Stabilization of solutions of the first mixed problem for the wave equation in domains with non-compact boundaries

In this paper we study the rate of decay, for large values of time, of the local energy of solutions of the first mixed problem for the wave equation in unbounded domains $\Omega\subset\mathbb R^n$, $n\geqslant 2$, with smooth non-compact boundaries. Under the assumption that the boundary surface satisfies a condition generalizing the condition of star-shapedness with respect to the origin we establish a power estimate of the rate of decay of the local energy as $t\to\infty$.
The proof is based on uniform estimates in the half-plane $\{\operatorname{Im} k>0\}$ of solutions of the corresponding spectral problem– the first boundary-value problem for the Helmholtz equation–obtained in the paper.