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 Mat. Sb., 1990, Volume 181, Number 10, Pages 1283–1305 (Mi msb1225)

Asymptotic problems connected with the heat equation in perforated domains

V. V. Zhikov

Abstract: 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,
\frac{\partial u}{\partial n}|_{\partial F}=0, \quad u|_{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.

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English version:
Mathematics of the USSR-Sbornik, 1992, 71:1, 125–147

Bibliographic databases:

UDC: 517.9
MSC: Primary 35K05, 35B40; Secondary 76S05

Citation: V. V. Zhikov, “Asymptotic problems connected with the heat equation in perforated domains”, Mat. Sb., 181:10 (1990), 1283–1305; Math. USSR-Sb., 71:1 (1992), 125–147

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Zhikov, “On passage to the limit in nonlinear variational problems”, Russian Acad. Sci. Sb. Math., 76:2 (1993), 427–459
2. Zhikov V., “Asymptotic Problems Related to Nondivergent Parabolic 2nd-Order Equation with Stochastically Uniform Coefficients”, Differ. Equ., 29:5 (1993), 735–744
3. Valikov K., “Pointwise Stabilization of Solutions to Parabolic Equations with Periodic Coefficients in a Perforated Space”, Differ. Equ., 30:8 (1994), 1235–1248
4. Jozef Telega, Wlodzimierz Bielski, “Stochastic homogenization and macroscopic modelling of composites and flow through porous media”, Theor. appl. mech. (Belgr.), 2002, no. 28-29, 337
5. Telega J.J., “Stochastic homogenization: Convexity and nonconvexity”, Nonlinear Homogenization and its Applications to Composites, Polycrystals and Smart Materials, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 170, 2004, 305–347
6. V. V. Zhikov, A. L. Piatnitski, “Homogenization of random singular structures and random measures”, Izv. Math., 70:1 (2006), 19–67
7. V. V. Zhikov, “Estimates of the Nash–Aronson type for the diffusion equation with non-symmetric matrix and their application to homogenization”, Sb. Math., 197:12 (2006), 1775–1804
8. Zhikov, VV, “Nash-Aronson estimates for solutions to some parabolic equations: Application to asymptotic diffusion problems”, Doklady Mathematics, 75:2 (2007), 247
9. O. V. Pugachev, “On the closability and convergence of Dirichlet forms”, Proc. Steklov Inst. Math., 270 (2010), 216–221
10. V. V. Zhikov, “Estimates of the Nash–Aronson type for degenerating parabolic equations”, Journal of Mathematical Sciences, 190:1 (2013), 66–79
11. Markus Schmuck, “Heterogeneous hard-sphere interactions for equilibrium transport processes beyond perforated domain formulations”, Applied Mathematics Letters, 2015
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