Regularity of the attractor for a weakly damped nonlinear Schrödinger equation on R Original Research Article

We study the long time behaviour of the solutions to a nonlinear Schrödinger equation, in presence of a damping term, and a forcing term, when the space variable x varies over View the MathML source. We show that the long time behaviour is described by an attractor which captures all the trajectories in View the MathML source. Our main result is concerned with the asymptotic smoothing effect for the equations. In other words, we prove that the attractor is included and compact in View the MathML source, generalizing results proven in [1] in the compact (bounded) case (see also [2]).