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Asymptotic behavior for a dissipative nonlinear Schrödinger equation (2007.13697v1)

Published 27 Jul 2020 in math.AP

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 $L2$ and $L\infty $, and asymptotic profile), for a class of arbitrarily large initial data.

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