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August  2020, 40(8): 4801-4819. doi: 10.3934/dcds.2020202

## Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation

 1 Sorbonne Université & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France 2 Department of Mathematics, Hangzhou Normal University, 2318 Yuhangtang Road, 311121 Hangzhou, China

* Corresponding author

Received  October 2019 Published  May 2020

Fund Project: ZH thanks NSFC 11671353, 11401153, Zhejiang Provincial Natural Science Foundation of China under Grant No. LY18A010025, and CSC for their financial support; and the Laboratoire JacquesLouis Lions for its kind hospitality

We study the time-asymptotic behavior of solutions of the Schrödinger 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
 $\frac2N$
.
Citation: Thierry Cazenave, Zheng Han. Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4801-4819. doi: 10.3934/dcds.2020202
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