Global smooth solutions for semilinear Schrödinger equations with boundary feedback on 2-dimensional Riemannian manifolds

Abstract

This paper considers the existence of global smooth solutions of semilinear schrodinger equation with a boundary feedback on 2-dimensional Riemannian manifolds when initial data are small. The authors show that the existence of global solutions depends not only on the boundary feedback, but also on a Riemannian metric, given by the coefficient of the principle part and the original metric of the manifold. In particular, the authers prove that the energy of the system decays exponentially.