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

This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by

$\left. \begin{array}{l} u_{tt} - \Delta_{\mathcal{M}}u+ a(x) g(u_... ...pace }\mathcal{M}\times \left] 0,\infty \right[ ,\smallskip \end{array} \right.$

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 $\underset{x\in V}\inf \vert\nabla f(x)\vert>0$.

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 $k \vert s\vert \leq \vert g(s)\vert \leq K \vert s\vert$ for all $\vert s\vert \geq 1$, then uniform and optimal decay rates of the energy hold.