Fast decay of solutions for wave equations with localized dissipation on noncompact Riemannian manifolds

In this paper, uniform energy and L2L2 decay for solutions of linear wave equations with an energy term and localized dissipation on certain noncompact Riemannian manifolds are considered. We prove that the total energy of the solutions decay like O(1/t2)O(1/t2) as tt goes to infinity under some assumptions on the curvature of the manifolds and initial data. It is shown that the decay depends not only on the initial data but also on the curvature properties of the manifolds. As an application, we obtain the decay rate for the solutions of the wave equation with variable coefficients on an exterior domain of RnRn.