Exponential decay for a class of non-local non-linear Schrödinger equations with localised damping

Abstract

In this paper we study the exponential decay of both the charge and the free energy for solutions of a family of non-linear, non-local Schrdinger equations with localised damping on the whole line. We first establish an observability inequality for the linear flow, from which we obtain the result in the linear case. Then we consider the non-linear case and by perturbative arguments we obtain the exponential decay for solutions with small initial data. Finally we discuss qualitative aspects of the dynamics and show that the stabilisation rate becomes smaller as the free damping region is chosen around the origin.