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

Let (M, g) be a n-dimensional (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n\geqq 2}$$\end{document}) 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left. \begin{array}{l} u_{tt} - \Delta_{{\bf g}}u+ a(x)\,g(u_{t})=0,\quad\hbox{on\ \thinspace}{M}\times \left] 0,\infty\right[ ,u=0\,\hbox{on}\,\partial M \times \left] 0,\infty \right[, \end{array} \right. $$\end{document}where ∂M represents the boundary of M and a(x) g(ut) 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_\ast \subset M}$$\end{document} 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.