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Remarks on scattering theory and large time asymptotics of solutions to Hartree type equations with a long range potential. (English) Zbl 0937.35167
The authors deal with a scattering problem and with large time asymptotics of solutions. The equation considered is of Hartree type, where the nonlinear interaction term is $$f(|u|^2) = V*|u|^2$$, $$V(x) = \lambda |x|^{-\delta}$$, $$\lambda \in {\mathbb R}$$, $$0 < \delta < 1$$. Initial data is supposed to be sufficiently small. The existence of a unique final state is proved for all $$t > 1$$ uniformly with respect to $$x \in {\mathbb R}^n$$. The regularity conditions are lower than in authors’ preceeding results.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35P25 Scattering theory for PDEs 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
scattering; large time asymptotics; Hartree equations