Exponential decay for the semilinear wave equation with localized frictional and Kelvin-Voigt dissipating mechanisms

In the present paper, we are concerned with the semilinear viscoelastic wave equation in an inhomogeneous medium Omega subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood omega 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 wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space.