Behaviors of the energy of solutions of two coupled wave equations with nonlinear damping on a compact manifold with boundary

Abstract: In this paper we study the behaviors of the the energy of solutions of coupled wave equations on a compact manifold with boundary in the case of indirect nonlinear damping . Only one of the two equations is directly damped by a localized nonlinear damping term. Under geometric conditions on both the coupling and the damping regions we prove that the rate of decay of the energy of smooth solutions of the system is determined from a first order differential equation .