Wave Decay on Manifolds with Bounded Ricci Tensor, and Related Estimates

Abstract

We investigate energy decay for solutions to the wave equation \(\partial_{t}^{2}u+a(x)\partial_{t}u-\Delta u=0\), with damping coefficient a≥0, where Δ is the Laplace–Beltrami operator on a compact Riemannian manifold M. We make a weak regularity hypothesis on the metric tensor of M, though one that guarantees the unique existence of the geodesic flow. We then establish exponential energy decay under the natural hypothesis that all sufficiently long geodesics pass through a region where a(x)≥a0>0, extending the scope of previous work done in the setting of a smooth metric tensor.