Finite time extinction for the strongly damped nonlinear Schrödinger equation in bounded domains

Abstract

We prove the finite time extinction property (u(t)≡0 on Ω for any t⩾T⋆, for some T⋆>0) for solutions of the nonlinear Schrödinger problem iut+Δu+a|u|−(1−m)u=f(t,x), on a bounded domain Ω of RN, N⩽3, a∈C with Im(a)>0 (the damping case) and under the crucial assumptions 0<m<1 and the dominating condition 2mIm(a)⩾(1−m)|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.