Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result

Let ( M , g ) be a n -dimensional ( ) compact Riemannian manifold with boundary where g denotes a Riemannian metric of class C ∞ . This paper is concerned with the study of the wave equation on ( M , g ) with locally distributed damping, described by where ∂ M represents the boundary of M and a ( x ) g ( u t ) is the damping term. The main goal of the present manuscript is to generalize our previous result in C avalcanti et al. (Trans AMS 361(9), 4561–4580, 2009), treating the conjecture in a more general setting and extending the result for n -dimensional compact Riemannian manifolds ( M , g ) with boundary in two ways: (i) by reducing arbitrarily the region where the dissipative effect lies (this gives us a totally sharp result with respect to the boundary measure and interior measure where the damping is effective); (ii) by controlling the existence of subsets on the manifold that can be left without any dissipative mechanism, namely, a precise part of radially symmetric subsets . An analogous result holds for compact Riemannian manifolds without boundary.