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Scattering theory for Hartree type equations. (English) Zbl 0634.35059
Summary: We study the asymptotic behavior in time of the solutions and the scattering theory for the following Hartree type equation $(1)\quad iu_ t+\Delta u=\lambda f(| u|^ 2)u,\quad (t,x)\in {\mathbb{R}}\times {\mathbb{R}}\quad n,$ $(2)\quad u(0,x)=\Phi (x),\quad x\in {\mathbb{R}}\quad n,$ where f($$| u|$$ $$2)=| x|^{- \gamma}*| u|$$ 2, $$0<\gamma <Min(4,n)$$, $$n\geq 2$$ and $$\lambda\in {\mathbb{R}}$$. $Let\quad \Sigma^{\ell,m}=\{v\in L^ 2({\mathbb{R}}\quad n);\| v\|^ 2_{\Sigma^{\ell,m}}=\sum_{| \alpha | \leq \ell}=D^{\alpha}v\|^ 2_ 2+\sum_{| \beta | \leq m}\| x^{\beta}v\|^ 2_ 2<\infty \},\ell,m\in {\mathbb{N}}.$ We prove that when $$(4/3)<\gamma <Min(4,n)$$ and $$\lambda >0$$, all solutions of (1)-(2) with $$\Phi \in \Sigma^{\ell,m}$$ are dispersive in $$\Sigma^{\ell,m}$$ and that when $$1<\gamma <Min(4,n)$$ and $$\lambda\in {\mathbb{R}}$$, the solutions of (1)-(2) with $$\Phi \in \Sigma^{\ell,m}$$ and $$\| \Phi \|_{\Sigma^{1,1}}$$ small are dispersive in $$\Sigma^{\ell,m}$$. This implies asymptotic completeness in $$\Sigma^{\ell,m}$$ of the wave operators for $$(4/3)<\gamma <Min(4,n)$$ and $$\lambda >0$$. Furthermore when $$\lambda >0$$, we show the existence of scattering states in L 2($${\mathbb{R}}^ n$$) for arbitrary data in $$\Sigma^{1,1}$$ if $$1<\gamma <(4/3)$$ and the non-existence of scattering states in L 2($${\mathbb{R}}^ n$$) for $$0<\gamma \leq 1$$.

##### MSC:
 35P25 Scattering theory for PDEs 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
 [1] J.E. Barab , Nonexistence of asymptotic free solutions for a nonlinear Schrödinger equation . J. Math. Phys. , t. 25 , 1984 , p. 3270 - 3273 . MR 761850 | Zbl 0554.35123 · Zbl 0554.35123 · doi:10.1063/1.526074 [2] P. Brenner , On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations . Math. Z. , t. 186 , 1984 , p. 383 - 391 . MR 744828 | Zbl 0524.35084 · Zbl 0524.35084 · doi:10.1007/BF01174891 · eudml:173453 [3] P. Brenner , On scattering and everywhere defined scattering operators for non-linear Klein-Gordon equations . J. Differential Equations , t. 56 , 1985 , p. 310 - 344 . MR 780495 | Zbl 0513.35066 · Zbl 0513.35066 · doi:10.1016/0022-0396(85)90083-X [4] J.M. Chadam and R.T. Glassey , Global existence of solutions to the Cauchy problem for time dependent Hartree equations . J. Math. Phys. , t. 16 , 1975 , p. 1122 - 1130 . MR 413843 | Zbl 0299.35084 · Zbl 0299.35084 · doi:10.1063/1.522642 [5] A. Friedman , Partial Differential Equations . Holt - Rinehart and Winston , New York , 1969 . MR 445088 | Zbl 0224.35002 · Zbl 0224.35002 [6] J. Ginibre and G. Velo , On a class of nonlinear Schrödinger equation I, II . J. Funct. Anal , t. 32 , 1979 , p. 1 - 32 , 33 - 71 ; III, Ann. Inst. Henri Poincaré , Physique Théorique , t. 28 , 1978 , p. 287 - 316 . Numdam | MR 533219 | Zbl 0397.35012 · Zbl 0397.35012 · numdam:AIHPA_1978__28_3_287_0 · eudml:75982 [7] J. Ginibre and G. Velo , On a class of nonlinear Schrödinger equations with non local interactin . Math. Z. , t. 170 , 1980 , p. 109 - 136 . Article | MR 562582 | Zbl 0407.35063 · Zbl 0407.35063 · doi:10.1007/BF01214768 · eudml:172905 [8] J. Ginibre and G. Velo , Sur une équation de Schrödinger non linéaire avec interaction non locale, in Nonlinear partial differential equations and their applications . Collège de France , Séminaire , vol. II , Pitman , Boston , 1981 . MR 652511 | Zbl 0497.35024 · Zbl 0497.35024 [9] J. Ginibre and G. Velo , Scattering theory in the energy space for a class of non-linear Schrödinger equations , J. Math. pures et appl. , t. 64 , 1985 , p. 363 - 401 . MR 839728 | Zbl 0535.35069 · Zbl 0535.35069 [10] R.T. Glassey , Asymptotic behavior of solutions to a certain nonlinear-Hartree equations , Comm. Math. Phys. , t. 53 , 1977 , p. 9 - 18 . Article | MR 486956 | Zbl 0339.35013 · Zbl 0339.35013 · doi:10.1007/BF01609164 · minidml.mathdoc.fr [11] N. Hayashi and M. Tsutsumi , L\infty -decay of classical solutions for nonlinear Schrödinger equations , preprint. · Zbl 0651.35014 · doi:10.1017/S0308210500019235 [12] N. Hayashi , K. Nakamitsu and M. Tsutsumi , On solutions of the initial value problem for the nonlinear Schrödinger equations , to appear in J. Funct. Anal. MR 880978 | Zbl 0657.35033 · Zbl 0657.35033 · doi:10.1016/0022-1236(87)90002-4 [13] N. Hayashi and Y. Tsutsumi , Remarks on the scattering problem for nonlinear Schrödinger equations , preprint. MR 921265 · Zbl 0633.35059 [14] W. Hunziker , On the space-time behavior of Schrödinger wavefunctions . J. Math. Phys. , t. 7 , 1965 , p. 300 - 304 . MR 193939 | Zbl 0151.43801 · Zbl 0151.43801 · doi:10.1063/1.1704932 [15] A. Jensen , Commutator methods and a smoothing property of the Schrödinger evolution group . Math. Z. , t. 191 , 1986 , p. 53 - 59 . Article | MR 812602 | Zbl 0594.35032 · Zbl 0594.35032 · doi:10.1007/BF01163609 · eudml:173656 [16] E.M. Stein , Singular Integral and Differentiability Properties of Functions , Princeton Univ. Press . Princeton Math. Series 30 , 1970 . MR 290095 | Zbl 0207.13501 · Zbl 0207.13501 [17] W.A. Strauss , Nonlinear invariant wave equations , in Invariant Wave Equations (Erice, 1977 ), Lecture Notes in Physics , t. 78 , Springer-Verlag , Berlin - Heidelberg - New York , 1978 , p. 197 - 249 . MR 498955 [18] W.A. Strauss , Nonlinear scattering theory at low energy . J. Funct. Anal. , t. 41 , 1981 , p. 110 - 133 . MR 614228 | Zbl 0466.47006 · Zbl 0466.47006 · doi:10.1016/0022-1236(81)90063-X [19] W.A. Strauss , Nonlinear Scattering theory at low energy: Sequel . J. Funct. Anal. , t. 43 , p. 281 - 293 , MR 636702 | Zbl 0494.35068 · Zbl 0494.35068 · doi:10.1016/0022-1236(81)90019-7 [20] R.S. Strichartz , Restrictions of Fourier Transforms to quadratic surfaces and decay of solutions of wave equations . Duke Math. J. , t. 44 , 1977 , p. 705 - 714 . Article | MR 512086 | Zbl 0372.35001 · Zbl 0372.35001 · doi:10.1215/S0012-7094-77-04430-1 · minidml.mathdoc.fr [21] Y. Tsutsumi , Global existence and asymptotic behavior of solutions for nonlinear Schrödinger equations , Doctor Thesis , Univ. of Tokyo , 1985 . · Zbl 0612.35104 [22] Y. Tsutsumi , Scattering problem for nonlinear Schrödinger equations . Ann. Inst. Henri Poincaré, Physique Théorique , t. 43 , 1985 , p. 321 - 347 . Numdam | MR 824843 | Zbl 0612.35104 · Zbl 0612.35104 · numdam:AIHPA_1985__43_3_321_0 · eudml:76303 [23] Y. Tsutsumi and K. Yajima , The asymptotic behavior of nonlinear Schrödinger equations , Bull. (New Series). Amer. Math. Soc. , t. 11 , 1984 , p. 186 - 188 . Article | MR 741737 | Zbl 0555.35028 · Zbl 0555.35028 · doi:10.1090/S0273-0979-1984-15263-7 · minidml.mathdoc.fr [24] K. Yajima , The surfboard Schrödinger equations . Comm. Math. Phys. , t. 96 , 1984 , p. 349 - 360 . Article | MR 769352 | Zbl 0599.35037 · Zbl 0599.35037 · doi:10.1007/BF01214580 · minidml.mathdoc.fr [25] J. Ginibre , Private communication . [26] T. Kato , On nonlinear Schrödinger equations , preprint, University of California , Berkeley , 1986 . MR 1037322 [27] Y. Tsutsumi , L2-solutions for nonlinear Schrödinger equations and nonlinear groups , to appear in Funkcialaj Ekvacioj . Article | MR 915266 | Zbl 0638.35021 · Zbl 0638.35021 · minidml.mathdoc.fr
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