Behaviors of the energy of solutions of the wave equation with damping and external force

In this paper we study the behavior of the energy of solutions of the wave equation with localized damping and an external force on compact Riemannian manifold with boundary. Under the Geometric Control Condition of Bardos et al. (1992) [4] and certain condition on the force, we prove that the energy goes to zero when the time goes to infinity and we give the rate of decay of the energy functional. More precisely, the behavior of the energy depends on the L2 norm of the force and is determined from a forced differential equation.