The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1018–1044 | Cite as

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

  • Michael TaylorEmail author


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)≥a 0>0, extending the scope of previous work done in the setting of a smooth metric tensor.


Damped wave equation Energy decay Microlocal regularity Ricci tensor 

Mathematics Subject Classification (2010)

35L10 35R01 



Thanks to anonymous referees for valuable suggestions to improve this paper.


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Authors and Affiliations

  1. 1.University of North CarolinaChapel HillUSA

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