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 Schr¨odinger equation when the nonlinear damping term corresponds to the limit cases of some “saturating non-Kerr law” F(|u|2)u = a "+(|u|2)α u, with a 2 C, " > 0, 2 = (1 − m) and m 2 [0, 1). Here we consider the sublinear case 0 < m < 1 with a critical damped coefficient: a 2 C is assumed to be in the set D(m) = z 2 C; Im(z) > 0 and 2pmIm(z) = (1−m)Re(z). 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.