Finite time extinction for a critically damped Schrödinger equation with a sublinear nonlinearity

Abstract

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schrödinger 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 $a\in{\mathbb{C}},$ ${\varepsilon}\ge0,$ $2\alpha=(1-m)$ 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 appropriate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.