# Asymptotic behavior for a Schr\"odinger equation with nonlinear subcritical dissipation

## TL;DR

This paper analyzes the long-term behavior of solutions to a nonlinear Schrödinger equation with subcritical dissipation, providing detailed decay rates and asymptotic profiles for large initial data when the nonlinearity is near the critical exponent.

## Contribution

It offers a precise description of the asymptotic behavior and decay rates of solutions to a nonlinear Schrödinger equation with dissipation, especially for large initial data and near the critical nonlinearity.

## Key findings

- Solutions exhibit specific decay rates in $L^2$ and $L^
Infty$ norms.
- Asymptotic profiles of solutions are characterized.
- Results hold for large initial data when $ ext{Re} \, ext{lambda} < 0$ and $ ext{alpha}$ is close to the critical exponent.

## Abstract

We study the time-asymptotic behavior of solutions of the Schr\"odinger equation with nonlinear dissipation \begin{equation*} \partial _t u = i \Delta u + \lambda |u|^\alpha u \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $\lambda\in {\mathbb C}$, $\Re \lambda <0$ and $0<\alpha<\frac2N$. 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 $\frac2N$.