Stabilization of the solutions of the second boundary value problem for a~second order parabolic equation

A. K. Gushchin

Sbornik. Mathematics, Tome 30 (1976) no. 4, pp. 403-440

Résumé

The paper is a continuation of work (RZhMat., 1973, 10B301) in which in the case of a “noncontracting” unbounded domain $\Omega$ there is distinguished a geometric characteristic $v(R)=\operatorname{mes}(\Omega\cap\{|x|$ of the domain $\Omega$ that determines (under the fulfillment of a certain condition of “regularity” of the domain) the rate of stabilization for $t\to\infty$ of the solution in $(t>0)\times\Omega$ of the following second boundary value problem for a parabolic equation: $$ u_t=\sum_{i,j=1}^n\bigl(a_{i,j}(t,x)u_{x_i}\bigr)_{x_j},\qquad\frac{\partial u}{\partial N}\Bigr|_{x\in\partial\Omega}=0,\quad u|_{t=0}=\varphi(x) $$ in which the initial function $\varphi(x)$ decreases sufficiently rapidly as $|x|\to\infty$. It is proved in the present paper that the same characteristic also determines the rate of stabilization of the solution in a class of “contracting” ($\lim_{R\to\infty}v(R)/R=0$) domains $\Omega$. In this case, as in the case of a “noncontracting” domain, $\|u(t,x)\|_{L_\infty(\Omega)}$ tends to zero as $t\to\infty$ like $1/v(\sqrt{t})$: there exist estimates of the function $\|u(t,x)\|_{L_\infty(\Omega)}$ from above and from below having such an order of decrease. Bibliography: 11 titles.