Large time behavior of solution for Hartree equation with long range interaction (Q1903382)

The asymptotic behaviour of the solutions of the Hartree equation \[ i\partial_t u= -\textstyle{{1\over 2}} \Delta_x u+ (|x|^{- \gamma}* |u|^2)u \] is investigated \((t\to \infty)\). Here, \(\gamma\leq 1\), \(u= u(t, x)\), \(x\in \mathbb{R}^n\). The nonlinear Schrödinger equation \[ i\partial_t u= -\textstyle{{1\over 2}} \Delta_x u+ |u|^{p- 1} u \] is considered also for the case \(n= 2\), \(p= 2\).