Bégout, Pascal and Diaz, Jesus Ildefonso (2020) Finite time extinction for the strongly damped nonlinear Schrödinger equation in bounded domains. Journal of Differential Equations, vol.268 (n°7). pp. 4029-4058.

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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 $\vi u_t+\Delta u+a|u|^{-(1-m)}u=f(t,x),$ on a bounded domain $\Omega$ of $\mathbb{R}^N,$ $\mathbb{N}\le 3,$ $a\in\mathbb{C}$ with $\mathrm{Im}(a)>0$ (the damping case) and under the crucial assumptions $0<m<1$ and the dominating condition $2\sqrt m\,\mathrm{Im}(a)\ge(1-m)|\mathrm{Re}(a)|.$ 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.