On uniform stabilization of solutions of the first mixed problem for a parabolic equation

The first mixed problem with a homogeneous boundary condition is considered for a linear parabolic equation of second order. It is assumed that the unbounded domain $\Omega$ satisfies the following condition: there exists a positive constant $\theta$ such that for any point $x$ of the boundary $\partial\Omega$
$$ \operatorname{mes}(\{y\colon|x-y|<r\}\setminus\Omega)\geqslant\theta r^n, \quad r>0. $$
For a certain class of initial functions $\varphi$, which includes all bounded functions, the following condition is a necessary and sufficient condition for uniform stabilization of the solution to zero: $\displaystyle r^{-n}\int_{|x-y|<r}\varphi (y)\,dy\to0$ as $r\to\infty$ uniformly with respect to all $x$ in $\Omega$ such that $\operatorname{dist}(x,\partial\Omega)\geqslant r+1$.
The proof of the stabilization condition is based on an estimate of the Green function that takes account of its decay near the boundary.