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

This article is cited in 11 scientific papers (total in 11 papers)

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,
\begin{gather*} 2\frac{\partial u}{\partial t}=\nabla u \quadin\quad \mathbf R^m\setminus F, \quad t>0,
\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
Received: 10.01.1990

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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\by V.~V.~Zhikov
\paper Asymptotic problems connected with the heat equation in perforated domains
\jour Mat. Sb.
\yr 1990
\vol 181
\issue 10
\pages 1283--1305
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\zmath{https://zbmath.org/?q=an:0774.35028}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992SbMat..71..125Z}
\transl
\jour Math. USSR-Sb.
\yr 1992
\vol 71
\issue 1
\pages 125--147
\crossref{https://doi.org/10.1070/SM1992v071n01ABEH002128}
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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  mathnet  crossref  mathscinet  zmath  isi
    2. Zhikov V., “Asymptotic Problems Related to Nondivergent Parabolic 2nd-Order Equation with Stochastically Uniform Coefficients”, Differ. Equ., 29:5 (1993), 735–744  mathnet  mathscinet  zmath  isi
    3. Valikov K., “Pointwise Stabilization of Solutions to Parabolic Equations with Periodic Coefficients in a Perforated Space”, Differ. Equ., 30:8 (1994), 1235–1248  mathnet  mathscinet  zmath  isi
    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  crossref
    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  isi
    6. V. V. Zhikov, A. L. Piatnitski, “Homogenization of random singular structures and random measures”, Izv. Math., 70:1 (2006), 19–67  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. Zhikov, VV, “Nash-Aronson estimates for solutions to some parabolic equations: Application to asymptotic diffusion problems”, Doklady Mathematics, 75:2 (2007), 247  crossref  zmath  isi  elib
    9. O. V. Pugachev, “On the closability and convergence of Dirichlet forms”, Proc. Steklov Inst. Math., 270 (2010), 216–221  mathnet  crossref  mathscinet  zmath  isi  elib
    10. V. V. Zhikov, “Estimates of the Nash–Aronson type for degenerating parabolic equations”, Journal of Mathematical Sciences, 190:1 (2013), 66–79  mathnet  crossref  mathscinet
    11. Markus Schmuck, “Heterogeneous hard-sphere interactions for equilibrium transport processes beyond perforated domain formulations”, Applied Mathematics Letters, 2015  crossref
  • Математический сборник - 1989–1990 Sbornik: Mathematics (from 1967)
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