Asymptotic problems connected with the heat equation in perforated domains

V. V. Zhikov

Sbornik. Mathematics, Tome 71 (1992) no. 1, pp. 125-147

Résumé

For the diffusion equation in the exterior of a closed set $F\subset\mathbf R^m$, $m\geqslant 2$, with Neumann conditions on the boundary, \begin{gather*} 2\frac{\partial u}{\partial t}=\nabla u \quad\text{in}\quad \mathbf R^m\setminus F, \quad t>0, \\ \frac{\partial u}{\partial n}\bigg|_{\partial F}=0, \quad u\big|_{t=0}=f, \end{gather*} pointwise stabilization, the central limit theorem, and uniform stabilization are studied. The basic condition on the set $F$ is formulated in terms of extension properties. Model examples of sets $F$ are indicated which are of interest from the viewpoint of mathematical physics and applied probability theory.