Stability for semilinear hyperbolic coupled system with frictional and viscoelastic localized damping

Abstract

In this paper we prove stability results for a semilinear hyperbolic coupled system subject to a viscoelastic localized damping acting in the first equation and a frictional localized one acting in the second equation of the system. We divide the proof into two parts. In the first part the equations are posed in a homogeneous medium Ω with the damping acting in a boundary neighbourhood. In the second part the equations posed in an inhomogeneous medium Ω with the damping acting in a collar of the boundary and in an appropriate mesh in the interior. Due to the assumptions on the nonlinear function involving the frictional damping, general decay rates are obtained. To prove the results we used the tools of the Microlocal Analysis Theory.