in the Repulsive Case In this chapter we continue the study of the global properties of the solutions of (4.1.1). We have seen in the preceding chapter that for certain nonlinearities and initial values, the solution of (4.1.1) satisfies u e L9(R, W1,r(RN)) for every admissible pair (o, r). See, e.g., Theorem 6.2.1. This implies that u(t) has a certain decay as t —» oo. If g is "sufficiently" superlinear near 0, for example if g(u) = X\u\au with a "sufficiently" large, then g(u) will have a stronger decay. One may then expect that the term g(u) becomes negligible in equation (4.1.1) and that the solution u(t) behaves as t —> oo like a solution of the linear Schrodinger equation. This turns out to be the case, under appropriate assumptions on 0, i.e., Tmax = oo. If the limit (7.1.1) u+ = lhn 7(-t)u{t) exists in X, we say that u+ is the scattering state of

£ X is such that the solution of (4.1.1) is defined for all t < 0; i.e., Tmin = oo, and if the limit (7.1.2) u_ - ^ lim 1(-t)u(t) 211 212 7. ASYMPTOTIC BEHAVIOR IN THE REPULSIVE CASE exists in X, we say that w_ is the scattering state of tp at — oo. We observe that saying that tp has a scattering state at ±00 is a way of saying that u(t) behaves as t —» ±00 like the solution 7(t)u± of the linear Schrodinger equation. We set (7.1.3) 71+ = {ip £ X : Tmax = 00 and the limit (7.1.1) exists} and (7.1.4) 11- = {tp EX : Tmin = 00 and the limit (7.1.2) exists} . In other words, 1Z± is the set of initial values tp which have a scattering state at ±00. We define the operators (7.1.5) U± : 1Z± —> X mapping tp t~» u± and we set (7.1.6) u± = u±(n±). If the mappings U± are injective, we set (7.1.7) «± = Ug1 : u± ->n±. The mappings Q± are called the wave operators. Next, we set (7.1.8) 0+ . In other words, u+ = Su_ if and only if there exists ip eY, such that jTmax = Tmin = 00 and such that 7(—t)u(t) —+ u± as t —► ±00. Remark 7.1.1. Note that the operators and the sets that we defined above depend on the space X in which the convergence (7.1.1) or (7.1.2) takes place. remark 7.1.2. We observe that for the linear Schrodinger equation; i.e., when g(u) = 0, all the operators U±, Cl±, S defined above coincide with the identity on X. Note, however, that, in the general case g ^ 0, these operators are nonlinear. Remark 7.1.3. Assume that g(u) = g(u) for all u € X. It follows that changing t to —t in the equation (4.1.1) corresponds to changing u to u, which means changing ip to Tp. So we see that = {ip g X :Tp e 71+} , =U^ = {v e X : v e U+}, 0_ = 0^= {v E X :vEO+}i and that U-tp — U+Tp and Sl-tp = £l+lp. 7.2. PSEUDOCONFORMAL CONSERVATION LAW 213 7.2. The Pseudoconformal Conservation Law Throughout this section we consider a nonlinearity g is as in Example 3.2.11. Therefore, we assume g(u) = Vu + f(;u(-)) + (W*\u\2)u, where V, /, and W are as follows. The potential V is real valued, V G LP(RN) + L°°(RN) for some p > 1, p > N/2, f : RN x R -» R is measurable in x € RN and continuous in u G R and satisfies (3.2.7), (3.2.8), and (3.2.17). Extend / to R^ x C by (3.2.10). The potential W is even and real valued, W G L^(RN) + L°°(RN) for some q > 1, q > N/4. In particular, g is the gradient of the potential G defined by (7.2.1) G(u) = J ^V(x)\u(x)\2 + F(x,u(x)) + ±(W*\u\2)^ and we set (7.2.2) E{u) = \ J \Vu(x)\2 dx - G(u) for all u G H1^). We recall (see Corollary 4.3.3) that the initial-value problem for (4.1.1) is locally well posed in iJ1(R7V) and that there is conservation of charge and energy. Moreover, if if G £ with £ defined by (6.7.3)-(6.7.4), then u G C7((-Tmin,Tmax), E) (see Remark 6.5.2). Also, if tp G ff^R*), then u G C((-rmin, T^^H^R*)) (see Remark 5.3.3). The following "pseudoconformal conservation law," discovered by Ginibre and Velo [133, 134], is essential for the study of the asymptotic behavior of solutions. Theorem 7.2.1. Let £ be defined by (6.7.3)-(6.7,4) and let g(u) = Vu + f(;u(-)) + (W*\u\2)u be as in Example 3.2.11. If

0, 0 < a < (0 < a < oo if JV = 1). 1 \ & and we refer to Section 7.9 and Ginibre and Velo [133, 132, 134, 139] for more general results. Note that in this case, (7.3.3) Tmin(^) = Tmax(^) = oo for all ip £ i71(R;v) (see Remark 6.8.1(i)). Furthermore, it follows from (7.2.8) that (7.2.3) is equivalent to (7.3.4) dx} = x 4/N, then for every 2 < r < ^ (2- < Cltr^-W1-6^ for all t e R, JO i/2 a + 4. Proof. If a > 4//V, we deduce from (7.3.4) that 8t2 J\Vv(t)\2dx < \\x 1, the argument being the same for t < -1. It follows from (7.3.4) that 8t2E(v(t)) = SE{v{l)) + 4v a 2 / 5 / b(s)|a+2da;ds. This implies that M*) a + 2. This completes the proof. □ Remark 7.3.2. Theorem 7.3.1 implies in particular that if

4/N, u has the same decay properties in Lr(M.N) as the solutions of the linear Schrodinger equation (see Proposition 2.2.3) for every 2 < r < 2N/(N - 2). When a < 4/7V, the decay properties are the same for r < a + 2. Corollary 7.3.4. Assume (7.3.1)-(7.3.2). Assume further that a > ao, where a0 is defined by (6.3.3). Let

0, let 9M = ~v 1+ l\u\«U fovueC- Let u£ be the maximal solution of (4.1.1) with g replaced by g£. Note that for every e > 0, gE is globally Lipschitz continuous C —> C. Note also (see, e.g., 218 7. asymptotic behavior in the repulsive case Corollary 6.1.2) that u£ is globally defined and that, by conservation of charge and energy, (7.3.10) sup||u£(i)||//i < oo. teE Next, we observe that \g(u) - gc(u)\ < \g(u)\ and that \g(u) - g£{u)\ —> 0 as £ j 0 for every u G C. It follows easily that uE u in L^R^)) as e | 0 for every 2 < p < 2N/(N - 2) (2 < p < oo if N = 1). (See, e.g., Step 3 of the proof of Theorem 4.4.6, and in particular the proof of (4.4.29).) Using the conservation of energy for both u and u£, one deduces easily that u£ —> u in Lj^c(R, JfiT1(RAr)) as £ J. 0. Therefore, we need only estimate the solutions u£ independently of e. It follows from Lemma 6.5.2 that v£(t) = (x + 2itV)u£(t) G C(R,L2(RN)). Furthermore, applying formula (2.5.5), we obtain (7.3.11) v£{t) = T(t)x

0, if (q,r) is any admissible pair, and if (7, p) is the admissible pair such that p — a + 2, then there exists a constant C independent of £ and r such that (7.3.12) KIIl.((-t,t),l-) < C\\xip\\L2 + Cr^f \\v£\\L,{^TiT)>LP). We first let (q,r) = (7, p). We deduce that if r > 0 is sufficiently small, then ||v£||l-?((-t,t),lp) ^ 2C||x T > 0, l|t*ll^((o,t),^.-) {{0fT)iW1,rf) +C|||u|au||Lg'((Tit)iiyl,r'), where C is independent of t and T. Since IIMau||Wl,r' ao, we have 2a > q ~- 2. Therefore for T large enough, (7.3.14) cf / ||u(s)||^da) ' <^ IT 2 On the other hand, u g L°°((0,r),lf ^M^)) n L9((0, T), W1,r(RJV)). Therefore, it follows from (7.3.13) and (7.3.14) that \\u\\L«{(0,t),W^) 0, NU«((0,tUr) - ^Mlx2 + CW(X + 2is^)\u\au\\L"'{{0,t),L-') > 220 7. asymptotic behavior in the repulsive case where C is independent of t. By applying (7.2.9), we obtain \\v\\Lo((Q,t),Ln ^ cIMIl* + c( f \\u{s)\\i;2 ds] HU^t),-^) (7-3.15) t 3=1 + IN«)# a0 defined by (6.3.3). The results of this section are due to Ginibre and Velo [133] for a > 4/JV and to Y. Tsutsumi [341] for a > ao. In Section 7.5 below, we will obtain a similar result for a = ao, but by a different method using explicitly the pseudoconformal transformation. Theorem 7.4.1. Assume (7.3.1)-(7.3.2). Assume further that a > ao, where a0 is defined by (6.3.3) and let E be the Hilbert space defined by (6.7.3)-(6.7.4). If (p € E and if u is the maximal solution of (4.1.1), then there exist u+ ,u~ € E such that \\7{-t)u(t) - u^Wv —> 0. t—► ±oo In addition, \\u+\\l* = = IMIl» and I j |Vn+|2 = \ f \Vu~\2 = E{ 0 t,r—>oo (see the proof of Corollary 7.3.4.) Therefore, x(v(t) - w+) —> 0 in L2 as t —► oo. One shows as well that x(v[t) — w~) —> 0 in L2 as t —> —oo. The other properties follow immediately from conservation of charge and energy. □ Remark 7.4.2. Theorem 7.4.1 means that the mappings U+ :

S. In fact, one can show with similar estimates that U+ and U- are continuous. Note that U± are nonlinear operators. Remark 7.4.3. By Corollary 2.3.6, /•±oo u± =

ao, where ao is defined by (6.3.3), and let E be the Hilbert space defined by (6.7.3)-(6.7.4). (i) For every u+ G E, there exists a unique ip G S such that the maximal solution u G C(U,H1(RN)) of (4.1.1) satisfies \\7{-t)u(t) - 0 as t —*• +oo. (ii) For every u~ G S, there exists a unique ip G £ sttc/i that i/ie maximal solution u G C^tf^R^)) o/ (4.1.1) satisfies \\7(-t)u(t) - -+ 0 as £ —* —oo. proof. We prove (i), the proof of (ii) being similar. The idea of the proof is to solve equation (7.4.2) by a fixed-point argument. To that end, we introduce the functions uj(t) = 7(t)u+ and z(t) = (x + 2itV)uj(t,x). Let (g,r) be the admissible pair such that r = a + 2. It follows from Strichartz's estimates and Corollary 2.5.4 that u G L«(R, W^R*)), 2 G L^R.i/rR*)), and that ||o>(i)JUr < C\t\-2/i. Let (7.4.3) if = |M|l9(r,w".*-) + PIUnK,^) + SUP 1*1* IK*)IUr • Given 5 > 0, set J = (S, 00), and let E = {ueI'(/,^1,r(RJV)) :(x + 2itV)u(t,x)Ll-) + sup |t|§ ||u(t)||Lr < 2K) tei 222 7. ASYMPTOTIC BEHAVIOR IN THE REPULSIVE CASE and d(u,v) = \\v - w|| for u,veE. It is easily checked that (E, d) is a complete metric space. Given u € E, \\9(u(t))\\w^' < Cl^CiXI^JI^Ci)!!^,,. < C(2ir)-i-^|jW(i)[fM,1,. , and so by Holder's inequality, \\9(u)\\L'\i,wi-r')^c(2K)a(fs s-T^dsJ " ||w||l,(/,^i.-) since 2a > q — 2. It then follows from Corollary 2.3.6 that J(u) defined by /oo 7(t - s)g{u(s))ds makes sense, that (7.4.5) J(u) e C([S, oo), H 1(RN)) n Lq(I, Wl < C||u(a)||£r ||(x + 2u>V)u(a)||L- , by (7.2.9), one concludes as above that (7.4.7) [x + 2itV)J(u) € C([5, oo), L2(RN)) n Lq{I, Lr(RN)), and that (7.4.8) ||(x + 2itV)J(u)\\Lg{I,Lr) < — for 5 large enough. Finally, it follows from (2.2.4) that /oo I* - spf HsJII^1 ds < C(2K)a+1S1-^^t-i since 2(q 4-1) > q. Therefore for S large enough, (7.4.9) sup {ii \\J(u)(t)\\Lr :t€[S, oo)} < y . Applying (7.4.3), (7.4.6), (7.4.8), and (7.4.9), we deduce that A defined by A(u)(t) = 7(t)u+ + J(u)(t), 7.4. scattering theory in the weighted L2 space 223 maps E to itself if S is large enough. One easily verifies with similar estimates that if S is large enough, then (7.4.10) d(A(u),A{v)) < ^ d(u, v) for all u,v £ E. Applying Banach's fixed-point theorem, we deduce that A has a fixed point u £ E that satisfies equation (7.4.2) on [S, oo). It follows from (7.4.5), (7.4.7), Strichartz's estimates, and Corollary 2.5.4 that u £ C([S, oo),H1(RN)) and that (x + 2itV)u £ C([S,oo),L2(RN)). In particular, ^ = u(S) £ E. Note also that u(£ + S) = T(t)^ +« / T(t - s)p(«(s + S))ds. Jo Therefore, u is the solution of the problem {iut + Au + g(u) = 0 u(S) = 0. Note that, by Remark 6.8.1, the solution u is global. In particular, u(0) is well defined, and by Proposition 6.5.1, w(0) £ E. It follows from equation (7.4.2) that /oo 7{s)g(u(s))ds. Since u £ E, it is not difficult to show with the above estimates that t)u(t) — —* 0 as t —» oo. Therefore, (p = u(0) satisfies the conclusions of the theorem. It remains to show uniqueness. Let —► ip and Q- : u~ h-+ ip are well-defined S —► E. In fact, one can show with similar estimates that ft+ and are continuous. By Theorems 7.4.1 and 7.4.4, U±Q± = £l±U± = I on E, where U± is defined by Remark 7.4.2. In particular, Sl± : E —► E is one-to-one with continuous inverse (f2-t-)_1 = U±. theorem 7.4.6. Assume (7.3.1)-(7.3.2). Assume further that a > ao, where c*o is defined by (6.3.3), and let E be the Hilbert space defined by (6.7.3)-(6.7.4). For every u~ £ E, there exists a unique u+ £ E with the following property. There exists (a unique) ip £ E such that the maximal solution u £ C(R, E) of (4.1.1) satisfies 7(—t)u(t) —> u± in E as t —> ±oo. The scattering operator S : E —► E mapping u~ i—► u+ is continuous, one-to-one, and its inverse is continuous E —» E. In addition, ||u+||L2 = ||u~i|l2 and I|Vw+||l2 = j|Vw~j|£2 for every u~ £ E. proof. The result follows from Theorems 7.4.1 and 7.4.4 and Remark 7.4.5, by setting S = U+Q-. Note that S_x = U-SI+. □ 224 7. asymptotic behavior in the repulsive case Remark 7.4.7. Since 7(t) is an isometry of JJ^R7"), the property \\J(~t)u(t) -i^Htfi —► 0 is equivalent to \\u(i) — 7(t)u±\\fj^ —> 0. In general, it is not known whether the property ||T(-t)w(£)-u±||2 —> 0 is equivalent to i|u(t)-D'(t)w± ||s —► 0. On this question, see Begout [18]. 7.5. Applications of the Pseudoconformal Transformation In this section we consider equation (4.1.1) in the model case (7.5.1) g{u) = \\u\au, where 4 (7.5.2) AgR, 00, (7.5.7) ||V«(*)|li3 = ^ll(y + 2»(l + fi)V)u(a)|||ai (7.5.8) ||Vu(5)||i9 = j||(x - 2f(l - OVMOHJ, . It follows from (7.5.6) and conservation of charge for (4.1.1) that (7.5.9) jt\\v(t)\\L2=0. 7.5. APPLICATIONS OF THE PSEUDOCONFORMAL TRANSFORMATION 225 Moreover, if we set £i(t) = i||Vt>(0ll£a - (i -O^^IWOIlSi, £?a(*) = (l-*)-»-£?i(0 = (i-*)^i||Vv(t)|iia-^Ht)||j2,, E3(t) = ilKar - 2i(l - t)V)i;(i)||£a - (1 - 0^~^IK0llj53, it follows that (7.5.10) ftEl{t) = -(1 (7.5.11) |ft(«) = (l- (7.5.12) !*w-o. 4 — Na X .. , .,.„19 4, V«(*)|| 2 L2 ' Indeed, (7.5.10) and (7.5.11) are equivalent, and both are equivalent to the pseu-doconformal conservation law for u, by (7.5.6) and (7.5.7). Similarly, the identity (7.5.12) is equivalent to the conservation of energy for u, by using (7.5.6) and (7.5.8). The results that we present in this section are based on the following observation. proposition 7.5.1. Assume (7.5.1)-(7.5.2) and let E be defined by equations (6.7.3)-(6.7.4). Let u E C([0, oo),E) be a solution of equation (4.1.1) and let v G C([0,1),E) be the corresponding solution of (7.5.5) defined by (7.5.3). It follows that 7(—s)u(s) has a strong limit in E (respectively, in L2(RN)) as s —> oo if and only ifv(t) has a strong limit in E (respectively, in L2(RN)) as t f 1, in which case (7.5.13) lim T(-s)u(s) = el * T(-l)w(l) in E (respectively, in L2(RN)). s—>oa proof. We define the dilation Dp, ft > 0, by Dpu(x) = j3%u(px) and the multiplier Ma, a G R, by Mffu(x) = e%~^~u(x). With this notation, and using the explicit kernel 7{t)u = / eii£^Lu(y)dy, (47Tt£)T J elementary calculations show that (7.5.14) V(t)Dp = Dp7(02r) for all r G R and (3 > 0, and that (7.5.15) T(r)MCT = M^_D_i_T( , T ) for all r, o G R such that 1 4* o~t > 0. We note that by (7.5.3), v(t)-M__i_D^u( —— ) for 0 < t < 1. 226 7. ASYMPTOTIC BEHAVIOR IN THE REPULSIVE CASE Therefore, we deduce from (7.5.14) and (7.5.15) that 7{-t)v(t) = M-&(-j^y(j±-^ for all £ G [0,1), which we rewrite in the form *i2_ / s 1(-s)u(s) = e'-r-T ---)v l + sj \l + sj Hence the result is established. □ The following result implies that if a < 2/JV, then no scattering theory can be developed for equation (4.1.1) (see Barab [17], Strauss [322, 325], and Tsutsumi and Yajima [348]). Theorem 7.5.2. Assume (7.5.1)-(7.5.2) and let S be defined by (6.7.3)-(6.7.4). Assume further 2 a < — (a < 1 ifN = 1). Let (fi G S and let u G C(R, £) 6e £/ie corresponding solution of (4.1.1). If(p^0, then 7(—t)u(t) does not have any strong limit in L2(RN) as either t —» oo or t —»■ —oo. 7n oi/ier words, no nontrivial solution of (4.1.1) has scattering states, even for the L2(RN) topology. Proof. We consider the case £ —► co, the argument for £ —* —oo being the same. We argue by contradiction and we assume 7(—t)u(t) —► u+ in L2(RN). In $—►00 particular, ||«+||La = \\u(t)\\L2 = H|L2 >o. On the other hand, we deduce from (7.5.13) that v(t) —► w in L2(RN) with t j"l w = 7{l)(e~ii^-u+) ^ 0. Since q + 1 < 2, we have \v(t)\av{t)—>\w\aw^0 inLsrr(RJV). tf 1 Let 9 G V(RN) be such that (7.5.16) (i\w\aw,8) = l It follows from (7.5.5) that jt(v(t),e) = (iAv,$) + A(l - £)^ ^[A|(l - i)-2^ - C for 1 - e < t < 1, £ > 0 small enough. Since (JVa - 4)/2 < -1, it follows that |{v(£),0)| -» oo as t \ 1, which is absurd. □ 7.5. APPLICATIONS OF THE PSEUDOCONFORMAL TRANSFORMATION 227 Remark 7.5.3. In the case N = 1 and 1 < a < 2, we have the following result. Let ip G E and let u G C(R, E) be the corresponding solution of (4.1.1) with g(u) = X\u\au. If ip 7^ 0, then T(-t)u(t) does not have any strong limit in E as either t —> oo or i -> — oo. The proof is similar. One needs only observe that, since v(t) is bounded in E (hence in H1(R)), v(t) —► u? as £f 1 in LP(R) for every 2 < p < oo, and so |t;(i)|Qt;(£) -» |wjQw as t T 1 in L2(R). If a > 2/N and if A < 0, then every solution in E of (4.1.1) with g(u) = A|u|Qw has scattering states in L2(RN), as the following result shows (see Tsutsumi and Yajima [348]). Theorem 7.5.4. Assume (7.5.1)-(7.5.2) and let E be defined by (6.7.3)-(6.7.4). Let ip G E and let u £ C(R, E) be the corresponding solution of (4.1.1). If X < 0, then there exist u± G L2(RN) such that 7(-t)u{t) —> u± inL2(RN). t—»±oo Remark 7.5.5. Here are some comments on Theorem 7.5.4. (i) If a > Qo? then it follows from Theorem 7.4.1 that u± G E and that the convergence holds in E. The same conclusion holds in some other situations: if a = ao, see Theorem 7.5.11; if a > 4/(iV+2) (a > 2 if N ~ 1) and if is small enough, see Theorem 7.5.7. On the other hand, if a < 4/(iV + 2), or if a < ojo and is large, then we do not know whether u± G E. (ii) Theorem 7.5.4 does not apply to the case A > 0. In fact, if a < 4/(AT + 2), there are arbitrarily small initial values

> 0, set oo in L2(WN). On the other hand, one easily verifies that if a < 4/(N + 2), then |Juw||£ —» 0 as a; J, 0. However, we will see below (Theorem 7.5.7) that if a > 4/(N + 2), then small initial values in E have scattering states in E at ±oo. Proof of Theorem 7.5.4. By Proposition 7.5.1, we need only show that v(t) has a strong limit in L2(RN) as t | 1- As observed above (Remark 7.5.5), there is a better result when a > ao. Therefore, we may assume that a < ao, and in particular a < 4/N. Therefore, it follows from (7.5.9) and (7.5.11) that (7.5.17) \\v(t)\\l2)L2\\\L2 J \\Vv\\L2ds + CMLa+2 J\l - s)t\\v\\aLtl2 ds < C||V0||L2 jT(l - s)^ ds + CU\\La+2 j\l - s)^ ds, by (7.5.19) and (7.5.18). Letting r | 1 and applying (7.5.20), we obtain (l-s)11^ ds + C\\i;\\La+2 J (l-s)Ej^Ads < C(l - t)^ || W||L» + C(l - t)^U\\La+2 . [w-v(t),v(t))L2\ 0, neither does it allow us to construct the wave and scattering operators, since the initial value (f and the scattering states u± do not belong to the same space. We will improve this result under more restrictive assumptions on a by solving the initial-value 7.5. APPLICATIONS OF THE PSEUDOCONFORMAL TRANSFORMATION 229 problem for the nonautonomous equation (7.5.5) and by applying Proposition 7.5.1 which relates the behavior of u at infinity and the behavior of v at t = 1. However, we want to solve the Cauchy problem for (7.5.5) starting from any time t £ [0,1], including t = 1 where the nonautonomous term might be singular. In order to do this, define the function (7.5.22) f(t) = iX{1~t)^ ^ NT2 (<*>2ifN = 1)- It follows that for every to € R and tp £ £, there exist Tm(to,f/>) < to < Tu(to,ip) and a unique, maximal solution v £ C((Tm,TM)^) of equation (7.5.23). The solution v is maximal in the sense that if Tm < oo (respectively, Tm > — oo), then \\u(t)\\ffi —► oo as 1f Tm (respectively, 11 Tm). In addition, the solution v has the following properties. (i) IfTM = I, ^enliminftTi{(l-i)<5||u(i)||Hi} > 0 vrithS= if N>3, S < 1 - i if N = 2, and 6 = ± - ± if N = 1. (ii) The solution v depends continuously on tp in the following way. The mapping ip h-> Tm is lower semicontinuous £ —> (0, oo], and the mapping ip i—*• Tm is upper semicontinuous H —+ [—oo,0). In addition, if ipn —► ip in E as n —► oo and if [S,T] £ (Tm,TM), then vn -* v in C([S,T],£), tu/iere vn denotes the solution of (7.5.23) with initial value ipn. Proof. The result follows by applying Theorems 4.11.1 and 4.11.2 with h(t) ~ f{t -10). □ We now give some applications of Theorem 7.5.6 to the scattering theory in E for (4.1.1). Theorem 7.5.7. Assume (7.5.1)-(7.5.2) and (7.5.24). Let £ be the space defined by (6.7.3)-(6.7.4). With the notation of Section 7.1 (corresponding to the £ topology), the following properties hold: (i) The sets 1Z± and U± are open subsets of £ containing 0. The operators U± : 1Z± —» U± are bicontinuous bisections (for the £ topology) and the operators Q± : U± —► 1Z± are bicontinuous bisections (for the £ topology). (ii) The sets 0± are open subsets of £ containing 0, and the scattering operator S is a bicontinuous bisection 0_ —► 0+ (for the £ topology). 230 7. ASYMPTOTIC BEHAVIOR IN THE REPULSIVE CASE Remark 7.5.8. Theorem 7.5.7 implies that there is a "low energy" scattering theory (i.e., scattering theory for small initial data) in E for the equation (4.1.1) with g(u) = X\u\au, provided 4/(TV + 2) < a < 4/(7V - 2) (2 < a < oo, if N = 1). As observed before (see Remark 7.5.5), if a < 4/(N + 2) and A > 0, then there is no low energy scattering. Proof of Theorem 7.5.7. Let ip £ E and let u be the corresponding solution of (4.1.1) with g(u) = A|u|Qu. Let v be the solution of (7.5.5) with the initial value ip defined by ip(x) = 1, and that in this case u+ = e% * 7(—l)v(l). Therefore, the open character of TZ+ and the continuity of the operator U+ follow from the continuous dependence of v on tp (property (ii) of Theorem 7.5.6). ' .Iff]? • l^P Let now y £ E, and set w = 7(l)(e~l 4 y), so that y = e*~4~T(—l)w. It follows from Proposition 7.5.1 and Theorem 7.5.6 that y = U+ip for some ip £ E (i.e., y £ U+) if and only if Tm(l,w) < 0. In this case,

0 and a < 4/N, then U± = E. Therefore, the wave operators fi± are bicontinuous bijections E —► 1Z±. Proof. Assume A < 0, or A > 0 and a < 4/7V. Let w £ E, and let z be the solution of (7.5.5) with the initial value z(l) = w. By Theorem 7.5.6, z is defined on some interval [1 - e, 1] with e > 0. Set <£(y) = e^e*£L*Lz(l — e,ey) £ E. Let u be the solution of equation (4.1.1) with the initial value (7.5.25) u(^-^j = 0 and a < 4/7V, we obtain that u is global. Therefore, we may ■Mi define

cvo with do defined by (6.3.3), then U± = 1Z± — £. In particular, U±, f2±, and S are bicontinuous bijections £ —> £. Here, we use the notation of Section 7.1 (corresponding to the E topology). Proof. By Theorem 7.5.9 and Remark 7.1.3, we need only show that 1Z+ = E. Let ifi g E, let u be the solution of (4.1.1), and let v be defined by (7.5.4). If a > A/N, then it follows from (7.5.10) that Ei(t) is nonincreasing, which implies that ||Vu(t)||z,2 is bounded as t | 1. Since ||^(£)||l2 is also bounded by (7.5.9), we deduce that Hv^lltf1 ls bounded as 11 1. By Theorem 7.5.6, this implies that v(t) has a limit as 11 1, and so

(£)||LQ+2 remains bounded as £ t 1- Set r = a + 2, and let (q,r) be the corresponding admissible pair. Given 0 Qo, / g L^(0,1). Therefore, if we choose to sufficiently close to 1 so that C\\f\\L_^^ < 1/2, then we deduce from (7.5.26) that IMlL~((to,t),.*P) + \\v\\L*uto,t),w*->-) < 2C||u(t0)||jfi for all t0 < t < 1. Therefore, v remains bounded in if1(RiV) as t f 1, and one concludes as above. □ Finally, we extend the asymptotic completeness result to the case a = ocq (see Cazenave and Weissler [72]). theorem 7.5.11. Assume N = 1 or N > 3, let E be defined by (6.7.3)-(6.7.4), and let Qo be defined by (6.3.3). If g(u) = X\u\au with X < 0 and a = ciq, then U± = 7Z± — E. In particular, U±, Q±, and S are bicontinuous bijections E —> E. Here, we use the notation of Section 7.1 (corresponding to the E topology). proof. By Theorem 7.5.9 and Remark 7.1.3, we need only show that 71+ — E. Let ip g E, let u g C(R, E) be the solution of (4.1.1) with g(u) = X\u\au, and let v 232 7. ASYMPTOTIC BEHAVIOR IN THE REPULSIVE CASE be defined by (7.5.4). Note that, since u is defined on [0, oo), v is defined on [0,1). By Proposition 7.5.1,

3 and N — 1. Case N > 3. By (7.5.28) and property (i) of Theorem 7.5.6, \\V*(t)\\h > ~-:Wr (l-i)"-s—« for some constant a > 0 and all t e [0,1). By applying (7.5.11), we obtain iw) < 6 for some constant b > 0. Since a = ao, the above inequality means ■E2(t) < dt ^ ' ~ (1 -t) ' which implies that E2(t) —> —oo. This is absurd, since E2(t) > 0. This completes the proof in the case JV > 3. Case N = 1. The argument is the same as above, except that we first need to improve the lower estimate of the blowup given by property (i) of Theorem 7.5.6. We claim that (7.5.29) Ht)\\Hi > (Q-2)(Q + 4) (1 — t) 4o for some constant a > 0 and all t e [0,1). Indeed, note first that by (7.5.11), jtW)ti). Since \\f\\mt0M\ ^ ll/lliKto.D ^ C{1 ~ we obtain (7-5.29). We now con- clude exactly as in the case N >3. This completes the proof. □ REMARK 7.5.12. Here are some comments concerning Theorem 7.5.11. (i) The conclusion of Theorem 7.5.11 holds in the case N = 2, but the result was established by a different method. See Nakanishi and Ozawa [263]. (ii) If 4/(N + 2) < a < a0, then we do not know whether 71+ -71- = £. Showing this property amounts to showing that no solution of (7.5.5) can blow up at t = 1. Remark 7.5.13. Ginibre and Velo [130] extended the construction of the wave operators (Theorem 7.5.7) to a wider range of ct's by working in the space Hs(RN)n Jr(Hs(RN)), where 0 < s < 2. The lower bound on a for that method is given by / i 4 2 1 a > max < s - 1, ———, — > . \ N + 2s N J If N < 3, one obtains the lower bound a > 2/N by letting s = 3/2. If N > 4, there is still a gap between the admissible values of a and the lower bound a > 2/N for the scattering theory given by Theorem 7.5.2. See also Nakanishi and Ozawa [263] for related results. 7.6. Morawetz's Estimate This section is devoted to the proof of Morawetz's estimate, which is essential for constructing the scattering operator on the energy space. See Lin and Strauss [230], and Ginibre and Velo [137, 138, 143]. We begin with the following generalized Sobolev's estimates. Lemma 7.6.1. Let 1 < p < oo. If q < N is such that 0 < q < p, then ^j," 6 L1(RN) for every u £ W1'p(RN). Furthermore, (7.6.1) J JH^C dx < {jfTa) VlljrilVullk for every u £ W^{RN). Proof. By density and Fatou's lemma, we need only establish (7.6.1) for u € V(RN). Let z(x) = \x\~qx. We have V • z — (N - q)\x\~q. Integrating the formula |u|PV - z = V • (\u\pz) - p\u\p-x V|u| 234 7. asymptotic behavior in the repulsive case ^)ip ^ ^ „ f \u(x)r'\vu(x)[ dx over the set {x G RN \ \x\ > r > 0}, we obtain that (7.6.2) (/V-g) f l4^dx

r} Fl9 Al*l>r} \Xr Applying Holder's inequality, it follows that J{\x\>r} \x\q \J{\x\>r} \X\Q / Since r > 0 is arbitrary, (7.6.1) follows. □ Corollary 7.6.2. If N > 4, then G L1{RN) for every u G H2{RN). Furthermore, there exists C such that (7.6.3) f ^j^- dx < C\\u\\2H2 for every u € H2(RN). J Fl Proof. Note that it suffices to establish (7.6.3) for u G T>(RN). Applying (7.6.2) with q = 3 and p = 2, we obtain |2 ,N_3)f J»W!l(fa<2/- M')\ |v«(«)l^ -/{M><-} M3 V{|»|>r} M M <2(7 M^W/ W{|*|>r} Fl / \i{|xl>r} Fl '{|x|>r} F|2 / \./{|z|>r} M Applying (7.6.1) with p = q — 2 to both u and Vtx, we obtain (N-S) f J{\x\>r} \u(x)\2 \x\3 dx 3, and we consider - yw + /(«(•)) + (W* |w|2)w, where V, f, and W are as follows. The potential V is real valued such that V, VV G L^R^) + L°°(RN) for some p > N/2. The function / : [0, oo) -* M is continuous and /(0) = 0. We assume that there exist constants C and a G [0, j^2~) such that - f(u)\ < C(l + \u\a + \v\a)\v - u\ for all u, v G M, and we extend / to C by setting f(z) = I-f(\z\) for all z GC, z^0. N We set rl*l F(z) = / f(s)ds for all z G C . Finally, W : RN —> R is an even, real-valued potential such that W, VW G L9(RN) + L°°(RN) for some q>l,q> N/4. 7.6. MORAWETZ'S ESTIMATE 235 We know (see Section 3.3) that g is the gradient of the potential G defined by G(u) = J ^V(x)\u(x)\2 + F(x,u(x)) + ^(W*\u\2)(x)\u(x)\2^dx. Finally, we set ~" E(u) = 1 j \Vu(x)\2dx - G(u) for all u £ H\RN). For u £ ifX(R ), we set (7.6.4) h(u) = \vr\u\2 + H^l{2Fiu) - uf(u)} + \\u\2^ • (W* \u\2). If u is such that h(u) £ LX(RN), we set (7.6.5) H(u) = J h(u)dx. We will use the following estimate. Lemma 7.6.3. Let N > 3, let g be as above, and set p = max{a + 2, ^r}- If h and H are defined by (7.6.4) and (7.6.5), then h(u) £ Ll(RN) for every u £ if ^R^) H W1'P(RN). Furthermore, there exists C such that (7 6 6) IH{V) ~ H(U)1 ~ C(1 + Mm + Mhi)U+2 x (||u||H1 + \\u\\Wi,P + \\v\\m + - u\\Hi for all u,v e H1(RN)nWl'f Proof. Let us write Vr - Vi + V2, where Vx £ Lt(Rn) and V2 £ L°°( WW = Zi + Z2, where Zx £ L<>(RN) and Z2 £ L00^); and f = fi + /2, where /i is globally Lipschitz continuous and \f2{v) - f2(u)\ < C(\u\ + \v\)Q\v - u\ for all u, v £ C. Set i(u) = |u|2~ • {Zi * |w|2) and ipi(u) = ~{2Fi(u) - ufi(u)} for % = 1,2. Consider w,t; £ V(RN). We have y |VH|«|2 - |u|2)| < j |Vl|(H + |u|)|t;-«| < CWVtWrf (\\v\\L7^ + Hull^JJIv - u\\l^ < C(\\u\\Hi + \\v\\H*)\\v - u\\w Also, J\V2(\v\2 - \u\2)\ < C(\\u\\L2 + HlOIIt; - u\\L2. 236 7. asymptotic behavior in the repulsive case Finally, j ivi(v) - i&iWi < c J ^L±^b - «i and Applying Holder's inequality and (7.6.1), we deduce that J hk(t>)-ife(tOI R* < C{\\v\\La+2 + \\u\\La + 2)a(\\Vv\\La+2 + ||Vu||La + 2)||v - U\\La + 2 < C(l + \\u\\m + \\v\\Hi)a+2(\\u\\Hi + \\u\\Wi,P + \\v\\Hi + \\v\\wx,P)\\v - U\\H1 . Also, J - ^i(u)| < C(||Vt;||L3 + ||V«||L»)||v - u||La. One obtains the same inequalities for 4>\ and 2 by applying Young's and Holder's inequalities. Therefore, (7.6.6) holds for all u, v G T>(RN). The result now follows easily by density. □ We are now in a position to state and prove the main result of this section. Theorem 7.6.4. (Morawetz's estimate) Assume N > 3 and let g be as in Lemma 7.6.3. If

Tmax),H2(RN)) nC1((-rmin,rmax),L2(RiV)). Therefore, the equation (4.1.1) makes sense in L2(RN) and we may multiply it by ur + (N — \)u/2r g C((-Tmln,Tm(iX),L2(RN)) (by Lemma 7.6.1). Therefore, / JV-1 \ (7.6.8) [iut + Au + g(u), ur + -^-uj ^ = 0 on (-Tmin, Tmax). We claim that (■ N-l \ Id' (7.6.9) {iut,ur + —^—uJ = - —(iMjUr)L2 • 7.6. MORAWETZ'S ESTIMATE 237 Indeed, by density, we need only establish the identity (7.6.9) for smooth functions u. In this case, it follows from integrating the identity Re \iut (ur H--2r^~} = ^e^u"r) + \^ ' (~ Re(iutu)j ■ We also claim that (7.6.10) ^Au,ur + N^ Xuj <0. Again by density, we need only establish (7.6.10) for u £ V(RN). Note that in this case, Re {Au(ur + } = V ■ Re {vu^ + - |;!V4, b= if/V>4. Note that b is well defined by Corollary 7.6.2. Inequality (7.6.10) follows immediately. Furthermore, (7.6.11) (vu,ur + ^^v)j ^ = -1 j Vr\u\2 . Also, we need only establish (7.6.11) for u £ V(RN). In this case, (7.6.11) follows from integrating the identity Note also that / JV-1 \ f N - 1 (7.6.12) (/(«),«r + a = - y -^{2F(W) - «/(u)} . Note that we need only establish (7.6.12) for u £ V(RN). In this case, (7.6.12) follows from integrating the identity Re |/(u)(ur + ^-u) 1 = V • (*^) ~ ^imu) - uRu)} . 238 7. asymptotic behavior in the repulsive case Finally, we claim that (7.6.13) (^(W*\u\2)u,ur+^^u^ =~\f \u\2-~-(VW*\u\2). L RN Also, we need only establish (7.6.13) for u £ V(RN). In this case, Re{(W*|W|2)W(V + Equation (7.6.13) is obtained by integrating the above equality. We deduce (7.6.7) from formulae (7.6.8)-(7.6.13). Step 2. Let tp £ H1(RN), let u be the corresponding maximal solution of (4.1.1), and let -Tmin < s < t < Tmax. Consider a sequence f H{u{T))dr. Js m-*°° Js The result now follows by applying Step 1. □ Corollary 7.6.6. Assume N > 3 and let g(u) = — r)\u\au for some 77 > 0 and 0 < a < 4/(JV - 2). For every

\B>N In addition, u(t) —0 in Hl(RN) as t —► ±00. Proof. It follows from Remark 6.8.1(i) that the solution u is global and bounded in H1(RN). Applying (7.6.7), we obtain a X ( Ma+2 U[SlX)l dxds < C{\\u(t)\\2Hr + \\u(-t)fH1) < C. Hence (7.6.14) follows by letting t | 00. In order to show the weak convergence to 0, we need to verify that for every i\> £ V(RN), (u(t),ift) —> 0 as t —* ±00. Note that \{u(tu)\< J Mt)\M< f ^.\x\^\ 3. (If TV = 1,2, then singular terms appear in the proof of (7.6.10).) This is the reason why the scattering theory in the energy space (the asymptotic completeness part) was developed only for TV > 3. Recently, Nakanishi [259] obtained a substitute for Morawetz's estimate in any dimension. More precisely, in the model case g(u) = — rj\u\au with n > 0 and 0 < a < 4/{N - 2) (0 < a < oo if TV = 1,2), (i + ^^)l*+4»p|-Wlli.; This is obtained by taking the scalar product of the equation with pu + T ■ Vit, where r(£, X) = —7====== , pit, X) = -. -\--r- . Note that (7.6.15) is weaker than (7.6.14) particularly because of the time dependence in the factor of |ti|Q+2. It is sufficient, however, to deduce the asymptotic completeness in dimensions N = 1 and N = 2 under the assumption a > 4/N, i.e., the analogue to Theorem 7.8.1. See [259]. Note that the proof, however, is much more delicate than the proof of Theorem 7.8.1. in particular, (7.6.15) 7.7. Decay of Solutions in the Energy Space Throughout this section we assume that TV > 3. We apply Morawetz's estimate to the study of the asymptotic behavior of solutions. For simplicity, we restrict our attention to the model case (7.7.1) 9(v) = ~vHau, where 4 (7.7.2) > 0, 0 3 and assume (7.7.1)-(7.7.2). If (7.7.3) a > 1, tfzen /or ewery <£> € H1^), tfie maximal solution u G C(R, iJ^R^)) o/ (4.1.1) satisfies 2N (7.7.4) [|**(*)IUr —> 0 for every 2 0, the case t < 0 being treated similarly. Note that we need only establish (7.7.4) for r = a+ 2, since the general case follows immediately from the boundedness of the solution in Hl(RN) and Holder's inequality. We proceed in several steps. Step 1. The estimate (7.7.5) / \u(t,x)\a+2dx —> 0 J{\x\>t\ogt} t^+co holds. Indeed, let M > 0 and let f tel if \x\ < M W I 1 if |z|>M so that 6M G W^°°(Rn) and |[Wm||l~ < Thus 6Mu g C(R,H^R^)) and (iut +Au +g(u),idMu)H-i,Hi =0. Note that and 1 d I {iut,i6Mu)H-i!Hi = - — / 9M\u\2 , (g{u),i9Mu)H-i,Hi = 0, = - Re J i iuVu ■ V6m 1......n C 7.7. decay of solutions in the energy space 241 It follows that j 6M\u(t,x)\2 dx < ^ + J $M\(p\2 dx for every £ Letting M = t log £, we obtain J \u{t,x)\2dx < + J et\ogt\if\2 dx, {|x|>tlogt} rn Applying the dominated convergence theorem to the last term in the right-hand side of the above estimate, we obtain that / \u(t,x)\2dx —► 0, J t—++oo {\x\>t\ogt} The result now follows from Holder's inequality and the boundedness of u in H1' Step 2. For every e > 0, t > 1, r > 0, there exists to > max{£, 2r} such that (7.7.6) f° f |u(s,x)|a+2dxds < s. {\x\ / —- / \u(s,x)\ Vi slogs J {|x| 0 for which ft+2(fc+l)r / / Ks,x)[Q+2<£ Jt+2kT J H+2kT {\x\ 0, a, b > 0, there exists £o > max{a, 6} such that (7.7.7) sup j|w(s)||Lo+2 < e. s£[to—b,t0] 242 7. asymptotic behavior in the repulsive case Consider t > r > 0, and write (7 ? g) u(t) = 7(t)ip + * jT j(t _ s)fl(«(s))da + »y" T(t - a)0(u(s))cfa = u(£) + iu(*, r) + z(i,r). It follows from Corollary 2.3.7 that (7.7.9) \\v(t)\\La+2 —» 0. t—»oo Let now f oo if a > 1 P=It\ ifa 1. Therefore, we deduce from (2.2.4) that IH*,r)||LP< /" \t-s)-N(^)\\u(s)\\ltl+1)p/ds JO -1) for all t > r > 0. On the other hand, note that w{t, r) = 7{r)u(t - t) - 7{t)y, and so it follows from conservation of charge that (7.7.11) M*,t)jjL2 <2|M|L2. Applying (7.7.10), (7.7.11), and Holder's inequality, we deduce that there exists K such that (7.7.12) \\w(t, r)|jLQ+2 < Kr ^+^> for all t > r > 0. Finally, by (2.2.4), (7.7.13) \\z(t,r)\\La+2 < / (t-s)-^\\u(s)\\aLtl2ds. Jt-r Note that Na < 2(a + 2), and let p <= (1, 2(^.+2)). It follows in particular that (a + l)p' > a + 2. Applying (7.7.13), Holder's inequality, and the boundedness of u in La+2(MJV), we obtain 7.7. decay of solutions in the energy space 243 for some 5, p: > 0. In particular, there exists L such that \\z(t,r)\\Lo+2 t-T J J {\x\>s\oes\ (7.7.14) t ^ +LtS{1 I \u(s>x)\a+2dxds^ ■ {\x\ max{a, 6} such that (7.7.15) \\v(t)\\La+2 <~ for t > tx. Next, let Ti > 6 such that (7.7.16) IMt,7i)||L-+2 < \ fort>0, which exists by (7.7.12). By Step 1, there exists t2 > t\ such that (7.7.17) Lr?+S ( sup / \u(s,x)\a+2dxY < -A for t > t2 • \«>t-n J J 4 {|x|>s log s} Finally, by Step 2, there exists to > t2 such that (7.7.18) lT*{Jt° J Hs,x)\a+2dxdsy < |. {|x| 0, |ju(t)||La+2 < e for t large. Let t > t > 0. It follows from (7.7.8) and (7.7.12) that No-2 inax{a,l} (7.7.20) |t«(t)||x,Q+2 < \\v(t)\\la+2 + Kt--2fz+2) + \\z(t,r)\\la+2 . Consider e > 0, and let re be defined by _ Na — 2 max{n,l) (7.7.21) Kre 2(a+2) = -. We deduce from (7.7.9) that there exists t\ > 0 such that (7.7.22) \\v(t)\\la+2 < | fort^h. Applying (7.7.20), (7.7.21), and (7.7.22), we obtain (7.7.23) \Ht)\\L^ < | + \\z{t,r£)\\La+2 for t > tx. 244 7. asymptotic behavior in the repulsive case Note also that by (2.2.4), ||*(*,re)HL.+3< / (t-s)-^\\u(s)\\lil2ds (7-7.24) Jt~Ts r£ . t-Te,t By Step 3, there exists to > max{r£,£i} such that (lu^H^a+a < £ for t £ [t0 — re,t0]-Therefore, we can define tE = sup{£ > t0 : \\u(s)\\la+2 < £ for all s £ [t0 ~r£,t}}. Assume that t£ < oo. It follows that (7.7.25) \\u(t£)\\Lo+2 =e. Applying (7.7.23) and (7.7.24) with t = t£, we obtain that which implies 1 No r£ ^^£a> 2M Applying (7.7.21), we see that (7.7.26) rp > 2M(4K)<* ' where - a(Na ~ 2 - 2 max{q, 1» + (Na - 4) 7 ~ ~~ 2(a + 2) ' Observe that when a < 1, we have 7 > 0 (remember that Na > 4). Therefore, (7.7.26) implies that r£ is bounded by a positive number. This is a contradiction when £ is small, since r£ —» 00 as £ j 0. When a > 1, one easily verifies that 7 > 0 when N > 4, or when N — 3 and a > v/^|,~1-, in which case we obtain the same contradiction. Therefore, t£ = 00, which is the desired estimate. □ Theorem 7.7.3. Let N > 3 and assume (7.7.1), (7.7.2), and (7.7.3). For every if £ H^R^), toe maximal solution u £ C(R, ^(R^)) 0/ (4.1.1) satisfies (7.7.27) :u e L9(R, W1,r(RJV)) /or every admissible pair (q,r). For the proof, we will use the following elementary lemma. Lemma 7.7.4. Let a,b > 0 and p > 1. Assume that b is small enough so that the function f(x) = a — x + bxp is negative for some x > 0, and let xq be the first (positive) zero of f. Let I c R be an interval and let £ C(I, R+) satisfy (t) (to) = 0 (or more generally (to) < xq) for some to £ I, then (t) < xo for all tel. 7.7. DECAY OF SOLUTIONS IN THE ENERGY SPACE 245 proof. By assumption, the set J = {x > 0;f(x) > 0} is of the form J = [0, y] U [z, oo) for some 0 < y < xq < z. Since { °- we must nave either {(f)(t) : t G 7} C [0,y], or else {<£(*) : t G 7} C [2,00). This proves the result. □ Proof of Theorem 7.7.3. Let (7, p) be the admissible pair such that p = a+2. For every S, t > 0, u(t + S) = 7(t)u{S) + i f 7{t - s)g{u{S + s))ds. Jo It follows from Strichartz's estimates, Remark 1.3.1(vii), and Holder's inequality that for every t > S > 0, / r* * ' \1/7' l|ti||L,((s,t))^-) < c\\u(S)\\m + c[j N*)H£ ll«(*)H^.pj 7, and so < sup{||w(s)||x,P : s > S}a+1~^\\u\\%{{sthwl.p). It follows from Theorem 7.7.1 that (^V(^)ii/:/1)y"7ii^)iii;7'ii^)ii^P)1/7 P(RN)), and so u G L'f(R,W1'P(RN)). This implies that g{u) G W1'P'(RN)), and the result follows from Strichartz's estimates. □ REMARK 7.7.5. One can add the following property to the statement of Theorem 7.7.3. If

3 and that g is given by (7.7.1)-(7.7.2). We refer to Section 7.9 and to Ginibre and Velo [137, 138] for more general results, and to Nakanishi [259] for the case N = 1,2. We first construct the scattering states. Theorem 7.8.1. Let N > 3 and assume (7.7.1), (7.7.2), and (7.7.3). If

u+ in Hl as £ —» co. One shows as well that there exists u~ e if1(RJV) such that —> u~ in il1 as £ —> — oo. The other properties follow from conservation of charge and energy. □ Remark 7.8.2. The mappings U+ : ip >-» u+ and t/_ : ip i-* u~ denned by Theorem 7.8.1 map H1(RN) —»■ if1(RAr). In fact, one can show with similar estimates that U+ and i/_ are continuous H^R1*) -» tf1! Remark 7.8.3, We deduce from Corollary 2.3.6 the following formula: »±oo u± = = ip + i T(-s)g(u(s))ds; Jo in particular, /±oo 7(t - s)g(u{s))ds for all* € R. We now construct the wave operators. 7.8. SCATTERING THEORY IN THE ENERGY SPACE 247 Theorem 7.8.4. Let N > 3 and assume (7.7.1), (7.7.2), and (7.7.3). (i) For every u+ E Hl{RN), there exists a unique if E i?'1(RAr) such that the maximal solution u € C'(R, H1(RA')) of (4.1.1) satisfies \\1(-t)u{t) - u+\\Hi -> 0 as t -* +oo. (ii) For every u~ E 7X1(RAr); there exists a unique

-co. Proof. We prove (i), the proof of (ii) being similar. The idea of the proof is to solve equation (7.8.1) by a fixed-point argument. To that end, we introduce the function w(t) = 7(t)u+. Let (q,r) be the admissible pair such that r = a + 2. It follows from Strichartz's estimates and Corollary 2.3.7 that u E L9{R, Whr(RN)) and that ||u;(t)||£,r —+ 0 as t —+■ oo. Consider S > 0 and let (7.8.2) Ks = ||w||i,((5,oo),wn.") + sup ||w(t)||Lr . Note that (7.8.3) #5—>0. Let £ = {u € L*((S, oo), W^R")) : ||w|U,((Sloo),wi.o + sup |M*)IU- < 2^s} , and d(w. v) = ||v - u||l9((s,oo),l«-) for u, v E £. It is easily checked that (E, d) is a complete metric space. Given u E E, we have (see the proof of Theorem 7.7.3) lb(«)llL.'((s>oo),Wi^) S 248 7. ASYMPTOTIC BEHAVIOR IN THE REPULSIVE CASE maps E to itself if S is large enough. One easily verifies with similar estimates that if S is large enough, one has (7.8.8) d(A(u),A{v)) < i d(«, v) for all u,v E E. It follows from Banach's fixed-point theorem that A has a fixed point u G E, which satisfies the equation (7.8.1) on [S,oo). Note that u G C([S,oo), H1(RN)) by (7.8.5); in particular, ip = u(S) G if 1(MiV). Note also that u(t + 5) = T(£)^ + i [ T(t- s)g(u(s + S))ds. Jo Therefore, u is the solution of the problem {iut + Ait + g(u) = 0 u(S) = V- Note that, by Remark 6.8.1, the solution u is global. In particular, u(0) G H1(RN) is well defined. It follows from the equation (7.8.1) that /oo 7{s)g(u(s))ds. Since u G it is not difficult to show with the above estimates that \\T(—t)u(t) — u+\\m —+ 0 as t ~*■ 00■ Therefore, (p = m(0) satisfies the conclusions of the theorem. It remains to show uniqueness. Let ipi, oo for j = 1,2. It follows from Remark 7.8.3 that u3 is a solution of (7.8.1). Furthermore, it follows from Theorems 7.7.1 and 7.7.3 that u3 G Lq(R, W1'r(RN)) and that ||uj(£)IUT' ^ 0 as £ —> oo. In a similar way to the proof of (7.8.8), one obtains that u\(t) = u2{t) for t sufficiently large. By uniqueness for the Cauchy problem at finite time, we conclude that tp\ = tp2. □ Remark 7.8.5. Note that the above proof of the construction of ip for a given uq only uses a fixed-point argument. In particular, it still works for TV = 1,2. It is not difficult to see that it also works in the limiting case a = 4/7V. The proof of uniqueness is more delicate and uses the decay estimate of Theorem 7.7.3. This is where we use the assumption JV > 3. As observed in Remark 7.8.9 below, uniqueness also holds in dimension N = 1 or 2. Remark 7.8.6. Nakanishi [260] has extended the existence part of Theorem 7.8.4 to the case a > 2/N when N > 3. The construction is by a compactness argument. Note that when a < 4/7V, uniqueness is an open problem (see Remark 7.8.5). remark 7.8.7. The wave operators Q+ : u+ h-+ tp and f2_ : u~ *-* ip defined by Theorem 7.8.4 map Hl(RN) -* if^R^). In fact, one can show with similar estimates that Cl+ and fi_ are continuous. By Theorems 7.8.1 and 7.8.4, U±Q± = Q±U± = I on if1(RN), where U± is defined by Remark 7.8.2. In particular, £l± : if^R^) -> H1(RN) is one-to-one with continuous inverse (fi±)_1 = U±. Theorem 7.8.8. Let N > 3 and assume (7.7.1), (7.7.2), and (7.7.3). For every u~ G if1(RAf), there exist a unique u+ G Hl(RN) and a unique (p G H1(RN), such 7.9. COMMENTS 249 that the maximal solution u € C(R,Hl(RN)) of (4.1.1) satisfies T(-t)u(t) -> in H1(RN) as t —► ±oo. The scattering operator S : H1^) -> H^R") mapping u~ ^ u+ is continuous, one-to-one, and its inverse is continuous H1(RN) —» i71(E7V). in addition, ||u+||L2 = ||u~||l3 and [jVw+H^ = ||Vu~||L2 for every u~ e H1(RN). PROOF. The result follows from Theorems 7.8.1 and 7.8.4, and Remark 7.8.7, by setting S = U+Q-. Note that S^1 = □ Remark 7.8.9. We note that the conclusion of Theorems 7.8.1, 7.8.4, and 7.8.8 also hold if N = 1 or N = 2. See Remark 7.6.7 and Nakanishi [259]. 7.9. Comments The estimates of Theorem 7.3.1 hold for more general nonlinearities. In particular, consider g(u) = f(u(-)), where / is as in the beginning of Section 7.2. Assume that F(s) < 0 for all s > 0, and that there exists 0 < 5 < 4/N such that -s~2~6F(s) is a nondecreasing function of s > 0. We have the following result. Proposition 7.9.1. Let g be as above. If ip e H1^1*) is such that j • \ 0 for some 4/N < n < u < 4/(N - 2). Assume finally that (7.9.3) 2F(s) - sf(s) > cmin{s"+2, sv+2} for all s > 0. We have the following result. Theorem 7.9.2. Assume N > 3 and let g be as above. For every p £ Hl the maximal solution u £ C(R,H1(RN)) of (4.1.1) satisfies \U(t)\\Lr —> 0, t—»±00 for every 2 < r < 2N/(N - 2). Proof. The proof is an adaptation of the proof of Theorem 7.7.1. We only prove the result for t —> -f oo, the case t —► — oo being similar. Note also that we need only establish the result for r — u + 2, the general case following immediately from the boundedness of the solution in H1(RN) and Holder's inequality. Step 1. We have the estimate 2N (7.9.4) J |«(t, x)\r dx —► 0 for all 2 < r < N-2 {M>tiogt} The proof is the same as that of Step 1 of the proof of Theorem 7.7.1. Step 2. For every e > 0, t > 1, r > 0, there exists to > max{t,2r} such that (7.9.5) f mm{\u\^+2,\u\v+2}dxdt < e. Jto-2r J {|ar|0, £>l,r>0 and 2 < r < 2N/(N - 2), there exists to > max{£, 2r} such that (7.9.6) / / \u(t,x)\r dxdt < e. Jto-2r J {|x|0, £ > 1, r > 0. Let f u(t,x) if |u(£,x)| < 1 vit.x) — < I 0 if \u{t,x)\ > 1, 7.9. comments 251 and w = u — v. It follows from Step 2 that for every e' > 0, there exists to > max{£, 2t} such that "to ,+2dxdt /° / \v(t,x)r J t0 —It J + f° f Iw^x^^dxdtb^s' Jtn-2r J {\x\1+2dxdt = f° I \v(t,x)\»+2dxdt Jt0~2T J , Jt0~2r J {\x\-2t {\x\ 0 and i, r > 0, there exists i0 > max{£, r} such that (7.9.11) sup \\u{s)\\L^2 < e. s€[to—r,to] Consider t > r > 0, and write (7.9.12) U(t) = J(t)^ + * I 7{t " s^u< r > 0 . 252 7. asymptotic behavior in the repulsive case Let now (7.9,5) ,= (t±Me(2,, + 2] Arguing as in the proof of Theorem 7.7.1, Step 3, one shows easily that there exist L < oo and a, 6, c > 0 such that r/ ft \£> (7.9.16) \\z{t,r)\\Lu+2 0, we have ||w(£)||l"+2 < £ for £ large. Consider e > 0, and let r£ be defined by (7.9.17) *Tr£ 3"<"+ai = |. It follows from (7.9.13) that there exists ti > 0 such that (7.9.18) \\v(t)\\l»+2 < s- fort>ti. Applying (7.9.12), (7.9.13), (7.9.14), and (7.9.17), we obtain (7.9.19) \\u{t)\\Lu+2 < | + ||z(£,te)||L,+2 for t > max{ti,t£}. Note also that, given t > re, we deduce from (2.2.4) that (7.9.20) \\z(t,r£)\\L^ < Crl'^ sup flMljJ1 + ||u||^a) , t — Ts,t where p is given by (7.9.15) (compare the proof of Theorem 7.7.1, Step 4). By Holder's inequality and conservation of charge, (7.9.21) UmII^1 < C\\u\\LV+»2 . Note that i/ — — and so it follows from (7.9.20), (7.9.21), and the boundedness of u in if^R-^) that there exists M such that (7.9.22) \\z(t,T£)\\l»+2 max{re, ti} such that |i«(£)||l^+2 < e for £ £ [to -r£, t0j. Therefore, we may define te = sup{£ > £o : liw(5)||i,^+2 < e for all s £ [to - T£,t}}. Assume that ts < oo. It follows that (7.9.23) ||u(tff)||L„+2 =£. Applying (7.9.22), (7.9.23), and (7.9.19) with t = t£, we deduce that £■ < g + Mre s-"-' 7.9. COMMENTS 253 which implies r€ 2{l'+2)£iLi^—" > 1/2M. Applying (7.9.17), we obtain that (7.9.24) > 1 2M{AKy where _ (fi(v + 2) - 2^)(7V/j - 2 - 2 max{/i, 1}) + y(N(i - 4) 7~ 2fi(v + 2) One can conclude as in the proof of Theorem 7.7.1, Step 4, provided 7 > 0. If /x < 1, then (vt{v + 2)-u)(Nti-4) 7 2/^ + 2) Note that \i > 2/N > vj{v + 2), so that \i{y + 2) - u > 0. Since also Np > 4, we see that 7 > 0. If // > 1, then 7 = ^f^(p - 4>{y)), where #r) = 2 (TV-2):r+ 4 {N-2)(x + 2) ' When N > 4, 0 is nondecreasing, and so 0. When N = 3, (x) is decreasing and (A/(N - 2)) = A/N. Since Nfi > A, there exists v < V < A/(N - 2) such that \i - 0(17) > 0. Observe that / satisfies as well assumptions (7.9.2) and (7.9.3) with v replaced by v. Therefore, in this case also, 7 > 0. This completes the proof. □ Remark 7.9.3. It is not difficult to extend the results of Theorems 7.7.3, 7.8.1, 7.8.4, and 7.8.8 to the case where g is as in Theorem 7.9.2. Therefore, one can construct a scattering theory in H1(RN) for such nonlinearities (see Ginibre and Velo [137, 138]). remark 7.9.4. Concerning the decay of solutions in L00, see Ginibre and Velo [132], Dong and Li [107] (one-dimensional case), Cazenave [57] (two-dimensional case), and Lin and Strauss [320] (three-dimensional case). Remark 7.9.5. When g(u) = \\u\~bu, A e R, a scattering theory can be constructed in a subset of L2(RN), containing, for example, all functions with small I? norm and also all functions in u g L2(UN) such that xu e L2(RN) in the case A < 0 (see Cazenave and Weissler [71] and also M. Weinstein [359] for a related result). A low energy scattering theory can also be constructed in HS(RN); see Nakamura and Ozawa [255]. Remark 7.9.6. When g{u) ~ \\u\au, with A > 0 and a > A/N, it is not difficult to adapt the proofs of Theorems 7.8.1, 7.8.4, and 7.8.8 (by using Theorem 6.2.1) in order to construct the scattering operator S on the set {u e Hl(^RN) : < e} for e small enough. Obviously, the scattering operator cannot be defined on the whole space H1(RN), since some solutions blow up in finite time (see Remark 6.8.1). The assumption a > A/N is optimal (see Cazenave and Weissler [72], Remark 4.4). Remark 7.9.7. The results of Sections 7.3 and 7.4 can be extended to Hartree-type nonlinearities. See Cazenave, Dias, and Figueira [61], Chadam and Glas-sey [76], Dias [103], Dias and Figueira [104], Ginibre and Velo [134], Hayashi [159], 254 7. asymptotic behavior in the repulsive case Hayashi and Ozawa [185, 188, 189, 186], Hayashi and Tsutsumi [194], Lange [221, 220], P.-L. Lions [233, 234], Nawa and Ozawa [271], and Pecher and Von Wahl [296]. The results of Sections 7.6, 7.7, and 7.8 can also be extended; see Ginibre and Velo [143] and Nakanishi [258]. Remark 7.9.8. It follows from Theorem 7.5.2 that if g(u) = X\u\au with AeR and a < 2/N, then no solution of (4.1.1) has a scattering state, even for the L2 topology. This means that no solution behaves as t —> ±oo like a solution of the Schrodinger equation iut + Au — 0. However, it may happen that some solutions behave as t —»■ ±oo like a solution of a different, linear Schrodinger-type equation. This is the theory of modified wave operators. See Ginibre and Ozawa [129], Ginibre and Velo [141, 142, 144], Hayaski, Kaikina, and Naumkin [172], Hayashi and Naumkin [181], Hayashi, Naumkin, and Ozawa [184], Nakanishi [262, 261], and Ozawa [287]. Remark 7.9.9. In the case N = 3 and g(u) = \\u\2u with A < 0, Colliander et al. [90] have shown that the Cauchy problem is globally well-posed in HS(RN) for s > 4/5 and constructed the scattering operator on all of HS(RN). The results are based on a new form of Morawetz's estimate. CHAPTER 8 Stability of Bound States in the Attractive Case In this chapter we study the stability of standing waves of the nonlinear Schro-dinger equation for a class of attractive nonlinearities. Throughout the chapter, we consider the problem (4.1.1) in the model case g(u) = X\u\au where A > 0 and 0 < a < 4/(N — 2) (0 < a < oo if N = 1,2), and we indicate references concerning more general nonlinearities. Without loss of generality, we may assume that A = 1. We have seen in the preceding chapter that when A < 0, all solutions converge weakly to 0, as t —► ±oo. When A > 0, we have a completely different situation. Indeed, in the case a > 4/N, all solutions with small initial data converge weakly to 0 as t —> ±oo, see Theorem 6.2.1; and, on the other hand, it follows from Remark 6.8.1 that solutions with "large" initial data blow up in finite time. In fact, in both the case a > 4/N and the case a < 4/N, we show in Section 8.1 the existence of a third type of solutions, that are global but do not converge weakly to 0. More precisely, we construct solutions of (4.1.1) of the form u(t,x) = eiuJt 4/N a class of standing waves is unstable, and in Section 8.3, we show that when a < 4/N a class of standing waves is stable. We apply purely variational methods, and we refer to Section 8.4 for other methods. 8.1. Nonlinear Bound States Throughout this section, we consider g of the form (8.1.1) g(u) = \u\au with 4 (8.1.2) 0 G R is a given parameter and

0. We begin with a regularity result. Theorem 8.1.1. Assume (8.1.2), a > 0, and b G R. If u G /^(R^) satisfies —Au + au = 6|w|aw in H~1(RN), then the following properties hold: (i) u G W3'P(RN) for every 2 < p < oo. In particular, u G C2(RN) and \D^u{x)\ —> 0 for all \/3\ < 2. |xj—»oo (ii) There exists e>0 such that e£^(\u(x)\ + |Vu(x)|) G L°°(RN). Proof. Changing u(x) to (\b\/y/a)~^u(x/y/a), we may assume that u satisfies (8.1.6) -Au + u = b\u\au with \b\ = 1. Note that (8.1.6) can be written in the form (8.1.7) ^-1((l-f-47r2|e|2)^ru) =b\u\au, where T is the Fourier transform and (8.1.7) makes sense in the space of tempered distributions S'(RN). (i) Note that if u G L^R^) for some a+1 < p < oo, then \u\au G (RN). It follows that u G H2'^(RN) = W2'^+t(Rw) (see Remark 1.4.1). Applying So-bolev's embedding theorem, this implies that (8.1.8) u£Lg(RN) for all q > such that i > _ JL . Consider the sequence q3- defined by (a + 1)J —— - — + g; \a + 2 Na Na{ct + l)3j' Since (TV - 2)a < 4, we see that ^ ~ W = ~^ witn ^ > °- We have 1 A = _(a + i)i<$< s, Qj+i Qj ig a exists k > 0 such that and so -f- is decreasing and -p- —► — oo. Since qo = a + 2, it follows that there Qi "i j—»oo — > 0 for 0 < I < k; —— < 0 . Qe Ofc+i We claim that u G L^fR"). Indeed, u G ff1(RiV) so that u G £*°(RN); and if u G (R*) for some £ < k - 1, then by (8.1.8), u G Lq(RN) for all g > such that - > a + 1 2 1 a + 1 q qe N qe+1 In particular, u G Z/9f+1(RJV). Hence the claim follows. Applying once again (8.1.8), we deduce that u G Lq(RN) for all q > qk/{oc + 1) such that 1/q > 1/qk+i- In 8.1. NONLINEAR BOUND STATES 257 particular, we may let q = oo. Therefore, \u\au g L2(RN) n L°°(RN), so that u g W2'p(RN) for all 2 < p < oo. Applying Remark 1.3.1(vii), we obtain that \u\au g W^P(RN) for all 2 < p < oo. In particular, it follows from (8.1.6) that for every j g {1,...,/V}, (-A + I)6> € LP(RN); i.e., .^((l + 47r2|£|2).F6>) g lp(RW). Thus dju g H2'p(RN) = W2'p(RN) (see Remark 1.4.1). Therefore, u g W3>p(Rn) for all 2 < p < oo. By Sobolev's embedding, u g C2>5(RN) for all 0 < 6 < 1, so that \D0u{x)\ -> 0 as |z| -> oo for all |/?| < 2. It) (ii) Let £ > 0 and 0E|Vu|2 - $£\u\\Vu\. Applying (8.1.9) and Cauchy-Schwarz's inequality, we obtain easily (8.1.10) J e£\u\2<2 J e£\u\a+2. RN RN By (i), there exists R > 0 such that \u(x)\a < 1/4 for jx| > R. Therefore, (8.1.11) 2 J 6£\u\a+2<2 J ela;l|u|Q+2 + ^ J 6£\u\2. RN {\x\ 0, and b g M. If u g Hi(RN) satisfies ~Au + au — b\u\au g H~l{RN), then the following properties hold: (i) ^N\Vu\2 + aJmN\u\2 = bfRN\u\«+2. 258 8. STABILITY OP BOUND STATES IN THE ATTRACTIVE CASE (ii) (Pohozaev's identity) (N-2)J \Vu\2 + Na J \u\2 = f^f |u|«+2. RN MN RN proof. Equality (i) is obtained by multiplying the equation by u, taking the real part, and integrating by parts. The identity (ii) is obtained by multiplying the equation by x ■ Vu and taking the real parts. Indeed, one obtains Re(~Au(x • Vu)) +oRe(«(i ■ Vu)) = bRe(\u\au(x ■ Vu)). Applying the identities TV - 2 / 1 \ Re{-Au(x • Vu)) =--— |Vu|2 + V- f - Re(Vu(z • Vu)) + -a;|Vu|2 J , Re(u(:r • Vu)) = -j\u\2 + ^V • {x\u\2), Re(|u|*u(s • Vu)) = -JL-\u\"+2 + -i-V • {x\u\«+2), a. + Z a + I and integrating over RN yields the result. Note that these calculations are justified by the regularity properties of Theorem 8.1.1. □ Before stating the main result of this section we need to introduce some notation. Assuming (8.1.2) and co > 0, we introduce the following functionals on (8.1.13) T(u) = J |Vu dx, (8.1.14) y(w) = ^T2 / \u\"+2dx-% J \u\2dx> (8.1.15) S(u) = l-T(u) - V(u), (8.1.16) E(u) = X- j | Vu|2 dx - —^ J \u\a+2 dx = S(u) - ~ J \u\2 dx One easily verifies that these functionals are in C1(i71(MAr), R), and that T'{u) -2Au, V'(u) = ju|Gu — cou. We introduce the sets A and G defined by (8.1.17) A = {u G H1^) : u ^ 0 and - Au + u/u = \u\au} (8.1.18) G = {ueA: S(u) < S(v) for all v e A}. We have the following result. 8.1. NONLINEAR BOUND STATES 259 Corollary 8.1.3. Assume (8.1.2) and u > 0. If u € J-f^R*) satisfies (8.1.4), then (8.1.19) S(u) = ±T(u), (8.1.20) (N - 2)7» = 2NV(u), (8.1.21) = ^ir(u), r^ proof. These identities follow immediately from Lemma 8.1.2. □ Our goal is to show that A and G are nonempty and to characterize G. For technical reasons, we consider separately the cases N > 3, N = 2, and N = 1. Theorem 8.1.4. Assume N > 3, (8.1.2), and w > 0. (i) A and G are nonempty. (ii) u £ G if and only if u solves the minimization problem { V(u) = AJ2l, (8-1-23) < \ \ S(u) = min{5(w) : V(w) = A#}, where A = ~^inf{T(i;) : V(v) = 1}. In addition, min{S(w) : V(w) = 2 (iii) Tftere exists a real-valued, positive, spherically symmetric, and decreasing function ipeG such that G = \J{ei6(p{- -y) : 9 GR,y € RN}. Theorem 8.1.5. Assume N = 2, (8.1.2), and u > 0. (i) A and G are nonempty. (ii) u € G if and only if u solves the minimization problem (8124) (ueN and fRN \u\2 = 7, 1 * ' 1 I S{u) = mm{S{w) :weN}, where N = {u e H^R") : ^(u) = 0 and u ^ 0} and 7 = ^ min™6JV 5(w). (iii) There exists a real-valued, positive, spherically symmetric, and decreasing function (p € G such that G = |J{e*V(" - y) : 9 eR,y e RN}. Theorem 8.1.6. Assume N = 1, (8.1.2), and u > 0. (i) j4 and G are nonempty. (ii) A = G. (iii) There exists a real-valued, positive, spherically symmetric, and decreasing function

2 and so (8.1.26) I^_^ + _1_M««=0 throughout the existence interval. It follows easily that

ac/2 < 0. Therefore, there exists a > 0 such that ip' < 0 on (0,a). We claim that ip' < 0 on (0, oo). Otherwise, there would exist b > 0 such that (b) = —c. Therefore, there would exist d £ (0,6) such that ip(d) = 0. Applying again (8.1.26), we would obtain ip'(d) = 0, which would imply that

0. Passing to the limit in (8.1.26), we obtain that which implies £ = 0. Therefore ip decreases to 0, as x —> +oo, and we deduce easily that the decay is exponential. Therefore ip" and hence ip' also decay exponentially to 0. Therefore,

Since v £ Hl(R), it follows that v(x) —* 0 as |x| —> oo. Therefore, by the equation, v"(x) —* 0 as \x\ —> oo, and so v'(x) —» 0 as \x\ —> oo. Letting jx| —> oo in (8.1.27), we deduce that K = 0, and so (8-1.28) I|^|2_|H2 + _l_N«+2=0i In particular, \v\ > 0, for if v would vanish, then by (8.1.28) v' would vanish at the same time and we would have v — 0. Therefore, we may write v = pet&, where p > 0 and p, 9 £ C2(R). Writing down the system of equations satisfied by p, 8, we see in particular that p9" + 2p'9' = 0, which implies that there exists K £ R such that p29' = K, and so 9' = K/p2. On the other hand, since \v'\ is bounded, it follows that p2#'2 is bounded. This means that K2 jp2 is bounded. Since p(x) 0 as \x\ —> oo, we must have K = 0. Therefore (remember that p > 0) 9 = #0 for some #o e R. Thus v = eieop. Since p € tf^R^), there must exist xQ £ R such that p'(xo) = 0; and, by (8.1.28), p(xo) = c. Let now w(x) = p(x - xq). It follows that w satisfies (8.1.25), w(0) = c, and w'(0) — 0. By uniqueness of the initial-value 8.1. nonlinear bound states 261 problem for (8.1.25), we have w = 3, and we begin with the following lemma. Lemma 8.1.7. Assume N > 3, (8.1.2), and u > 0. It follows that the minimization problem { V(u) = 1 (8.1.29) <^ V ' \ T(u) = min{7» : V(w) = 1} has a solution. Every solution u of (8.1.29) satisfies the equation -Au + Aoju - A\u\au, where (8.1.30) A=^inf{T(t;):^) = l}. Proof. We repeat the proof of Berestycki and Lions [25]. We recall the definition of the Schwarz symmetrization. If u e L2(RN) is a nonnegative function, we denote by u* the unique spherically symmetric, nonnegative, nonincreasing function such that |{a: 6 RN : u*(x) > X}\ = \{x 6 RN : u(x) > X}\ for all A > 0. We refer to Berestycki and Lions [25], appendix A.III for the main properties of the Schwarz symmetrization. In particular, (8-1.31) J |«T = / Hp for all 1 < p < oo such that u e LP(RN), and (8.1.32) J |Vw*|2 < J |Vw|2 if uei/^). The proof proceeds in four steps. Step 1. Selection of a minimizing sequence. Let u e H1(RN). One can easily find A > 0 such that V(Xu) = 1. Therefore, the set {u e H1(RN) : V(u) = 1} is nonempty. Let (vm)m€N be a minimizing sequence of (8.1.29). Let um = \vm\*. It follows from (8.1.31) and (8.1.32) that (wm)meN is also a minimizing sequence of (8.1.29). Step 2. Estimates of (um)m€N- By definition, |jVitm||£2 is bounded, and by Sobolev's inequality, (wTO)meN is bounded in L17^(RN). On the other hand, V(um) = 1 implies that By Holder's inequality, this implies that w,, ,|2 1 II 11^ II |ia+2-Na/2 262 8. stability of bound states in the attractive case 2 Since a + 2 — Na/2 < 2, it follows that (um)m^ is bounded in L (R^), hence in iif1(EiV). step 3. Passage to the limit. By Step 2 and Proposition 1.7.1, there exist u e H1(RN) and a subsequence, which we still denote by (wm)meN, such that um —> u as m —» oo, weakly in if1(MiV) and strongly in La+2(RN). By the weak lower semicontinuity of the I? norm, 2N (8.1.33) V(u) > 1 and T{u) < liminf T(um) = ——-A, m—*oo JV — 2 where A is defined by (8.1.30). Since V(u) > 1, it follows that u 0. We claim that in fact V(u) = 1. Indeed, if V{u) > 1, then there exists A > 1 such that v(x) = u{Xx) satisfies V(v) = 1. It follows that 9 N T(v) = X2-NT(u) 2NA/{N - 2). Comparing with (8.1.33), we see that T(u) = 2NA/(N - 2). Therefore, u satisfies (8.1.29). Step 4. Conclusion. Let u be any solution of (8.1.29). There exists a Lagrange multiplier A such that (8.1.34) -Au = X(\u\au-uiu). Taking the L2-scalar product of (8.1.34) with u, we obtain T(u) = A ((a + 2)V{u) + f M2) = A/i with \i > 0. Therefore, A > 0. Applying Lemma 8.1.2(h), we deduce that T(u) = #^AV(n) = #^A. v ' N-2 v ' N-2 Since T(u) = 2NA/(N - 2), it follows that A = A. This completes the proof. □ Corollary 8.1.8. Assume N > 3, (8.1.2), andu > 0. If A is defined by (8.1.30), then the minimization problem [ V(u) = At (8 1 35) < \ T{u) = min{J» : V(w) = At} has a solution. Every solution u of (8.1.35) satisfies the equation (8.1.4). In addition, (8.1.36) mm{T(w) : V(w) = A%} = -t^^At . 1 V ~~ At Proof. Given u e H1(RN), let Au € H^R") be defined by u(x) = Au(A^x). One quite easily verifies that u satisfies (8.1.29) if and only if Au satisfies (8.1.35). Therefore, it follows from Lemma 8.1.7 that (8.1.35) has a solution. Finally, given a 8.1. nonlinear bound states 263 solution u of (8.1.35), let v be defined by Av — u. It follows that v satisfies (8.1.29), and so T(v) = 2JVA/(/V - 2), by (8.1.30). This implies that T(u) = k%~lT{v) = 2 TV At/(TV - 2). Hence (8.1.36) follows. Furthermore, since v satisfies -Au + Auu = A\u\au, it follows that u satisfies (8.1.4). This completes the proof. □ Corollary 8.1.9. Assume TV > 3, (8.1.2), anduj > 0. If A is defined by (8.1.30), then the minimization problem [ V(u) = At (8.1.37) { W [ S(u) = min{S{w) : V(tu) = At} /ms a solution. Every solution u of (8.1.37) satisfies the equation (8.1.4). In addition, (8.1.38) min{5h : V(w) = At} = —L-At . jN' 2 Finally, u satisfies (8.1.37) if and only if u satisfies (8.1.35). Proof. Let u e H1^) be such that V(u) = At. We have S(u) = l-T{u) - At , so that u satisfies (8.1.35) if and only if u satisfies (8.1.37). Therefore, (8.1.37) has a solution by Corollary 8.1.8. Finally, let u satisfy (8.1.37). It follows that u satisfies (8.1.35), and by Corollary 8.1.8, u satisfies (8.1.4). Furthermore, (8.1.38) is a consequence of (8.1.36) and (8.1.19). □ Corollary 8.1.10. Assume TV > 3, (8.1.2), and u > 0. It follows that G is nonempty. Furthermore, u £ G if and only if u satisfies (8.1.37); Proof. Consider a solution u of (8.1.37). It follows from Corollary 8.1.9 that u satisfies (8.1.35) and (8.1.4). In particular, we deduce from (8.1.36) and (8.1.38) that (8.1.39) V(u) = A*. T(U) = ^Af, 5(w) = _1_At . Applying Corollary 8.1.9, we deduce that A is nonempty. Consider any v e A. It follows from Corollary 8.1.3 that if (8.1.40) V(v) = 7^ , then (8.1.41) t{v) = -^-^^ and s(v) = j^i* • Let a = A/7, and let v(x) = w(o%x). We have V(w) ~ At, and so by (8.1.36), m/ n 2TV yv (8.1.42) T{w) > —^AT . 264 8. STABILITY OF BOUND STATES IN THE ATTRACTIVE CASE By (8.1.41), Applying (8.1.42), we deduce that 7 > A. By (8.1.39) and (8.1.41), this implies that (8.1.43) S(v)>S(u), and so u G G. In particular, G is nonempty. If we assume further that v G G, then we must have S(v) < S(u), since u satisfies (8.1.4). In view of (8.1.43), this means that S(v) = S(v). Applying (8.1.39), (8.1.40), and (8.1.41), we obtain that V(v) = A^ and S(u) = Jfz^^ ■ By Corollary 8.1.9, v satisfies (8.1.38), which completes the proof. □ Finally, before completing the proof of Theorem 8.1.4, we need the following lemma. Lemma 8.1.11. Let a : M.N —> R be continuous and assume that a(x) —+ 0 as \x\ —» 00. If there exists v G H1(RN) such that (8.1.44) J (\Vv\2 -a\v\2)dx < 0, then there exist A > 0 and a positive solution u G i?1(]RJV) 0 C(RN) of the equation (8.1.45) -Au + Xu = au. In addition, if w G if J(R ) is nonnegative, w ^ 0, and i/ ihere exists v G R swc/i ifeai —Aw + uw = aw, then there exists c > 0 s«c/i that w — cu. In particular, /i = A. proof. We claim that the minimization problem (8 146) fNU» = l I = min{J(y) : ||u||L2 = 1}, where J(u) = J(|Vu|2 - a\u\2)dx, 8« has a nonnegative solution. Indeed, let (fm)mGN be a minimizing sequence of (8.1.45), and let um — \vm\. Since |wm| = \vm\ and |Vwm| < |Vvm|, we see that (um)m€N is also a minimizing sequence. Since a G L00^^) by assumption, we deduce easily that (um)meN is bounded in ii1(RJV). Therefore, there exists a subsequence, which we still denote by (um)meNi and there exists u G H1(RN) such 8.1. NONLINEAR BOUND STATES 265 that um u in H1(RN). Note that u > 0 and let us show that u satisfies (8.1.46). For every r > 0, / \a\\um - «2| ^ y* \a\(um +u)\um - u\ +sup{|a(x)| : \x\ > r} j (u2m + u2). {Mr} It follows that y lallw^-u2] < 2||a||Loc^ J \um - + 2sup{|a(rr)| : \x\ > r] . RN {\x\ 0. There exists r > 0 such that 2sup{ja(a;)| : |ar| > r} < e/2 . Since the embedding H1(RJV) «-* L2(Br) is compact, we deduce that, for m large enough, 2NU-( y Itim-ul2^9 0O /|.| R* Using the weak lower semicontinuity of the L2 norm, we obtain that J{u) < -ji and ||w||l2 < 1, where ~[x = inf{J(u) : \\v\\L2 = 1}. Note that by (8.1.44), \x > 0, and so u 0. We have ||u||z,2 = 1, since, otherwise, there would exist A; > 1 such that w = ku satisfies ||u>||L2 = 1. We would obtain J(w) = k2J(u) < -fj,, which is a contradiction by definition of fi. Therefore, \\u\\Lz = 1, and, again by definition of ^, we must have J(u) = —\x. This proves the claim. Therefore, there exists a Lagrange multiplier A such that (8.1.47) -Au + Xu = au. On taking the L2-scalar product of the equation with u, we obtain (8.1.48) A = /x>0. It follows easily from (8.1.47) that u € H2(RN) n C(RN) (see the proof of Theorem 8.1.1); and since u > 0, we deduce from the strong maximum principle (Gilbarg and Trudinger [127], corollary 8.21, p. 199) that (8.1.49) u>0. 266 8. STABILITY OF BOUND STATES IN THE ATTRACTIVE CASE So far, we have proved the first part of the statement of the lemma. Let now i/el be such that there exists a solution w £ 7f1(]R;v), w > 0, of the equation (8.1.50) -Aw + uw = aw. We may assume that w 0. On multiplying (8.1.47) by w, (8.1.50) by u, and computing the difference, we obtain (A — v) J wu = 0. Since wu > 0 and wu ^ 0 by (8.1.49), this implies that v — X. We now claim that there exists c > 0 such that w — cu, for if this were not the case, there would exist c> 0 such that z — w — cu takes both positive and negative values. Note that —Az + Xz = az. On multiplying the equation by z, we see that Therefore, y defined by z satisfies (8.1.46). It follows that \y\ also satisfies (8.1.46). Repeating the argument that we made for u, we deduce that \y\ satisfies (8.1.47), and that \y\ > 0. Therefore, z has a constant sign, which is a contradiction. This completes the proof. □ Proof of Theorem 8.1.4. Parts (i) and (ii) follow immediately from Corollary 8.1.10. It remains to show (iii). Consider u £ G, so that u satisfies (8.1.37). Let / = |Rew[, g — |Imuj, and v = / + ig. We have \v\ = |w| and |Vv| = |Vu|. It follows that v also satisfies (8.1.37). Applying Corollary 8.1.10, this implies that —Av + uv = \v\av, and so f -Af + wf = af \ ~Ag + u;g = ag, where a = \v\a. Applying Theorem 8.1.1, we deduce that a satisfies the assumption of Lemma 8.1.11. Furthermore, J(v) = -u\\v\\2La<0. It follows from Lemma 8.1.11 that there exist a positive function z and two non-negative constants /i, v such that / = fxz and g = vz. In particular, Reu and Imu do not change sign, and so there exist c, d £ R such that u = cz + i dz. This implies that there exist a positive function tp and 9 £ R such that u = ei6ip. Therefore, ip also satisfies (8.1.37), hence (8.1.4) follows by Corollary 8.1.10. By Theorem 8.1.1, ip £ C^R^) and ip(x) ^ 0 as |x| -»• oo. Applying Gidas, Ni, and Nirenberg [125, theorem 2, p. 370], we obtain that there exist a positive, spherically symmetric solution ip of (8.1.4) and y £RN such that tp(-) = 3 does not apply to this case, since by Corollary 8.1.3, V(u) = 0 for every u € A. Proof of Theorem 8.1.5. We proceed in four steps. We define (8.1.51) N = {u € Hl{RN) : V(u) = 0 and u ^ 0}, (8.1.52) c = inf{S(u>) : w e N} , and (8.1.53) 7 = -— inf{S(w) : w 0. Indeed, consider u € N. We have / \u\2 <—-?-— f \u\a+\ J 1 1 - u(a + 2) J 1 1 On the other hand, it follows from Gagliardo-Nirenberg's inequality that there exists C independent of u such that j MQ+2 < C(T(w))f j \u\2. This implies that there exists a > 0 such that T(u) > a/2 for all u € TV, which implies 7 > 0. Step 1. The minimization problem (8.1.24) has a solution. We repeat the proof of Berestycki, Gallouet, and Kavian [24]. It is clear that N =£ 0. Let {vm)m^fq be a minimizing sequence. In other words, vm 0, V(vm) — 0, and S(vm) —> c. Let wm = \vm\* (see the beginning of the proof of Lemma 8.1.7), so that (wm)m€^ 1 /2 has the same properties as (vm)meN- Define now (uTn)m€M by um(x) = wm(Xm x), where \Wr |2 7 We have (8.1.54) Ju2m = j, RN (8.1.55) V{um) = 0, and (8.1.56) S(um) = S(wm) —► c m—»00 268 8. stability of bound states in the attractive case In particular, (um)meN is also a minimizing sequence. It follows from (8.1.54), (8.1.55), and (8.1.56) that (wm)meN is bounded in //"1(R-Ar). Therefore, there exists a subsequence, which we still denote by (um)m^, and there exists u £ HX{RN) such that um —^ u in H1(RN) as m —> oo. In particular (see the proof of Lemma 8.1.7), / <+2 — / U«+2 , / u2 < liminf / t& = 7, 1N T(u) < liminfr(um). and Therefore, V(u)>0 and 5(u) 0, then in particular u ^ 0, so that there exists A £ (0,1) such that v — Xu satisfies V{v) = 0. Thus v e N. Furthermore, T(v) = X2T(u) < T(u), so that S{v) < S(u), which implies that S(v) < c. This contradicts the definition of c. Therefore, V(u) = 0. It follows that V(um) —* V(u), which implies that / u2 = lim / u2m = 7, and so u satisfies (8.1.24). Step 2. Every solution of (8.1.24) belongs to A. Indeed, consider a solution u of (8.1.24) (which exists by Step 1). There exists a Lagrange multiplier A such that —Aw = X(\u\au — uju). On taking the L2-scalar product of the equation with u, we obtain RN Since u satisfies (8.1.24), this implies that Au>Q7 and so A = 1. Therefore, u satisfies (8.1.4). Step 3. u satisfies (8.1.24) if and only if u € G. Consider any solution u of (8.1.24) and any v G A (A ^ 0, by Step 2). It follows from Corollary 8.1.3 that v € N and 4 c v S(v) (8.1.57) / \v\2 = ^~S(v)=j J uia S(u)- Since v G N, we deduce that S(v) > S(u), and so u € G 0. Assume further that v £ G. Since u £ G also, we have S(u) = S(v). It follows from (8.1.57) that \v\2 = 7, / RN which means that ^satisfies (8.1.24). Hence the result is established. 8.2. an instability result 269 Step 4. Conclusion. Properties (i) and (ii) follow from Step 3. We establish (iii) by following the argument from the proof of Theorem 8.1.4. □ Definition 8.1.13. A function u 6 A is called a bound state of (8.1.4). A function u 6 G is called a ground state of (8.1.4). By definition, this is a bound state that minimizes the action S among all other bound states. Remark 8.1.14. Note that the ground state is unique, modulo space translations and multiplication by et0, as follows from Theorems 8.1.4 to 8.1.6. Remark 8.1.15. In the literature, one sometimes calls any positive solution of (8.1.4) a ground state. It follows from Theorems 8.1.4 to 8.1.6 that these two definitions are equivalent, modulo multiplication by e%e'. Remark 8.1.16. In the case N = 1, every u e A is a ground state, since A — G. This is not true anymore when N > 2. Indeed, in this case, it follows from Berestycki and Lions [25] and Berestycki, Gallouet, and Kavian [24] that there exists a sequence (um)m£^ C A such that S(um) —» oo as m —> oo. This implies that for m large, um G. Remark 8.1.17. Let u be the (unique) positive, spherically symmetric ground state of (8.1.4) with lo — 1. For oj > 0, let u^x) = wl/au(u)%x). It follows that uw satisfies (8.1.4), and so uw is the unique positive, spherically symmetric ground state of (8.1.4). We have IMIffi = (Ja 2 / « +wq 2 / |Vl/|. Therefore, if a > 4/N, there exists a > 0 such that > a for all u > 0. On the other hand, if a < 4/N, then —► 0 as u; —► 0. In particular, there exist ground states of (8.1.4) of arbitrarily small H1 norm (when uj varies). 8.2. An Instability Result We begin with the following result of M. Weinstein [356]. Theorem 8.2.1. Assume (8.1.1) with a = A/N and let uj > 0. If

0 and t < 0. Proof. We have E{ 1. On the other hand, it follows from Theorem 8.1.1 that | • [ 0 and t < 0. The result follows by letting, for example, fm — (1 + ~ Jv7- D 270 8. stability of bound states in the attractive case In the case a > 4/iV, we have the following result of Berestycki and Cazenave [23] (see also Cazenave [59, 60]). Theorem 8.2.2. Assume (8.1.1), (8.1.2), and u > 0. Suppose further that a > A/N. IfipeG (cf. Theorems 8.1.4, 8.1.5, and 8.1.6), then u{t,x) = eiuty{x) is an unstable solution of (4.1.1) in the following sense. There exists (^m)mEN c H1(RN) such that —>

0 and t < 0. Remark 8.2.3. As we will see, the proof of Theorem 8.2.2 is much more complicated than the proof of Theorem 8.2.1. On the other hand, the result is much weaker (except when N — 1), since it only concerns the ground states (see Remark 8.1.16). It is presently unknown whether the other stationary states are unstable. Let us define the functional Q e C1(H1(RN),R) by (8.2.1) Q(u) = J \Vu\2 - ^7^2) / H*+2 for u e Fl(RiV)' and let (8.2.2) M = {u e H^R") : u ^ 0 and Q(u) = 0} . The proof of Theorem 8.2.2 relies on the following result. Proposition 8.2.4. Let a,w be as in Theorem 8.2.2. Ifu e H1(RN), then u e G if and only if u solves the following minimization problem: ( ueM, ' (8*2'3) I S(u) = min{5(t;) : v € M}. For the proof of Proposition 8.2.4, we will use the following lemma. Lemma 8.2.5. Given u e H1^), u^0, and A > 0, set V(X,u)(x) = X%u(Xx). The following properties hold: (i) There exists a unique X*(u) > 0 such that V(X*(u),u) e M. (ii) The function X i-» S(V(X,u)) is concave on (A*(u),oo). (in) X*(u) < 1 if and only if Q(u) < 0. (iv) X*(u) = 1 if and only if u G M. (v) S(V(X,u)) < S(V(X*(u),u)) for every X > 0, X ^ A*(u). (vi) £S(V(X,u)) = xQ(V(X,u)) for every A > 0. (vii) \V{X,u)\* = V(X, \u\*) for every A > 0, where * is the Schwarz symmetriza-tion. (viii) If um —> u in Hl(RN) weakly and in La+2(RN) strongly, then V(X,um) —> V(X,u) in H1^) weakly and in La+2(RN) strongly for every A > 0. 8.2. AN INSTABILITY RESULT 271 Proof. Let u € ^(R^), u ^ 0, and let ux = V{\, u). We have (8-2.4) S{ux) = ^J \Vu\2 + | J \u\2 - ^ / MQ+2 • RJV ]gN jjAf Property (vi) follows easily. Let A*(u) be defined by ^ = ^(/i-e)(/i<+f. Elementary calculations show that with \*(u) defined as above, properties (i), (ii), (hi), (iv), and (v) are satisfied. Property (vii) follows easily from the definition of Schwarz's symmetrization (see the beginning of the proof of Lemma 8.1.7). Finally, given A > 0, the operator u t—> V(\,u) is linear and strongly continuous H1^1*) -> i/^R^). Therefore, it is also weakly continuous. The La+2 continuity is immediate. Hence (viii) follows. □ Corollary 8.2.6. The set M is nonempty. If we set (8.2.5) m = \ni{S(u) : u € M} , then Q{u) < S(u) — m for every u e i?1(RAr) such that Q(u) < 0. Proof. It follows from Lemma 8.2.5(i) that M is nonempty. Let u e iJ^R^) be such that Q(u) < 0, and let /(A) = S(P(\,u)). By Lemma 8.2.4(iii), \*{u) < 1, and, by (ii), / is concave on (A*(u), 1). Therefore, /(l)>/(A*(tt)) + (l-A'(u))/'(l). Applying Lemma 8.2.5(vi), we obtain S(u) > /(A») + (1 - \*(u))Q(u) > /(A») + Q(u). Since by Lemma 8.2.5(i) V(\*(u),u) € M, we deduce that /(A*(u)) > m, and so S(u) >m + Q(u), which completes the proof. □ Proof of Proposition 8.2.4. We proceed in three steps. Step 1. The minimization problem (8.2.3) has a solution. We know that M / 0 by Corollary 8.2.6, so that (8.2.3) has a minimizing sequence (vm)me^. In particular, Q{vm) = 0 and S(vm) —* m, where m is defined by (8.2.5). Let «im = \vm\*, and um = V(\*(wm),wm)- It follows from Lemma 8.2.5(i) that um £ M. Furthermore, it follows from Lemma 8.2.5(vii) that um = \V(\*(wm),vm)\*-Therefore, S(um) < S(V(y(wm),vm)) < S(V(\*(vm),vm)) < S(vm), 272 8. stability of bound states in the attractive case where the last two inequalities follow from Lemma 8.2.5(v) and (i). In particular, (wm)mgN is a nonnegative, spherically symmetric, nonincreasing minimizing sequence of (8.2.3). Furthermore, note that =-Mta-yi7"»i + 2y Kw R* It follows that («m)mgn is bounded in H1(RN). Since Q(um) = 0, we deduce from Gagliardo-Nirenberg's inequality and the boundedness of (uTO)m€N in L2(RN) that there exists C such that ||Vum||L2 4, we obtain that ||Vum|[7y2 is bounded from below, and since Q(um) = 0, there exists a > 0 such that (8.2.6) ||um||LQ+2 > 0. By Proposition 1.7.1, there exist v 6 H1(RN) and a subsequence, which we still denote by (um)meN, such that um —> v as m —► oo, in H1(RN) weakly and in La+2(RN) strongly, and so by (8.2.6), v ^ 0. Therefore, we may define u = V(X*(v),v), By Lemma 8.2.5(i), u e M and Lemma 8.2.5(vii), V(X*(v),um) -» u in ^(M^) weakly and in L0*2^) strongly. Therefore, S(u) < liminf S{V(X*{v),um)) < liminf 5(P(A*(«m),um)) = liminf S(um) = m, TO—>00 where the last three inequalities follow from (v), (iv), and (8.2.5), and so u satisfies (8.2.3). Step 2. Every solution of (8.2.3) satisfies (8.1.4). Consider any solution u 2 of (8.2.3). For a > 0, let u(x) = a° ua(ax). One easily verifies that Q(ua)=crN-2-^Q(u)=0, and so ua £ M. Since u = wi satisfies (8.2.3), we deduce that f(a) = S(ua) satisfies /'(l) =0. One computes easily, by using the property ua e M, that //(l) = <5,(w),«)ff-i,Jifi, where S' is the gradient of the C1 functional .5" (i.e., S'(u) — —Au + oju — \u\au). It follows that (8.2.7) (S'(u),u)H-liHi = 0. On the other hand, Q'{u) = —2Au - ^|w|qm, and so since u € M, we obtain (8.2.8) = -aT(tt) < 0. Finally, since u satisfies (8.2.3), there exists a Lagrange multiplier A such that S'(u) = XQ'(u). Applying (8.2.7) and (8.2.8), we deduce that A = 0, and so S'(u) = 0, which means that u satisfies (8.1.4). 8.2. an instability result 273 Step 3. Conclusion. Consider (8.2.9) £ = mm{S(u) : u G A} . Let u G G. In particular, S(u) = L Applying Corollary 8.1.3, one obtains easily that u G M. Therefore, S(u) > m, where m is defined by (8.2.5). In particular, (8.2.10) £>m. Consider now a solution u of (8.2.3). By Step 2, u G A. Since S(u) = m, it follows from (8.2.9) that m > £. Comparing with (8.2.10), we obtain m = L The equivalence of the two problems follows easily. □ Proof of Theorem 8.2.2. Let

0. It follows from Proposition 8.2.4 and Lemma 8.2.5 that (8.2.11) Q(<^a) < 0 and (8.2.12) S( 1. Let u\ be the maximal solution of (4.1.1) with the initial value 4/7V. When a < 4/7V, it follows from the results of Section 8.3 that the ground states are, to the contrary, stable. Remark 8.2.8. The method of proof of Theorem 8.2.2 can be adapted to more general nonlinearities. See Berestycki and Cazenave [23], Fukuizumi and Ohta [118], and Ohta [282]. 274 8. STABILITY OF BOUND STATES IN THE ATTRACTIVE CASE 8.3. A Stability Result Our goal in this section is to establish the following result of Cazenave and Lions [66] (see also P.-L. Lions [235, 236] and Cazenave [58]) Theorem 8.3.1. Assume (8.1.1), (8.1.2), and w > 0. Suppose further that a < A/N. IfipGG (cf. Theorems 8.1.4, 8.1.5, and 8.1.6), then u(t,x) = eiuJt 0, there exists 8(e) > 0 such that if yj € if1(RAr) satisfies W^-iPWh1 < then the corresponding maximal solution v of (4.1.1) satisfies (8.3.1) sup inf inf \\v(t, ■) - eie 0 and y E R^ such that \y\ = 1, let tp£(x) = eiex-yip{x) and u£(t, x) = eie{x-y-£t^eiu}t 0, ten ff€K On the other hand, it is clear that if

0, let tpE(x) = (1 +£)1/CV((1 + and uE(t,x) = eiuj{1+£)t(l + e)^ 0, sup inf \\ue(t, ■) - \\tp\\Hi . ten j/eKN tern ye&N 8.3. a stability result 275 Remark 8.3.5. Theorem 8.3.1 only asserts the stability of ground states. Except when N = 1, where A — G, one does not know whether the other standing waves are stable. The proof of Theorem 8.3.1 relies on the following result. Proposition 8.3.6. Assume 0 < a < 4/7V and u> > 0. Let r > 0, and let E be defined by (8.1.16). If (8.3.3) r = jw then there exist a subsequence (umfc)fceN and a family (t/fc)fceN C RN such that (umk(--yk))keN has a strong limit u in if^M^). In particular, u satisfies (8.3.5). Proof. The proof relies on the concentration-compactness method introduced by P.-L. Lions [235, 236] in the form of Proposition 1.7.6. We proceed in three steps. Step 1. 0 < u < oo. It is clear that T ^ 0. Let u € T and A > 0; set n u\(x) = A 2 u(\x). It follows easily that u\ € T and that l2 ^)-^-/|v.|'-^/l«l-«. Since No. < 4, we have E(u\) < 0 for A small, and so u > 0. Next, we claim that there exist 8 > 0 and K < oo such that (8.3.6) E(u) > S\\ufH1 -K for all u € T. This follows immediately from Gagliardo-Nirenberg's inequality /|<« -K > — oo. Step 2. Every minimizing sequence of (8.3.5) is bounded in H1(RN) and bounded from below in La+2(RN). Let [un) >o be a minimizing sequence. Since un 6 T, (un)n>o is bounded in L2(RN), then by (8.3.6) (un)>o is bounded in 276 8. stability OF BOUND states IN the attractive case if^M^). This proves the first part of the statement. Furthermore since v > 0, we have E(un) < —u/2 for n large enough. It follows that (8.3.7) J K|«+2 > ^±*„. R« Hence the result is established. Step 3. Conclusion. We need only prove (ii). Let (wn)n>o satisfy ||un||L2 —> ■y/Ji and E(un) —► —v. Setting Un ~ \\un\\L*Un' we deduce that (un)n>o is a minimizing sequence of (8.3.5). Note that by rescaling, we may assume that r = 1. We now apply Proposition 1.7.6 to the minimizing sequence (un)n>o (note that a = 1). We claim that (8.3.8) At = 1, where fi is defined by (1.7.6). Note first that, since (un)n>o is bounded from below in La+2(RN), we have p, > 0 by Proposition 1.7.6(h). Suppose now by contradiction that (8.3.9) 0 < fjt < 1. We use the sequences (vk)k>o and (wjt)fc>o introduced in Proposition 1.7.6(iii). It follows from (1.7.15)-(1.7.16) that liminf(£(unJ - E(vk) - E(wk)) > 0, k—>oo so that (8.3.10) limsup(E(ufe) + E(wk)) < -v. k—*oa Next, observe that, given u e H1(UN) and a > 0, we have E(u) = ±E(au) + ^1 J \ur2 . Applying the above inequality with and ak = l/Ht^Hx,2? and since akVk € T, we obtain that a*k a + 2 J Similarly, ia+2 Rw with bk = l/|jt0fc||L2, and so E(Vk)+E{wk) > -v{a~2+b~2) + J J Kr2 RN 8.3. a stability result 277 2 . .. „„j l—2 Finally, note that -* n and —» 1 - \x by (1.7.14). In particular, by (8.3.9) 0 := min{A*~',(l - m)-^} > 1 • Therefore, using (1.7.16), then (8.3.7) we deduce that 9 — 1 ľ liminf(J5ľ(t;fc) + E(wk)) >-u +-- liminf / k—»-oo Oĺ -f- 2 fc—»oo J Unfcl°+2 0-1 which contradicts (8.3.10). Therefore, the proof of the claim (8.3.8) is complete. We finally apply Proposition 1.7.6(1), and we deduce that for some sequence {yk)k>o C RN and some u e H1^), unk{- - yk) -+ u in L2(RN) (and in particular, u~eT) and in La+2(RN). Together with the weak lower semicontinuity of the Hl norm, this implies E(u) < lim E(unk) = —v. k—foo By definition of u, we have E{u) = —v. In particular, E{unk) —> E(u), and it follows that ||VunJ|x,2 —> ||Vu||£2, which implies that wnfc(" — Vk) —> u strongly in H1(RN). □ Lemma 8.3.7. Let 0 < a < A/N and uj > 0. There exists > 0 suc/i tooi (8.3.11) y \u\2 = ^ /or every ground state u of (8.1.4). proof. The result follows from uniqueness of the ground state up to translations and rotations (cf. Theorems 8.1.4, 8.1.5, and 8.1.6). Alternatively, when N > 2 the result follows from (8.1.19), (8.1.22), and property (ii) of Theorems 8.1.4 and 8.1.5. □ corollary 8.3.8. Let 0 < a < A/N, u > 0, and let p, be defined by (8.3.11). If u € ií1(RAľ), then u is a ground state of (8.1.4) if and only if u solves the minimization problem (8"3'12) I S(u) = mm{S{v) : v e T}, where T is defined by (8.3.3). In addition, the problems (8.3.12) and (8.3.5) are equivalent. Proof. We proceed in four steps. STEP 1. Problem (8.3.12) is equivalent to problem (8.3.5), which has a solution by Proposition 8.3.6. Indeed, if u e T, then S(u) = E(u) + ojp/2, and so problem (8.3.12) is equivalent to problem (8.3.5). Step 2. We have k < £, where £ is defined by (8.2.9) and k is defined by (8.3.13) k = inf{S(t>) : v e ľ} . 278 8. STABILITY OF BOUND STATES IN THE ATTRACTIVE CASE Indeed, consider u € G. We have S(u) = £, and by Lemma 8.3.7, u £ T. By definition of k, this implies k < L Step 3. Every solution of (8.3.12) belongs to A. Consider a solution u of (8.3.12), and let N u\(x) = A 2 u(Xx) for A > 0. We have ux TW = 2(^2) / Now, since u satisfies (8.3.12), there exists a Lagrange multiplier A such that S'(u) — Xu, and so there exists 8 such that (8.3.15) -Au + Suu = \u\au. On taking the L2-scalar product of (8.3.15) with u and applying (8.3.14), we obtain from which it follows that 8 > 0. Define now v by u(x) = 8°v(8?x). We deduce from (8.3.15) that v € A, which implies (8.3.16) S{v)>£. One computes easily that S(u) = 5—35—S{v) + =^(1 - 5) . Applying (8.3.16) and Step 2, we obtain that 4-(N-2)o (jJU £>$-53- On the other hand, it follows from Corollary 8.1.3 that £ > 0, and by (8.3.11) and Corollary 8.1.3, up 4 - (TV — 2)oj 2 2a and so 1>5 2q +--—(1-5). 2a This means that f(5) < 0, where 4-(N-2)a 4 — (n — 2)a 4-Na One checks easily that f(s) > 0, if s ^ 1. Therefore, <5 = 1, which implies in view of (8.3.15) that u e A. 8.3. A STABILITY RESULT 279 Step 4. Conclusion. It follows in particular from Steps 2 and 3 that £ = k. Therefore, if u £ G, then u £ T and S(u) = k, which implies that u satisfies (8.3.12). Conversely, let u be a solution of (8.3.12). We have u £ A by Step 3, and since S(u) = k = £,it follows that u £ G. □ proof of theorem 8.3.1. Assume by contradiction that there exist a sequence (ipm)m€N C Hl(RN), a sequence (£m)m<=N C M, and e > 0 such that (8.3.17) lltfVn-vllzri —> 0, and such that the maximal solution um of (4.1.1) with initial value tpm (which is global, cf. Remark 8.3.2) satisfies (8.3.18) MMN\\um(tmr)-eiee. Let us set (8.3.19) vm - um{tm) ■ It follows from Corollary 8.3.8; Theorems 8.1.4, 8.1.5, and 8.1.6; and (8.3.19) that (8.3.18) is equivalent to (8.3.20) inf ||vm - u\\jri > e. Applying Corollary 8.3.8, we deduce from (8.3.17) that l^ml2 —* V> and S(ipm) —* k, I where k is defined by (8.3.13). From conservation of charge and energy, we deduce that l^ml2 —► A*, and S(vm) —* k I as well. Therefore, (vm)m£n is a minimizing sequence for the problem (8.3.12), hence of the problem (8.3.5) (see Corollary 8.3.8). From Proposition 8.3.6(h) it follows that there exist (ym)meN C RN and a solution u of the problem (8.3.5) such that \\vm —u(- — ym)iJHi —»• 0. But ueGby Corollary 8.3.8, and so u(- — ym) £ G, which contradicts (8.3.20). □ remark 8.3.9. Note that the proof of Theorem 8.3.1 only makes use of the following two properties. The conservation laws of (4.1.1) (charge and energy), and the compactness of any minimizing sequence. Therefore, the method is quite general and may be applied to many situations. See, e.g., Cazenave [58], Cazenave and Lions [66], P.-L. Lions [235, 236]), and Ohta [282, 283, 284]. Remark 8.3.10. One does not know in general about the functions 9(t) and y(t) of (8.3.2). If both tp and tp are spherically symmetric, one may let y(t) = 0 (see Remark 8.3.4). Remarks 8.3.3 and 8.3.4 display examples for which one may let 6 and y be linear in t. One does not know whether this is true in general. Concerning this question, see the remarkable papers of Soffer and Weinstein [316, 317]. They consider in particular a one-dimensional equation with a potential. In this case, y = 0, but they also show that one may let 6 be linear in t. 280 8. STABILITY OF BOUND STATES IN THE ATTRACTIVE CASE 8.4. Comments Remark 8.4.1. There are other methods to study the stability of standing waves, based on the study of a linearized operator. See Shatah and Strauss [311], Gril-lakis, Shatah, and Strauss [154, 155]. See also Goncalves Ribeiro [151], Blanchard, Stubbe, and Vazquez [33], M. Weinstein [358, 357], Rose and Weinstein [303], and Cid and Felmer [81]. The stability of excited states has also been studied, in particular by Jones [198] and Grillakis [153]. By using the techniques of Section 8.1, one can establish the following useful result of M. Weinstein [356] relating the ground states of (8.1.4) with the best constant in a Gagliardo-Nirenberg inequality. Lemma 8.4.2. Let R be the (unique) spherically symmetric, positive ground state of the elliptic equation (6.6.3), i.e., -AR + R=\R\aR inRN with a = 4/N (see Definition 8.1.13 and Theorems 8.1.4, 8.1.5, and 8,1.6). It follows that the best constant in the Gagliardo-Nirenberg inequality a + 2 isC = IIÄI i-a iL2 ■ proof. We follow the argument of M. Weinstein [356]. We need to show that (8.4.1) inf J(u) = ?!!3f, where We set „,, MhMh a = inf J(u), u€tf\u#0 and we consider a minimizing sequence (un)n>0. We observe that by Gagliardo-Nirenberg's inequality, a > 0. We consider vn defined by vn(x) ~ pnUn(Xnx) with N-2 L2 so that \\vn\\L2 = \\Vvnh2 = 1 and \\vn\\~Li%t2) = J(vn) = J(un) —> a > 0. By symmetrization (see the proof of Lemma 8.1.7), we may assume that vn is spherically symmetric, and so there exist a subsequence, which we still denote by (vn)n>o, and v e HX(RN) such that vn -* v in HX(UN) weakly and in La+2(RN) strongly (see Proposition 1.7.1). Since = limn-*oo H^nlU^2 = > 0, it follows that v 7^ 0. This implies that (8.4.2) J(v) = a and |ju.||L2 = j|Vv||L2 = 1. 8.4. COMMENTS 281 In particular, -^J(v + tw)\t=o — 0 for all w e Hl{RN) and, taking into account (8.4.2), we obtain a a + 2 —Av -i—v ~ a-\v\ v. Let now u be defined by v(x) = au(bx) with a = (a/a(a + 2))° and 6 = (a/2)?, so that tf is a solution of (6.6.3) and J(u) — J(v) — a. Since u satisfies equation (6.6.3), we deduce from Pohozaev's identity (see Lemma 8.1.2) that ^liVu|ll2 = -^Nl^+22, and that 2||Vu||5,a = N\\u\\2L2 (see formulae (8.1.21) and (8.1.22)), and so TV ■ 4 2 (8-4.3) J(u) = — I = _ ||„||«a. Since R also satisfies equation (6.6.3), it satisfies the same identity. Since u minimizes J, we must have J(R) > J(u), which implies that ||ujjz,2 < ||#||l2- On the other hand, R being the ground state of (6.6.3), it is also the solution of (6.6.3) of minimal L2-norm by (8.1.19) and (8.1.22), so that \\R\Il* <.\\u\\L2r Therefore, \\R\\l2 = ||u||l2, and the result now follows from (8.4.3). □ CHAPTER 9 Further Results In Sections 9.1 and 9.2 we present some results that follow easily from the techniques that we developed in the previous chapters. On the other hand, we describe in Sections 9.3 and 9.4 two results that do not fall into the scope of these methods. Finally, we briefly describe in Section 9.5 some further developments. 9.1. The Nonlinear Schrodinger Equation with a Magnetic Field In this section we study the nonlinear Schrodinger equation in R3 in the presence of an external, constant magnetic field. Given b E R, b ^ 0, we consider the (vector-valued) potential $ defined by B = (0,0,&). We define the operator A on L2(R3) by D(A) = {uG L2{RZ) : Vu + i$u g L2(R3) and Au + 2i$ • Vu - |$|2w € L2(R3)} , and we refer to Avron, Herbst, and Simon [5, 6, 7], Combes, Schrader, and Seiler [93], Eboli and Marques [109], Kato [202], Reed and Simon [301], and B. Simon [313] for its physical relevance. We begin with the following observation. Lemma 9.1.1. A is a self-adjoint, < 0 operator on L2(R3). PROOF. Since £>(R3) c D(A), D(A) is dense in L2(R3). Furthermore, given u,v g D(A), (Au,v)L2 = -(Vu + i$u, Vt; + i$v)L2. Therefore, A is < 0 and symmetric. It now remains to solve the equation Au — Xu = / for every / g L2(R3) and A > 0. This follows easily by applying Lax-Milgram's lemma in the Hilbert space H = {u g L2(R3); Vu + i$u g X2(R3)}, equipped with the scalar product (u,v)h = (Vw + i$u, Vu + i$v)L-2 + X(u,v)L2. □ = -(-x2,ii,0) for x - (x1,X2,xs) g M3, which is the vector potential of the (constant) magnetic field B = curl($), that is, (9.1.1) 283 284 9. further results We then may apply the results of Section 1.6. In particular, D(A) is a Hilbert space when equipped with the norm \Hl(A) = \\M\h + HU» , and %A generates a group of isometries (T(£))tera on the Hilbert space (D(A))*. The operators (T(t))teu. restricted to any of the spaces D(A), XA, L2(R3), or X*A are a group of isometries, where XA is defined by XA = {u £ L2(R3) : Vw + i$u £ L2(R3)} , and IMlL =\\Vu + i$u\\li + \\ufL2. In addition, A can be extended to a self-adjoint, < 0 operator on (D(A))* (which we still denote by A), and A is bounded XA -> X\ and L2(R3) -> (D(A))*. Furthermore, we have the following result. lemma 9.1.2. The following properties hold: (i) XA LP(R3) for every 2 < p < 6. (ii) Lq(Rs) ^ X\ for every § < q < 2. (iii) D(A) ^ LP(R3) for every 2(R3) C XA, from which we deduce that the embedding XA <-+ LP(R3) is dense, and so (ii) follows from (i) by duality. Finally, let u £ D(A) and set f — Au £ L2(R3). For every j £ {1,2,3}, let Vj = djU + i$>jU. We have (9.1.3) Avj - v3 = ~(dj - i$j)f - 2i(Vu + i$u) ■ (dj$ - ~ vd . Next, observe that \dj$ — < b. Furthermore, V+i$ is by definition a bounded operator XA —► L2(R3), and so, by duality, V - i$ is bounded X2(R3) -> XA. In particular, the right-hand side of (9.1.3) belongs to X*A and \\Avj — < Cllullx,^). It follows easily that Vj £ XA and Ji^H^ < l7||u||£»(j4). Letting successively j = 1,2,3 we obtain the inequality ||Vw + < Ap- plying (i), we deduce that |jVu + i<&ti||£,e < C||u||£)(,4). Therefore, by (9.1.2), lll^d^DIII-l6 < C||"IId(j4)- Claim (iii) follows by Sobolev's embedding theorem. □ Lemma 9.1.3. Ife>0 and 1 < p < oo, then (I - eA)~x is continuous LP(R3) —► LP(R3) and - eA)-l\\c{LPtLv) < 1. PROOF. Let 9 £ C1(R+,R+) be such that both 9 and & are bounded, 9 > 0, 9' > 0, and 9(0) = 0. By applying the method of proof of Proposition 1.5.1, we need only show that (9.1.4) (Au,9(\u\2)u)l2 < 0 for all u € £>(4). 9.1. the nonlinear schrodinger equation with a magnetic field 285 Consider p e Z>(R3) such that 0 < p < 1 and p(x) — 1 for |x| < 1, and set Pm(x) = p(x/m) for m > 1. Let u e D(A). We have (9.1.5) [Au,9(\u\2)u)l2 = lim (Au, pm9{\u\2)u)L2. m—*oo In addition, since pm6(\u\2)u has compact support, (Au,pm9{\u\2)u)L2 = ~ReJ Vu.V(pm9{\u\2)u) 3 Jpm9(\u\2)u ■ Vu- Jpm\$\29{\u\2)\u\ E3 -21m ^ n q(U.ft\Txih . v7„. /* „ ljf,|2fl/'|..|2M..|2 R3 It follows from the Cauchy-Schwarz inequality that 2 ~2Lm j' pm9(\u\2)u-Vu< Jpm\^\29(\u\2)\u\2 + Jpm9(\u\2)\Vu, . K3 R3 I3 and so {Au,pm9{\u\2)u)L2 <-Rej Vu-V(pm6{\u\2)u) + jpm6(\u\2)\Vu\2 . R3 R3 An elementary calculation shows that - Re (Vm • V(Pm9(\u\2)u)) + Pm0{\u\2)\Vu\2 = -pm9'(\u\2)(\u\2\Vu\2 - Re(u2Vu2)) - l-Vpm • V0(M2) 1 /*s < -xVpm • V6(|uj2) a.e. where 0(s) = / 0( 0 and C < oo suc/i £/tai T(t) is continuous LJ(R3) -» L°°(R3) /or every t e (-<5,<5) andt^O. Moreover, mt)u\\L~ < -S-HuIIli /or every u € LX(R3) and £ e (-(5, S),t^O. Proof. For every t such that sin(6t) ^ 0, the following formula holds (see Avron, Herbst, and Simon [5]). 4-K(4iritp sm(b£) J R3 286 9. further results where F{xty,t) = (X3 ~ y3) + ^ ({xi - yi)2 + (x2 - y2)2) cotg(6i) - ^(xlV2-x2yi) Therefore, 1*1 l^(*)l|£(Li,L«) < |í|*|sin(6í)| ' from which the result follows easily. □ Consider now g as in Example 3.2.11 with TV — 3; i.e., with V e LP(R3) + L°°(R3) with p > 3/2, W e LX(R3) + L°°(R3) real-valued potentials, W even, and f(x, is locally Lipschitz in u, uniformly in x, and satisfying 1/(3, u) - f(x,v)\ < C(l + |uja + |*;|a)|u-i;| for some 0 < a < 4. We set G(u) = J {^V(:c)ju(x)|2+F(^ with r\z\ i r F(x,z) = / f(x,s)ds and E(u) = - \Vu + i$u\2 dx ~ G(u). ° K3 We have the following result (see Cazenave and Esteban [62]; see also de Bouard [96] for related results for a more general equation). Theorem 9.1.5. If g is as above, then the following properties hold. (i) For every ip e XA, there exist Tmln(ip),Tmax(ip) > 0, and a unique, maximal solution u E C((-Tmin,Tmax), XA) nC1 ((-Tmin,Tmax), XA) of problem (9.1.1). The solution u is maximal in the sense that if Tmax < oo (respectively, Tmin < oo), then \\u(t)\\a —» oo as t | Tmax (respectively, as t | ~^min)- (ii) There is conservation of charge and energy; that is, \\u(t)\\L2 = \\ 0 such that if ||^||x/i < <^ then the maximal solution u of (9.1.1) is global and sup{|JM(ť)||x,4 : t e R} < oo (compare the proof of Corollary 6.1.2). 9.2. THE NONLINEAR SCHRODINGER EQUATION WITH A QUADRATIC POTENTIAL287 Remark 9.1.7. In addition to the assumptions of Theorem 9.1.5, suppose that W+ e Lq{R3) + L°°(R3) for some q > 3/2, and that there exists 0 < 5 < 4/3 such that F(x,u) < C(l + |u|*)|«|2 for all u G C. It follows that for every

3/2. Let f „ 2P 2? r = max < a + 2,-, — I p—1 q _2q_\ and let (q, r) be the corresponding admissible pair. It follows that for every

u in u G L^C(R,LP(R3)) for every admissible pair (7,p). proof. One adapts easily the proofs of Theorem 4.6.4 and Corollary 4.6.5. □ remark 9.1.9. Under certain assumptions on g, one can adapt the methods of Section 6.5 and show that some solutions of (9.1.1) blow up in finite time (cf. Gongalves Ribeiro [150]). Remark 9.1.10. For a certain class of nonlinearities, equation (9.1.1) has stationary states of the form u(t,x) = eluJttp(x) (cf. Esteban and Lions [110]). One obtains stability results that are similar to those of Sections 8.2 and 8.3. For some nonlinearities, the ground states are stable (cf. Cazenave and Esteban [62]), and for other nonlinearities, the ground states are unstable (cf. Gongalves Ribeiro [151]). 9.2. The Nonlinear Schrodinger Equation with a Quadratic Potential We already studied the nonlinear Schrodinger equation in R-^ with an external potential V, with V G LP(RN) +L°°{RN) for some p > 1, p > N/2. Here we extend these results to the case of potentials U that are not localized, but have at most a quadratic growth at infinity, the model case being U(x) = \x\2. More precisely, consider a real-valued potential U G Cco(RN) such that U > 0 and DaU G L°°(RN) for all a eNN such that I a I > 2. We define the operator A on L2(RN) by J D(A) = {u6 tf^R*) : U\u\2 G LX{RN) and Aw - Uu G £2(RN)}, \ Au = Au-Uu for u G D(A). We consider the nonlinear Schrodinger equation r iut + Au + g(u) = 0 \ u(0) = 0. This is done easily by applying Lax-Milgram's lemma in the Hilbert space H = {u € Hl(RN) : U\u\2 € Ll(RN)}: equipped with the norm defined by \H% = \\Vu\\h + J U\u\2 + X\\u\\2Li for all ueH. □ We may then apply the results of Section 1.6. In particular, D(A) is a Hilbert space when equipped with the norm \H\l(A) = \\M\h + Ml* , and %A generates a group of isometries (T(t))tm on the Hilbert space (D(A))*. The group (T(t))t€R restricted to either of the spaces D(A), XA, L2(RN), XA is a group of isometries, where XA is defined by XA = {u € H1{RN) : U\u\2 e ^(R^)} and \HxA = llVu||£» + \\u\\h + J U\u\2. RN In addition, A can be extended to a self-adjoint, < 0 operator on (D(A))* (which we still denote by A), and A is bounded XA -» X\ and L2(RN) (D(A))*. Furthermore, we have the following result. Lemma 9.2.2. The following properties hold: (i) XA^H'(RN). (ii) H~1(RN) ^X*A. (hi) D(A) ^ LP(RN) for every 2 < p < oo such that J > \ - f. Proof. Claim (i) follows from the definition of XA, and (ii) then follows by duality. We now prove (hi) for N > 3, the proof for N = 1,2 being easily adapted. Let u e D(A) and let / = Au. Consider p > 2 and take the L2-scalar product of the equation Aw — JJu — f with jw|p-2u. (In fact, a rigorous proof would require a regularization; see the proof of Lemma 9.2.3 below.) One obtains easily J Mp" W < ll/MI<7(u • 9.2. THE NONLINEAR SCHRODINGER EQUATION WITH A QUADRATIC POTENTIAL289 Since p2\u\p~2\Vu\2 = 4|V(|u|p/2)|2, it follows from Sobolev's inequality that Nrv 5 and taking any p < oo if N < 4, then applying Holder's inequality. □ Lemma 9.2.3. If e > 0 and 1 < p < oo, tfien (7-eA)^1 is continuous LP(RN) —> LP(RN), and \\(I - eA)-l\\C[LP>LP) < 1. proof. Let d £ c71(R+,R+) be such that both 6 and 8' are bounded, 6 > 0, 6>' > 0, and #(0) = 0. By applying the method of proof of Proposition 1.5.1, we need only show that (Au,$(\u\2)u)L2 < 0 for all u € D(A). We have (Au, 6(\u\ )u)l2 = (Au,9(\u\2)u)L2 - J U0{\u\2)\u\2 < (Au,0(\u\2)u)L2, and we already know that (see the proof of Proposition 1.5.1) (Au,0(\u\2)u)L2 < 0. The result follows. □ Finally, we have the following estimate of (1(t))t£R. Lemma 9.2.4. There exist 8 > 0 and C < oo such that T(t) is continuous £i(R^) -» L°°(RN) for every t e (-8,8), t ^ 0, and (9.2.2) \\7(t)u\\L~ <-^\\u\\Li \t\ 2 for every u e L1^) and \t\ <6,t^0. Proof. This is a delicate result, based on a calculation of the kernel associated to T(t). See Oh [277], proposition 2.2. □ Remark 9.2.5. Estimate (9.2.2) holds for |i| < 8. In fact, (9.2.2) does not in general hold for all t ^ 0. This can be seen in the special case U(x) = u;2j:r|2/4, where there is the following explicit formula (Mehler's formula; see Feynman and Hibbs [112]) 7{t)u(x)=(-/ c(*rafeT((l»l9+lyl9)co.(«t)-2,.y))u( )d We see that |j3"(t)u||£(i,i)i,oc) < (u>/47n|sin(o;£)|)"^ if sin(o;i) ^ 0 and that this estimate is optimal. Consider now a real-valued potential V : RN -» R such that V e LP(RN) + L°°(RN) for some p>ltp> N/2. Let g(u) = Vu + f(;u(-)) + (W*\u\2)u 290 9. further results as in Example 3.2.11, and set G(u)= J l±V(x)\u(x)\2 + F(x,u(x)) + \(W*\u\2)(x)\u(x)\2 2Ux and We have the following result (see Oh [277, 278] for a similar result in the case where g(u) = — X\u\Qu). Theorem 9.2.6. If g is as above, then the following properties hold; (i) For every ip G Xa, there exist Tm[n(ip),Tma.x(tp) > 0 and a unique, maximal solution u G C((-Tmin,rmax),Xj4) n ^((-Tmin^max), JC^) of problem (9.2.1). The solution u is maximal in the sense that if Tmax < oo {respectively Tm-m < oo), then \\u(t)\\a —► oo as t | Tmax (respectively, as (iii) There is continuous dependence of the solution on the initial value in the sense that both functions Tmin(■ tp in XA and if [-7i,T2] c (-Tmin(y?),Tmax((£>)), then Um u in C([—Ti,T2],Xa), where Uffi is the maximal solution of (9.2.1) with initial value 0 such that if ||vl|x/i ^ ^> then the maximal solution u of (9.2.1) is global and sup{||w(t)||x/1 : t G R} < oo (compare the proof of Corollary 6.1.5). Remark 9.2.8. In addition to the assumptions of Theorem 9.2.6, suppose that W+ G Lq(RN) + L^iR*) for some q > 1, q > N/2, and that exists 0 < 8 < A/N such that F(x, u) < C(l + Iw^jul2 for all u G C. It follows that, for every tp G X&-, the maximal solution u of (9.2.1) is global and sup{||tt(£)||xi4 : t G R} < oo (compare the proof of Corollary 6.1.2). Concerning the existence of solutions of (9.2.1) for initial data in L2(RN), we have the following result. theorem 9.2.9. Let g be as in Theorem 9.2.6 and assume further that a < A/N and that W G Lg(RN) + L°°(RN) for some q>l,q> N/2. Let max of Theorems 4.12.1 and 5.7.1. □ 9.3. THE LOGARITHMIC SCHRODINGER EQUATION 291 and let (q, r) be the corresponding admissible pair. It follows that for every

u in u £ L\oc(R,Lp(RN)) for every admissible pair (7,p). Proof. One adapts easily the proofs of Theorem 4.6.4 and Corollary 4.6.5. □ Remark 9.2.10. Under certain assumptions on g, one can adapt the methods of Section 6.5 and show that some solutions of (9.2.1) blow up in finite time. More precisely, if we assume that \x- VC/| < C(\x\2 + U) and that g satisfies the assumptions of Proposition 6.5.1, one can show that (with the notation of Proposition 6.5.1) /"(*) = 16£(v?) + J(8(N + 2)F(u) - ANRe{f(u)u))dx + 8 J (v +^x-VV^\u\2dx + 4 J (Jw+ ^x- VW^j *|u|2^|w|2dx -8 J (u+ ^x- VU^j\u\2dx. RN The proof of the above inequality is similar to that of Proposition 6.5.1. Assume further that g satisfies (6.5.24), (6.5.25), and (6.5.26), and that U+^x-VU>0. If ip £ XA is such that j • \ip(-) £ L2(RN) and E((p) < 0, then Tm&x(tp) < 00 and Tmm( 0 and 4//V < a < 4/(JV-2) (4//V < a < 00, if N = 1), it follows from Remark 9.2.10 that if

0 : J a(^\ =inf |fc>0: Ja*(^J<1^ fOY u eX'. n We have the following results (see Cazenave [58], lemmas 2.1 and 2.5, and Kra-nosel'skii and Rutickii [218]). Lemma 9.3.1. The spaces X and X' are linear spaces. The inner product spaces (X, || • ||x) and (X', || • ||x') are reflexive Banach spaces and X' is the topological dual of X. Furthermore, the following properties hold: (i) Ifum —► uinX, thenA{\um\) —► A(\u\) in V-itt). m—*oo m—*oo (ii) If um —► u a.e. and if f A(\um\) — f A(\u\) J 7Tl-*00 J < CO , then um —► u in X. 771—*0o Lemma 9.3.2. The operator u i-> a(u) maps continuously X —* X'. The image under a of a bounded subset of X is a bounded subset of X'. Finally, consider the Banach space W = H^(Q) n X equipped with the usual norm. It follows from Proposition 1.1.3 that W* = H~1{n)+X'. 9.3. the logarithmic schrodinger equation 293 Define E(u) = \Vu\2 - ^ y V|u|2 - ~ J M2log(|u|2) for every uEW, n n a where the potential V E Lp(tt) + L°°(f2) for some p > 1, p > N/2. We have the following result. Lemma 9.3.3. The operator L : u h-» Au + Vu + ulog(|uj2) maps continuously W —> . The image under L of a bounded subset of W is a bounded subset of W*. The operator E is continuous W —► R. Proof. One easily verifies that for every e > 0, there exists Ce such that (9.3.2) \b(v)-b{u)\ < C£{\u\£ + \v\s)\v - u\ forallu,ueC. Integrating inequality (9.3.2) on $7, and applying Holder's and Sobolev's inequalities, we obtain easily that u > b(u) maps continuously Hq(£1) —> #_1(f2) and that the image under b of a bounded subset of Hq(Q) is a bounded subset of if-1(Q). The same holds for A, and also for u <—> Vu (by Holder's inequality), and so the first part of the statement follows from Lemma 9.3.2. Finally, (9.3.3) E(u) = i | |Vu|2 - \ J V\u\2 + 1 J A(\u\) - \ JB{\u\). q o. n n The first term in the right-hand side of (9.3.3) is continuous Hq(Q,) —■» R, and it follows from Lemma 9.3.1(i) that the third term is continuous x —*• x'. Furthermore, \B(v) - B{u)\ < Cs(\u\l+e + \v\l+e)\v - u\ by (9.3.2). Integrating the above inequality on Q, and applying Holder's and Sobolev's inequalities, we deduce that J\B(v)-B(u)\ 1, p > N/2. The following properties hold: (i) For every tp E W, there exists a unique, maximal solution u E C(R, W) D CX(R, W*) of problem (9.3.1). Furthermore, supteK ||u(£)||vk < oo. 294 9. further results (ii) There is conservation of charge and energy; that is, \\u(t)\\l2 = Ml2 and E(u{t)) = E{ ip in W, then um —»■ u in W uniformly on bounded intervals, where um is the maximal solution of (9.3.1) with initial value q>nd there exists u G L°°(R, Hq(ft)) such that the following properties hold: (i) u\nk G Wl^(R,H~1(ftk)) for every keN. (ii) um(t) —1 u(t) in H^ft) as m —► oo for every t G R. (iii) For every t G R, there exists a subsequence mj such that umi(t,x) —> u(t, x) as k —> oo for a.a. x G fl. (iv) um(t,x) —» u(t,x) as m —> co for a.a. (t,x) G R x fl. Proof. Fix k G N. (um|nfc)m€N is a bounded sequence of L°°((-k,k), if1 (fife)) n W1,OG((—k, k),H~1(flk), so that (by Proposition 1.1.2) there exist a subsequence (which we still denote by (um)m€^) and u G L00((-k,k),H1(ftk)) such that um(t)\nk —1 u(t) in Hl(flk). Letting A; —► co and considering a diagonal sequence, we see that there exist a subsequence (which we still denote by (um)m€^) and u G Lco(R,H1(ft)) such that um(t)\nk u(t) in iJ1(£7fe) for every k G N and every t G R. This implies in particular that um(t) —^ u(t) in iJx(fi). Therefore, u G L°°(R,H^(fl)), and (ii) holds. In addition, since the embedding if1 (fife) L2(ftk) 9.3. the logarithmic schrodinger equation 295 is compact, we have um(t)\nk —*• u(i)jofe in L2(£lk) for every A; G N and every tel. Applying the dominated convergence theorem, we deduce that / \um -u\2 —► 0 for every k G N. , J m—>oo In particular, there exists a subsequence rrij for which umj —> u a.e. on (—A;, fc) x as j —+ co. Letting A; —> oo and considering a diagonal sequence, we see that (iv) holds. Furthermore, given t G R and fc e N, there exists a subsequence rrij for which umj(t) —* u(t) a.e. on fi^ as j —+ oo. Letting k —► oo and considering a diagonal sequence, we see that (iii) holds. Finally, it follows from (i) and Remark 1.3.13(i) that u\Qk e W1'00(R,if-1(fijfe)). Hence (i) is established. □ Proof of Theorem 9.3.4. We apply a compactness method, and we proceed in four steps. Consider ip G W. Step 1. Construction of approximate solutions. We have V = V\ + V2 with Vi G LP(Q) and V2 G L°°(fi). Given m e N, define the potentials and V2m by „___. , , I r n \**s I 11 I r o 1 %C- I ^» M t _ ^•(x) ifj^(x)|m Define the functions am and bm by J a{z) if |2| > i J &(*) if |*| < m I mza(±) if |*| < I ^6(m) if |*| > m. Finally, set gm(u) = V^u + V2mu - am(u) + £>m(«) for u G (ft). Since V™,V2m G L°°(Q) and both am and fcTO are (globally) Lipschitz continuous C —> C, we see that gm is Lipschitz continuous L2(fl) —> L2(f]). It follows from Corollary 3.3.11 that there exists a unique solution um G C(R,H0l(Q)) n C^R,^-1^)) of the problem r ^r + ^m + ^m(^) = o (9-3-5) 1 „«(0) = In addition, (9.3.6) ||um(i)||L» - IMU» and £m(um(t)) = Em(i2 -1 j v?\u\2+1 y *ra(i„i) -1 y *m(m), o. n n n n and the functions $TO and \I/m are defined by I r\z\ i r\A $m(z) = o / am(s)d$ and *m(z) = - / 6m(s)ds for all z G C. ^ Jo ^ Jo 296 9. further results Step 2. Estimates of the approximate solutions. It follows from (9.3.6) that um is bounded in L°°(R, L2(f2)). Note that, by the dominated convergence theorem, (9.3.7) Em( E( 0 and compare inequality (9.3.4)) \\um(t)\\^ < C + C||^ri|l,||ura(t)|||ri + ||*m(um(t))||Li. Note that ||V7"||lp < Note also that we may assume that ||V"i||iv is arbi- trarily small, by modifying V2. In particular, we may assume that C\\V™\\lp < 1/4. Finally, one easily verifies (see the proof of Lemma 9.3.3) that there exists C such that \\ym(um(t))\\L1 <-\\um(t)fHl + c\\um{t)\\l., therefore, (9.3.8) um is bounded in L°°(R,Hi(Q)). Finally, it follows from elementary calculations that for every e > 0, there exists C£ such that \9m(u)\ < |V7»IN + |V2m|M + CM1'8 + \u\1+£) . We deduce easily from Holder's and Sobolev's inequalities and (9.3.8) that, given k g N, (9.3.9) gm{um) is bounded in £°°(R, (ftfc)), where fifc = D {x g : |x| < k}. In particular, (pm(um))meN is bounded in L°°{R1H~1(Qk)), and it follows from (9.3.5) that (mm|nJmeN is bounded in Step 3. Passage to the limit. By Step 2, (um)m£^ satisfies the assumptions of Lemma 9.3.6. Let u be its limit. It follows from (9.3.5) that, for every tp g V(Sl) and every g 2>(R), J {iu? + Aum + gm(um), ip)v,p4>{t)dt = 0. r This means that (9.3.10) j (-(ium,i>)ct>,(t) + (um,A'il>}(t))dt +j Jgm(um)4;(f>dxdt = Q. r an It follows easily from (9.3.8) and from property (ii) of Lemma 9.3.6 that / (-(ium,ip)\t) + {um,A1>)(t))dt —+ J x m—t-oo (9.3.11) R y (-){t))dt Furthermore, the function hm(t,x) = grn(um)ij;(x)(f)(t) has compact support. We therefore deduce from (9.3.9) that hm is bounded in L^(R x fi). By property (iv) of Lemma 9.3.6, hm —> + v>\og(\u\2))ip a.e. on R x Q,. Since hm has compact 9.3. the logarithmic schrodinger equation 297 support, it follows from Proposition 1.2.1 that hm —► (Vu+u\og(\u\2))ipdxdt = 0, which implies that J (iut + Au + Vu + u log |-uj2, ip)v,tV(t)dt = 0. It follows that, for all teR, (9.3.12) iut + Au + Vu + ulog{\u\2) = 0 in H~1(Qk) for every k eN. In addition, «(0) =

u(t) in L2(f2). Therefore, by boundedness of um in H^(Q.) and Holder's and So-bolev's inequalities, um(t) —> u(t) in Lq(Q,) for every 2 < q < (2 < q < oo if N = 1,2). We now may pass to the limit in (9.3.6). We apply the weak lower semicontinuity of the H1 norm for the gradient term, we apply property (iii) of Lemma 9.3.6 and Fatou's lemma to the term R. Thus, by conservation of energy, (9.3.14) j |Vw|2 + J A(\u\) is continuous R->R. Since both terms in (9.3.14) are lower semicontinuous R R (the second one by Fatou's lemma), we deduce easily (see Cazenave and Haraux [63], lemma 2.4.4) that they are in fact continuous R -* R. In particular, u e C(R,Hq(£1)) and u e C(R,X) (by Lemma 9.3.1(h)). Therefore, u e C(R,W), and by the equation and Lemma 9.3.3, u e C1(R, W*). Finally, one proves continuous dependence by a similar argument (compare Step 3 of the proof of Theorem 3.3.9). This completes the proof. □ Remark 9.3.7. Strangely enough, one can apply the theory of maximal monotone operators to the equation (9.3.1). In particular, one can obtain stronger regularity if the initial value is smoother, and one can construct solutions of (9.3.1) for initial data in L2(Q) (see Cazenave and Haraux [63] and Haraux [157]). Note that one does not know whether the 1? solutions are unique. Remark 9.3.8. At least in the case where fl = RN and V = 0, equation (9.3.1) has standing waves of the form u(t, x) = elojt0. t€M l 0 and a > 0. Consider the following problem: f iiH 4- Aw - t)\u\au ~ 0, 9-4.1 ,m I u{0) = ip. We already know that if a < 4/(7V - 2) (a < oo, if N = 1,2), the problem (9.4.1) has a solution u e L°°(K,^(fi)) n W1,O0(R, if _1(Q)) for every <^ e H$(Q) (see Section 3.4). In addition, if Q = RN, or if iV = 1, or if JV = 2 and a < 2, the solution is unique (see Corollary 4.3.3 and Remarks 3.5.4 and 3.6.4). However, those results do not apply when a > 4/(N - 2). We present below a result of Strauss [324] (see also [321]) that applies for arbitrarily large a's. Before stating the result, we need some definitions. Let us denote by V the Banach space equipped with the usual norm (see Proposition 1.1.3). Since V(Sl) is dense in both Hb(Q) and La+2{Q), where the Banach space H (fi) + La+1 (fi) is equipped with its usual norm (see Proposition 1.1.3). Since A is continuous Hq(Q) —► H~1(Q) and u \u\au is continuous La+2(Q) —► La+l (O), it follows that the operator ( V —> V* \ u h-> Aw — rj\u\au is continuous. Therefore, if u € L°°(R,K) n W1'°°(R, V*), then equation (9.4.1) makes sense in V*. Finally, we define E(u) = I f \Vu\2 + -2— f \u\a+2 for all ueV. 2 7 a -H 2 7 We have the following result (see Strauss [324]). Theorem 9.4.1. Let n > 0 and a > 0. It follows that for every y e V, there exists a solution u 6 X°°(R,F) n W^li0O(R, V*) of equation (9.4.1) t/»o< satisfies (9.4.2) N*)IU* = IMU* and (9.4.3) < for all t € R. Remark 9.4.2. Note that, in particular, u <= C(R, V*), and so u is weakly continuous R -ff,j(ft) and R -+ LQ+2(ft); in particular, € V for all t € R. Therefore, tt(0) makes sense (in V) and E(u(t)) is well defined for all tel. Remark 9.4.3. Note that when a < 4/(TV - 2) (a < oo, if N = 1,2), then H£{Q) ^ LQ+2(0), therefore, V = H^Q). 300 9. further results REMARK 9.4.4. As observed before, when a < Aj{N - 2) (a < oo, if N = 1), Theorem 9.4.1 follows from the results of Section 3.4. Before proceeding to the proof of Theorem 9.4.1, we establish the following two lemmas. Lemma 9.4.5. v and v* are reflexive. Proof. We need only show that v is reflexive. By Eberlein-Smulian's theorem, we need to show that, given any bounded sequence (um)men C V, there exist a subsequence mk and u E v such that umk —u in v as k —> oo. Let p — a 4- 2. We recall that if u E v and

v in D'(^); hence u = v E v. It follows that for every oo /or a. a. x G £2. (hi) um(i, en) —»■ x) as m —» oo for a.a. (t, x) G R x £2. Proof. Let £ N and let fit=ftn{iefi; |x| < for k G N. Consider an integer o > AT/2. It follows from Sobolev's embedding theorem that JY^fifc) <—> La+2(Ofc), from which we obtain by duality L^r(fifc) ^ H~q(Q,k). Therefore, um\nk is bounded in fc, fc), H^fifc)) D W1'°c((—k,k),H~Q(Qk))- Therefore (by Proposition 1.1.2), there exist a subsequence (which we still denote by (um)me^) and u G L°°((-k,k),H1^)) such that um(t)\nk u(i) in i?1^). Letting A; —* oo and considering a diagonal sequence, we see that there exist a subsequence (which we still denote by (um)men) and u G L°°(R,if such that um{t)\^k -± u(t) in i?1(fi/c) for every G N and every i G R, This implies in particular that um(t) -± u(t) in H^Q). Therefore, u G L°°(R,H&(Q)), and (i) holds. In addition, since the embedding H1^) ^ L2(Qk) is compact, we have um(t)\ak -» u(t)|nfc 9.4. existence of weak solutions for large nonlinearities 301 in Z/2(f2fc) for every k G N and every teR. Applying the dominated convergence theorem, we deduce that —» 0 for every fceN. m—»oo In particular, there exists a subsequence rrij for which umi —> u a.e. on (-fc, k) x fi/, as j —► oo. Letting A; —> oo and considering a diagonal sequence, we see that (hi) holds. Furthermore, given ( e K and A; € N, there exists a subsequence rrij for which umj(t) —► u(t) a.e. on flk as j> —> oo. Letting k —► oo and considering a diagonal sequence, we obtain (ii). Finally, it follows from (i) and Lemma 9.4.5 that um{t) ->> u(t) in V (hence in V*) for all t G R. By Theorem 1.2.4 and Remark 1.3.13(i), w G X°°(R, V) n W1,00(R, V*). This completes the proof. □ Proof of Theorem 9.4.2. We construct the solution u by a compactness method, and we proceed in three steps. Step 1. Construction of a sequence of approximate solutions. Given an integer m > 1, let J -t]\z\az if \z\ < m [ —rymrz if \z\ > m. In particular, fm is globally Lipschitz continuous C —► C. Let Gm(z) = / fm{s)ds . Jo Given u€ H&(Q), let gm(u)(x) = fm{u(x)) for a.a. x G Q and ^m(«) = ^ / |Vu|2 + J Gm(u). Applying Corollary 3.3.11, we see that there exists a unique solution um g C(R,#. Furthermore, (9-4-5) \\um{t)\\v = IMU» and (9.4.6) £m(um(£)) = £m(<^) for every £ G R. Step 2. Estimates of um. Since GTO > 0, it follows from (9.4.5) and (9.4.6) that (9.4.7) um is bounded in L°°(R, #d(fi)) and (9.4.8) Gm(um) is bounded in L°°(R, L1^)). (9.4.4) {«u 302 9. FURTHER RESULTS On the other hand, one easily verifies that \gm(z)\^ < (a + 1)Gm(z) for all z £ C and all m £ N. Applying (9.4.8), we deduce that (9.4.9) gm(um) is bounded in L°°(E,(ft)). Therefore, it follows from (9.4.4) that (9.4.10) is bounded in L°°(R, V*). Step 3. Conclusion. It follows from (9.4.7) and (9.4.10) that we may apply Lemma 9.4.6 to the sequence um. Let u be the limit of um. By Lemma 9.4.6(i) and (ii), the weak lower semicontinuity of the H1 norm and Fatou's lemma, we deduce that u(t) £ La+2(Q) for every t £R and that (9.4.3) holds. In particular, u £ L°°(R,La+2(n)), and so u £ L°°(R,V). Furthermore, it follows from property (i) that u(0) = (p. Finally, we deduce from the equation (9.4.4) that for every

(ft), J (in? + Aum + gm(um), ^^{fidt = 0. r This means that (9.4.11) j (-(ium,tp)'(t) + (um,Ail;)(f>(t))dt + J J gm(um)ijxt>dxdt = 0. R R fi It follows easily from (9.4.7) and from property (i) of Lemma 9.4.6 that J (-{ium^)(j>'(t) + (um,AiP)(t))dt (9.4.12) J (-(iu, ip)(t))dt. Furthermore, the function hm(t,x) = gm(um)%p(x)4>{t) has compact support. Therefore, it follows from (9.4.9) that hm is bounded in L°+1(R x ft). By property (iii) of Lemma 9.4.6, hm —> —r}\u\aw -rj\u\auip^) in LX(R x ft). Applying (9.4.11) and (9.4.12), we thus obtain J (~{iu^)4>'{t) + (u,Ai))(j)(t))dt~r] J J \u\aip(f>dxdt = Q, R R ^ which implies that (iut + Au- r)\u\Qu, i))v,vcf>{t)dt = 0. / Since u £ L°°(R,V), we obtain easily that ut £ L°°(R:V*) and that u satisfies (9.4.1). It remains to establish conservation of charge. This follows easily by taking the V - V* duality product of the equation with iut £ V*. This completes the proof. □ 9.5. comments 303 Remark 9.4.7. In the case where a > 4/(iV - 2), it is not known whether the solution given by Theorem 9.4.1 is unique or not, even when Q = WN. We do not know either whether the energy is conserved. Remark 9.4.8. Remember that Theorem 3.3.5 applies to the case 77 < 0 and a < 4/(iV — 2). On the contrary, in the case a > 4/(N — 2), the method of proof of Theorem 9.4.1 does not apply when 77 < 0. We do not know whether it is possible to construct (local) solutions of (9.4.1) in this case. 9.5. Comments The conservation laws that we used in these notes are conservation of charge and energy, and the pseudoconformal conservation laws. They are related to the invariance of the equation for some groups of transformations. On this subject, consult Ginibre and Velo [139], Olver [285]. When N = 1 and g(u) = \\u\2u, there are infinitely many conservation laws (cf. Zakharov and Shabat [367]), while in general, there do not seem to be other useful conservation laws (cf. Serre [310]). In relation with the invariance properties of nonlinear Schrodinger equations, one can construct families of explicit solutions for some nonlinearities (cf. Fushchich and Serov [121, 122, 123]). Unfortunately, these solutions do not in general belong to the energy space. Nonlinearities of different types than those studied here were also considered. See Baillon, Cazenave, and Figueira [9], Cazenave [57], Stubbe and Vazquez [328, 329], Adami, and Teta [2] and Adami, Dell'Antonio, Figari, and Teta [1], and Colin [85, 86]. Quasilinear Schrodinger equations require in general completely different methods for proving the existence of solutions, making an essential use of the smoothing properties of the Schrodinger group. See, for example, Biagioni and Linares [29], Chang, Shatah, and Uhlenbeck [77], Chihara [79], Colliander et al. [88, 91], Hayashi [166], Hayashi and Hirata [170], Hayashi and Kaikina [171], Hayashi, Kaik-ina, and Naumkin [173], Hayashi and Kato [176], Hayashi and Naumkin [180], Hayashi and Ozawa [190, 191], Katayama and Tsutsumi [201], Kenig, Ponce, and Vega [212, 215], Klainerman and Ponce [217], Ozawa and Tsutsumi [292], Takaoka [331, 333], and Y. Tsutsumi [346]. See also Lange [222] for a suggestive numerical study. Systems of Schrodinger equations or coupled systems with other equations (Klein-Gordon, for example) are also of a great interest. See, for example, Cipo-latti and Zumpichiatti [84], and Colin and Weinstein [87] (systems of Schrodinger equations); Castella [54] (Schrodinger-Poisson system); Baillon and Chadam [10], Bachelot [8], and Ozawa and Tsutsumi [290] (Schrodinger-Klein-Gordon system); Ginibre and Velo [145], Guo, Nakamitsu, and Strauss [156], Nakamitsu and Tsutsumi [254], and Y. Tsutsumi [345, 347] (Maxwell-Schrodinger system); Schochet and Weinstein [307], Lee [224], Ozawa and Tsutsumi [289, 291], Glangetas and Merle [146, 147], Kenig, Ponce, and Vega [213], Merle [247], Ginibre, Tsutsumi, and Velo [131], Bourgain [36], Bourgain and Colliander [40], Colliander and Staffi-lani [92], Masselin [241], Takaoka [332], and Tzvetkov [349] (Zakharov system); and Ghidaglia and Saut [124], Cipolatti [82, 83], Ozawa [288], Hayashi [167], Hayashi and Hirata [168, 169], and Ohta [279, 280, 281] (Davey-Stewartson system). 304 9. FURTHER RESULTS Stochastic nonlinear Schrodinger equations (i.e., with a probabilistic noise) were also considered. They display interesting phenomena, in particular concerning blowup. See de Bouard and Debussche [98, 99, 100, 101], and de Bouard, Debussche, and Di Menza [102]. Bibliography [1] Adami, R., DelPAntonio, G., Figari, R., and Teta, A. The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity. 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