Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control

In this paper, we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear Schrödinger equation with an inhomogeneous Dirichlet boundary control. First of all, we prove the global existence of weak solutions at the H¹-energy level together with the stabilization in the same sense. It is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate. Secondly, we prove some regularity and stabilization results for the strong solutions in H²-sense. The proof uses the direct multiplier method combined with monotonicity and compactness techniques. The result for weak solutions is strong in the sense that it is independent of the dimension of the domain, the power of the nonlinearity, and the smallness of the initial data. However, the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data.