Asymptotics as $t\to\infty$ of solutions of a problem of mathematical physics

V. V. Skazka

Sbornik. Mathematics, Tome 54 (1986) no. 1, pp. 1-37

Résumé

Solutions are considered of the mixed problem of S. L. Sobolev $$ \frac{\partial^2}{\partial t^2}\biggl(\frac{\partial^2u}{\partial x^2_1}+\frac{\partial^2u}{\partial x_2^2}\biggr)+\frac{\partial^2u}{\partial x_2^2}=0 \quad\text{in}\quad \Omega,\qquad u\big|_{\partial\Omega}=0, $$ $u|_{t=0}=u_0$, $u_t|_{t=0}=u_1$, where $\Omega$ is the complement of a simply connected, compact, convex set in $R^2$. Asymptotic representations are given for a solution of this problem as $t\to\infty$. A boundary-layer phenomenon is discovered in a neighborhood of $\partial\Omega$. Bibliography: 15 titles.