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

F. Kh. Mukminov

Sbornik. Mathematics, Tome 71 (1992) no. 2, pp. 331-353

Résumé

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 $\partial\Omega$ $$ \operatorname{mes}(\{y\colon|x-y|\}\setminus\Omega)\geqslant\theta r^n, \quad r>0. $$ 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: $\displaystyle r^{-n}\int_{|x-y|$ 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.