Stabilization of solutions of the first mixed problem for a parabolic equation of second order

The behavior for large time of the solution $u(t,x)$ in an unbounded domain $\Omega\subset R_n$ of the first mixed problem for the parabolic equation
\begin{gather} u_t=(a_{ij}(t,x)u_{x_j})_{x_i},\qquad(t,x)\in(t>0)\times\Omega,\\ \gamma^{-1}|y|^2\leqslant a_{ij}(t,x)y_iy_j\leqslant\gamma|y|^2, \end{gather}
with initial function $\varphi$, $\operatorname{supp}\varphi\subset K_{R_0}$, $K_r=\{|x|<r\}$, is investigated. It is shown that the function $\lambda(r)$, which for each fixed $r$ is the first eigenvalue of the Dirichlet problem for the operator $-\Delta$ in $\Omega_r=\Omega\cap K_r$, for a certain class of domains determines the rate at which the solution $u(t,x)$ tends to zero as $t\to\infty$. Namely, let $r(t)$ be the function inverse to the monotone increasing function $F(r)=r/\sqrt{\lambda(r)}$. Then for all $t\geqslant T$ and all $x$ in $\Omega$
\begin{equation} |u(t,x)|\leqslant M\exp\biggl(-\varkappa\,\frac{r^2(t)}t\biggr)\|\varphi\|_{L_2(\Omega)}. \end{equation}
Here the constant $\varkappa$ depends only on $n$ and $\gamma$ of (2), while $T$ and $M$ depend on $\Omega$, $\gamma$, and $R_0$. It is proved that for a certain class of domains the estimate (3) is in a sense best possible.
Bibliography: 13 titles.