Weyl formula for the eigenvalues of the dissipative acoustic operator

We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition ${\mathrm{\partial }}_{\nu }u-\gamma (x){\mathrm{\partial }}_{t}u=0$ on the boundary $\mathrm{\Gamma }$ and $\gamma (x)>0.$ The solutions are described by a contraction semigroup $V(t)=e^{tG},t\ge 0.$ The eigenvalues ${\lambda }_k$ of $G$ with $\mathrm{R}\mathrm{e}{\lambda }_k<0$ 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 ${\min }_{x\in \mathrm{\Gamma }}\gamma (x)>1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $G\mathrm{.}$