Communications in Mathematical Sciences

Volume 20 (2022)

Number 6

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

Pages: 1685 – 1701

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n6.a10

Author

Mariano De Leo (Departamento de Matemática, Instituto de Matemática de Bahía Blanca (INMABB–CONICET), Universidad Nacional del Sur, Buenos Aires, Argentina)

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 Schrödinger 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.

Keywords

stabilisation, localised damping, nonlinear Schrödinger, Hartree potential

2010 Mathematics Subject Classification

35Q55, 93B05, 93D15

Full Text (PDF format)

Received 2 September 2021

Received revised 12 January 2022

Accepted 17 January 2022

Published 14 September 2022