Uniform stabilization for a strongly coupled semilinear/linear system

Abstract

Abstract In this manuscript, we analyze the exponential stability of a strongly coupled semilinear system of Klein-Gordon type, posed in an inhomogeneous medium Ω \Omega , subject to local dampings of different natures distributed around a neighborhood of the boundary according to the geometric control condition (GCC). The first one is of the type viscoelastic and is distributed around a neighborhood ω \omega of the boundary ∂ Ω \partial \Omega of Ω \Omega , according to the GCC. The second one is a frictional damping and we consider it hurting the GCC. The third dissipation, which acts only in the second equation (according to the GCC), is of the frictional type. 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. We also prove the exponential decay for the linear problem associated with this same system, and in this case, no restrictions are made with respect to dimension of the space nor with respect to the limitation of the initial data in the phase space.