Uniform stabilization of the Klein-Gordon system

Abstract

We consider the Klein-Gordon system posed in an inhomogeneous medium \begin{document}$ \Omega $\end{document} with smooth boundary \begin{document}$ \partial \Omega $\end{document} subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood \begin{document}$ \omega $\end{document} of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. 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 [ 5 ]. Although the present problem has some similarity to the reference [ 6 ] it is important to mention that due to the Kelvin-Voigt dissipation character associated with the nonlinearity of the problem the approach used is completely new, which is the main purpose of this paper.