Asymptotic problems connected with the heat equation in perforated domains. (English. Russian original) Zbl 0774.35028

Zhikov, V. V.

Math. USSR, Sb. 71, No. 1, 125-147 (1992); translation from Mat. Sb. 181, No. 10, 1283-1305 (1990).

Summary: For the diffusion equation in the exterior of a closed set \(F\subset\mathbb{R}^ m\), \(m\geq 2\), with Neumann conditions on the boundary, \[ 2\partial u/\partial t=\Delta u\quad\text{in }\mathbb{R}^ m\backslash F,\quad t>0,\quad \partial u/\partial n|_{\partial F}=0,\quad u|_{t=0}=f, \] 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.

MSC:

35K05	Heat equation
35B40	Asymptotic behavior of solutions to PDEs
76S05	Flows in porous media; filtration; seepage

Keywords:

diffusion equation; Neumann conditions; pointwise stabilization; central limit theorem; uniform stabilization