On uniform quasistabilization of solutions of the second mixed problem for a~second-order hyperbolic equation

Yu. A. Mikhailov

Sbornik. Mathematics, Tome 57 (1987) no. 1, pp. 243-262

Résumé

The problem \begin{gather*} u_{tt}(x,t)=\operatorname{div}_x(A(x)\nabla_xu(x,t)),\qquad x\in\Omega,\quad t>0; \\ \frac{\partial u}{\partial N}\bigg|_{\partial\Omega}=0;\quad u|_{t=0}=\varphi(x);\quad u_t|_{t=0}=0 \end{gather*} is considered in the cylindrical region $\Omega\times(0,+\infty)$. A criterion for uniform stabilization (with respect to $x$ in $\Omega$) of the mean over $t$ of order $\alpha$, $\alpha>[n/2]+1$, of the solution $u(x,t)$ of this problem is proved for a rather broad class of unbounded domains $\Omega\subset\mathbf R^n$ (determined by conditions of isoperimetric type). Bibliography: 15 titles.