Attractor for a damped cubic-Schrödinger equation on a two-dimensional thin domain

Abstract

In this article we prove the existence of an attractor for a dissipative nonlinear Schrödinger equation in the critical case for a two-dimensional thin domain. Moreover we prove that this attractor is smooth, i.e., made of smooth functions when the forcing term is smooth enough. The proofs use a splitting of the Fourier series of the solutions according to the geometry of the domain, together with anisotropic Sobolev inequalities.