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¨NONLINEAR SCHRODINGER EQUATIONON REAL HYPERBOLIC SPACESJEAN–PHILIPPE ANKER & VITTORIA PIERFELICE
Abstract.WeconsidertheSchro¨dingerequationwithnoradialassumptiononrealhyperbolic spacesHn. We obtain in all dimensionsn2 sharp dispersive and Strichartzestimates for a large family of admissible pairs. As a first consequence, we obtain strongwell–posedness results for NLS. Specifically, for small initial data, we proveL2andH1global well–posedness for any subcritical power (in contrast with the Euclidean case)and with no gauge invariance assumption on the nonlinearityF. On the other hand,ifFis gauge invariant,L2charge is conserved and hence, as in the Euclidean case, itis possible to extend localL2solutions to global ones. The corresponding argument inH1requires conservation of energy, which holds under the stronger condition thatFisdefocusing. Recall that global well–posedness in the gauge invariant case was alreadyproved by Banica, Carles and Staffilani [4], for small radialL2data or for large radialH1data. The second application of our global Strichartz estimates isscatteringfor NLSboth inL2and inH1, with no radial or gauge invariance assumption. Notice that, onEuclidean spacesRn, this is only possible for the critical powerγ=1+4nand can be falsefor subcritical powers while, on hyperbolic spacesHn, global existence and scatteringof smallL2solutions holds for all powers 1< γ1+4. If we restrict to defocusingnnonlinearitiesF, we can extend theH1scattering results of [4] to the nonradial case.Also there is no distinction anymore between short range and long range nonlinearities :the geometry of hyperbolic spaces makes every power–like nonlinearity short range.
2000Mathematics Subject Classification.Primary 35Q55, 43A85 ; Secondary 22E30, 35P25, 47J35,58D25.Key words and phrases.NonlinearSchro¨dingerequation,hyperbolicspace,dispersiveinequality,Strichartz estimate, wellposedness, scattering.1
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J.–PH. ANKER & V. PIERFELICE
Abstract.Nous´etudionsle´quationdeSchr¨odingersurlesespaceshyperboliquesr´eelsHn,sansaucunehypoth`esederadialite´.Nouscommenc¸onspar´etablirunein´egalit´edispersive optimale en toute dimensionn2,ainsiquuneine´galit´edeStrichartzpourunegrandefamilledepairesadmissibles.Nousende´duisonsquel´equationsemilin´eaireestfortementbienpos´eedansL2ou dansH1,pourdesdonn´eesinitialespetitesetpourdesnonlinearite´srelativementg´ene´rales,enparticulierpourtouteslespuis-´sancesouscritiques(contrairementaucaseuclidien)etsanshypoth`esedinvariancepar changement de jauge. Dans ce dernier cas, on a conservation de la charge et lessolutionsL2locales se prolongent en solutionsL2globales;lephe´nom`eneanaloguedansH1reposesurlaconservationdel´energie,quiestv´eri´eepourdesnonlin´earite´s´de´focalisantes.RappelonsqueBanica,CarlesetStalani[4]avaientpre´cdemmentmontr´equel´equationsemilin´eaire´etaitglobalementbienpos´eepourdesnonline´arit´esinvariantesparchangementdejaugeetpourdesdonn´eesradialespetitesdansL2ou ar-bitraires dansH1. Comme seconde application, nous montrons qu’il y a diffusion (scat-tering) dansL2et dansH1,a`nouveausanshypothe`sederadialit´eoudinvarianceparchangement de jauge. Rappelons que dansRnceci n’est possible que pour l’exposantcritiqueγ= 1 +n4etpeuteˆtrefauxpourdesexposantssouscritiques,tandisquesur l’espace hyperboliquesHn, on a existence globale et diffusion pour tout exposant11+n4(et pour des conditions initiales petites dansL2).Danslecasde´focalisant,nouspouvonse´tendreaucasnonradiallesr´esultatsdediusionH1de [4]. Observonsegalementque,surlespacehyperbolique,touteslesnonlin´earite´sdetypepuissance´nontquuneeta`courteporte´e,contrairementaucaseuclidien.
1.IntroductionThenonlinearSchro¨dingerequation(NLS)inEuclideanspaceRn(1)(i ∂tu(t x) + Δu(t x) =F(u(t x))u(0 x) =f(x)has motivated a number of mathematical results in the last 30 years. Indeed, this equa-tion (especially in thecubiccaseF(u)=±u|u|2) seems ubiquitous in physics and appearsin many different contexts, including nonlinear optics, the theory of Bose–Einstein con-densates and of water waves. In particular a detailed scattering theory for NLS has beendeveloped.An essential tool in the study of (1) is thedispersive estimatekeitΔfkL(Rn)C|t|2nkfkL1(Rn)for the linear homogeneous Cauchy problem(2)(iu(t0u)+=fΔu= 0 tR xRnThis estimate is classical and follows directly from the representation formula for thefundamental solution. A well known procedure (introduced by Kato [15], Ginibre &Velo [9], and perfected by Keel & Tao [16]) then leads to theStrichartz estimates(3)kukLp(I;Lq(Rn))CkfkL2(Rn)+CkFkLp˜(I;Lq˜(Rn))
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