RT Journal Article
SR 00
ID 10.1016/j.jde.2019.10.016
A1 Bégout, Pascal
A1 Diaz, Jesus Ildefonso
T1 Finite time extinction for the strongly damped nonlinear Schrödinger equation in bounded domains
JF Journal of Differential Equations
YR 2020
FD 2020-03-20
VO vol.268
IS n°7
SP 4029
OP 4058
K1 damped Schrödinger equation
K1 existence
K1 uniqueness
K1 finite time extinction
K1 asymptotic behavior
AB 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.
PB Elsevier
SN 0022-0396
LK https://publications.ut-capitole.fr/id/eprint/43586/