Asymptotic behavior of cubic defocusing Schrödinger equations on compact surfaces.

We are concerned with the asymptotic behavior of two different cubic, defocusing and damped nonlinear Schrödinger equations on compact Riemannian manifolds without boundary. Two mechanisms of locally distributed damping are considered: a weak damping and a stronger one. In the first problem, we consider a two-dimensional case and prove that the corresponding energy functional goes to zero as time goes to infinity. The proof is based on a result of propagation of singularities due to Dehman et al. (Math Z 254(4):729-749, 2006) and Strichartz type inequalities due to Burq et al. (Am J Math 126(3):569-605, 2004), combined with new ingredients which come from the observability inequality associated with the linear problem. When a stronger damping is in place, we show that the energy functional decays exponentially to zero and for this purpose a forced smoothing effect due to Aloui (Collect Math 59(1):53-62, 2008) plays an essential role in the proof. [ABSTRACT FROM AUTHOR]