Stabilization of solutions of the third mixed problem for a~second order parabolic equation in a~noncylindrical domain

V. I. Ushakov

Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 87-105

Résumé

This paper studies the behavior for large values of time $t$ of the solution of the third mixed problem in a noncylindrical domain $D\subset\mathbf R^{n+1}$ that expands as $t$ increases, for a linear second order parabolic equation in selfadjoint form without lower terms. In this connection the boundary condition is chosen so that the “energy conservation law” holds. For a very large class of domains a simple geometric characteristic of the domain is singled out-the function $V(t,\sqrt t)=\operatorname{mes}_n(D_t\cap\{|x|\sqrt t\})$, where $D_t$ is the intersection of the domain $D$ with the hyperplane $t=\operatorname{const}$ – determining the stabilization speed of the solution. Namely, it is proved that a solution $u(t,x)$ of the above problem with initial function $\varphi$ from $L_1(D_0)$ satisfies the estimate $$ \|u(t,x)\|_{L_\infty(D_t)}\leqslant\frac C{V(t,\sqrt t)}\|\varphi\|_{L_1(D_0)},\qquad t>0, $$ and the accuracy of this estimate is of the order of the convergence to zero as $t\to\infty$. Bibliography: 6 titles.