Turker Ozsari, University of Virginia


Stabilization of Weakly Damped Defocusing Semilinear Schroedinger Equation with Inhomogeneous Dirichlet Boundary Control

Abstract: n this study, we consider the stabilization as well as the existence of solutions of the weakly damped defocusing semilinear Schrödinger equation with an inhomogeneous Dirichlet boundary control. We prove that weak solutions decay to zero in $H^1\cap L^{p+2}$-sense ($p$ is the power of the nonlinearity) under the assumption that the boundary control also decays to zero in a similar sense. Moreover, we show that the decay rate of the boundary data controls the decay rate of the solutions up to the decay rate of the initial data. The proof of our main result demonstrates how the multiplier method combined with monotonicity and compactness techniques can be used to solve inhomogeneous boundary value problems. In addition, we note that the result is strong in the sense that it is independent of the dimension of domain, the power of the nonlinearity or the smallness of the initial data. Secondly, we study the global existence and stabilization of the H^2-solutions for the same problem.

Mentor: Irena Lasiecka (University of Virginia)