This article is cited in 5 papers

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

V. V. Skazka


Abstract: 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.

UDC: 517.9

MSC: Primary 35Q20, 35B40; Secondary 76U05

Received: 25.11.1983

 English version: Mathematics of the USSR-Sbornik, 1986, 54:1, 1–37

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