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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Identification Number : 10.1016/j.jde.2019.10.016


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.

Item Type: Article
Language: English
Date: 20 March 2020
Refereed: Yes
Place of Publication: Amsterdam
Uncontrolled Keywords: damped Schrödinger equation, existence, uniqueness, finite time extinction, asymptotic behavior
Divisions: Institut de mathématiques de Toulouse
Site: UT1
Date Deposited: 08 Jun 2021 13:35
Last Modified: 08 Jun 2021 13:35
URI: https://publications.ut-capitole.fr/id/eprint/43586
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