Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation

Abstract

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