Asymptotic Stability for the Damped Schrödinger Equation on Noncompact Riemannian Manifolds and Exterior Domains

Abstract

The Schrödinger equation subject to a nonlinear and locally distributed damping, posed in a connected, complete, and noncompact n dimensional Riemannian manifold (ℳ, g) is considered. Assuming that (ℳ, g) is nontrapping and, in addition, that the damping term is effective in ℳ \\ Ω, where Ω ⊂ ⊂ ℳ is an open bounded and connected subset with smooth boundary ∂Ω, such that is a compact set, exponential and uniform decay rates of the L 2 − level energy are established. The main ingredients in the proof of the exponential stability are: (A) an unique continuation property for the linear problem; and (B) a local smoothing effect for the linear and nonhomogeneous associated problem.