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Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation

  • * Corresponding author

    * Corresponding author

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

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  • 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 $.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B40.


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