Asymptotic behavior for a dissipative nonlinear Schr\"odinger equation

TL;DR

This paper analyzes the long-term behavior of solutions to a nonlinear Schrödinger equation with dissipative effects, providing decay rates and asymptotic profiles for large initial data in various dimensions.

Contribution

It offers a detailed description of the asymptotic behavior of solutions with nonlinear dissipation, including decay rates and profiles, for a broad class of initial data.

Findings

Solutions decay in $L^2$ and $L^$ norms over time.

Asymptotic profiles are characterized explicitly.

Results hold for arbitrarily large initial data within specified nonlinearities.

Abstract

We consider the Schr\"odinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +\Delta u=\lambda|u|^{\alpha}u \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $\lambda\in {\mathbb C} $ with $\Im\lambda<0$. Assuming $\frac {2} {N+2}<\alpha<\frac2N$, 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.