Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-A sharp result

Abstract:

This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by \begin{equation} \left . \begin {array}{l} u_{tt} - \Delta _{\mathcal {M}}u+ a(x) g(u_{t})=0 \; \text {on \thinspace }\mathcal {M}\times \left ] 0,\infty \right [ , \end{array} \right . \nonumber \end{equation} where $\mathcal {M}\subset \mathbb {R}^3$ is a smooth oriented embedded compact surface without boundary. Denoting by $\mathbf {g}$ the Riemannian metric induced on $\mathcal {M}$ by $\mathbb {R}^3$, we prove that for each $\epsilon > 0$, there exist an open subset $V \subset \mathcal M$ and a smooth function $f:\mathcal M \rightarrow \mathbb R$ such that $meas(V)\geq meas(\mathcal M)-\epsilon$, $Hess f \approx \mathbf {g}$ on $V$ and $\underset {x\in V}\inf |\nabla f(x)|>0$.

In addition, we prove that if $a(x) \geq a_0> 0$ on an open subset $\mathcal {M}{\ast } \subset \mathcal M$ which contains $\mathcal {M}\backslash V$ and if $g$ is a monotonic increasing function such that $k |s| \leq |g(s)| \leq K |s|$ for all $|s| \geq 1$, then uniform and optimal decay rates of the energy hold.