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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.

35Q55 NLS equations (nonlinear Schrödinger equations)
35P25 Scattering theory for PDEs
35B40 Asymptotic behavior of solutions to PDEs