Asymptotic behavior for a dissipative nonlinear Schrödinger equation

Abstract

We consider the Schrödinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +Δu=λ|u|^αu \end{equation*} in ${\mathbb{R}}^N$, $N\ge 1$, where $\lambda \in \mathbb{C}$ with $\Im \lambda <0$. Assuming $\frac{2}{N+2}<\alpha <\frac{2}{N}$, we give a precise description of the long-time behavior of the solutions (including decay rates in $L^2$ and $L^{\infty }$, and asymptotic profile), for a class of arbitrarily large initial data.