Finite time extinction for a critically damped Schr{\"o}dinger equation with a sublinear nonlinearity

Abstract

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schr{\"o}dinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{\varepsilon+(|u|^2)^\alpha}u,$ with $a\in\mathbb{C},$ $\varepsilon\geqslant0,$ $2\alpha=(1-m)$ and $m\in[0,1).$ Here we consider the sublinear case $0<m<1$ with a critical damped coefficient: $a\in\mathbb{C}$ is assumed to be in the set $D(m)=\big\{z\in\mathbb{C}; \; \mathrm{Im}(z)>0 \text{ and } 2\sqrt{m}\mathrm{Im}(z)=(1-m)\mathrm{Re}(z)\big\}.$ Among other things, we know that this damping coefficient is critical, for instance, in order to obtain the monotonicity of the associated operator (see the paper by Liskevich and Perel'muter [16] and the more recent study by Cialdea and Maz'ya [14]). The finite time extinction of solutions is proved by a suitable energy method after obtaining appropiate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.