Asymptotic stability for a strongly coupled Klein-Gordon system in an inhomogeneous medium with locally distributed damping

We consider a strongly coupled Klein-Gordon system posed in an inhomogeneous medium Omega with smooth boundary partial derivative Omega subject to a local damping distributed around a neighborhood omega of the boundary according to the Geometric Control Condition. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gerard [16]. By using sharp L-p - L-q -Carleman estimates we prove a unique continuation property for coupled systems. We also consider a strongly coupled system of wave equations with localized nonlinear damping acting only on one equation. Considering a compact Riemannian manifold, we show that the energy of the coupled system goes uniformly to zero. (C) 2019 Elsevier Inc. All rights reserved.