Global existence and open loop exponential stabilization of weak solutions for nonlinear Schrödinger equations with localized external Neumann manipulation

In this paper, we consider the nonlinear Schrödinger equations (NLS) with (focusing/defocusing) interior source and (possibly nonlinear) damping on a bounded regular domain in the Euclidean space. Moreover, it is assumed that the solutions are subject to external Neumann boundary manipulation on one portion of the boundary. Our aim is to obtain global existence of the weak solutions under various assumptions on the sign of the source and powers of the nonlinearities. In the case of a linear damping, we also prove that solutions decay exponentially under the assumption that the Neumann type control at the boundary decays in a similar manner.