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

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 $t\to\infty$.
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