Weyl formula for the eigenvalues of the dissipative acoustic operator

Abstract

We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $\partial_{\nu} u - \gamma(x) \partial_t u = 0$ on the boundary $\Gamma$ and $\gamma(x) > 0.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \: t \geq 0.$ The eigenvalues $\lambda_k$ of $G$ with ${\rm Re}\: \lambda_k < 0$ yield asymptotically disappearing solutions $u(t, x) = e^{\lambda_k t} f(x)$ having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case $\min_{x\in \Gamma} \gamma(x) > 1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $G.$