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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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