Finite time extinction for the strongly damped nonlinear Schr{ö}dinger equation in bounded domains

Abstract

We prove the \textit{finite time extinction property} $(u(t)\equiv 0$ on $\Omega$ for any $t\ge T_{\star },$ for some $T_{\star }>0)$ for solutions of the nonlinear Schrödinger problem $\mathrm{i}{u}_t+\Delta u+a|u{|}^{-(1-m)}u=f(t,x),$ on a bounded domain $\Omega$ of ${\mathbb{R}}^N,$ $N\le 3,$ $a\in \mathbb{C}$ with $\Im (a)>0$ (the damping case) and under the crucial assumptions $0<m<1$ and the dominating condition $2\sqrt{m}\Im (a)\ge (1-m)|\Re (a)|.$ We use an energy method as well as several a priori estimates to prove the main conclusion. The presence of the non-Lipschitz nonlinear term in the equation introduces a lack of regularity of the solution requiring a study of the existence and uniqueness of solutions satisfying the equation in some different senses according to the regularity assumed on the data.

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