On stabilization of the solution of the third mixed problem for the wave equation in a cylindrical domain

Necessary and sufficient conditions are obtained for stabilization as $t\to\infty$ of the solution of the third mixed problem for the wave equation in the exterior of an infinite closed cylindrical surface in space variables, in the presence of an influx of energy into the region through the boundary $\bigl(\frac{\partial u}{\partial n}+g(x)u|_{\partial\Omega}=0$, $g(x)$ of arbitrary sign$\bigr)$. An asymptotic expansion as $t\to\infty$ is established for the solution.
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