Abstract
Let (M, g) be a n-dimensional (\({n\geqq 2}\)) 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 Cavalcanti 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 \({M_\ast \subset M}\) 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.
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Communicated by C. Dafermos
Research of M. M. Cavalcanti partially supported by the CNPq Grant 300631/2003-0.
Research of V. N. Domingos Cavalcanti partially supported by the CNPq Grant 304895/2003-2.
Research of J. A. Soriano partially supported by the CNPq Grant 301352/2003-8.
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Cavalcanti, M.M., Domingos Cavalcanti, V.N., Fukuoka, R. et al. Asymptotic Stability of the Wave Equation on Compact Manifolds and Locally Distributed Damping: A Sharp Result. Arch Rational Mech Anal 197, 925–964 (2010). https://doi.org/10.1007/s00205-009-0284-z
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DOI: https://doi.org/10.1007/s00205-009-0284-z