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 Izv. RAN. Ser. Mat., 2007, Volume 71, Issue 6, Pages 119–182 (Mi izv711)

Multiphase homogenized diffusion models for problems with several parameters

G. V. Sandrakov

National Taras Shevchenko University of Kyiv

Abstract: We deal with the homogenization of initial-boundary-value problems for parabolic equations with asymptotically degenerate rapidly oscillating periodic coefficients, which are models for diffusion processes in a strongly inhomogeneous medium. The solutions of these problems depend on a finite positive parameter and two small positive parameters. We obtain homogenized initial-boundary-value problems (whose solutions determine approximate asymptotics for solutions of the problems under consideration) and prove estimates for the accuracy of these approximations. The homogenized problems are initial-boundary-value problems for integro-differential equations whose solutions depend on additional positive parameters: the intensity of diffusion exchange and the impulse exchange. In the general case, the homogenized equations form a system of equations coupled through the exchange coefficients and define multiphase mathematical models of diffusion for a homogenized (limiting) medium. We consider the spectral properties of some homogenized problems. We also prove assertions on asymptotic reductions of the homogenized problems under additional hypothesis on the limiting behaviour of the exchange parameters.

DOI: https://doi.org/10.4213/im711

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English version:
Izvestiya: Mathematics, 2007, 71:6, 1193–1252

Bibliographic databases:

UDC: 517.956.8
MSC: 35B27
Revised: 06.12.2006

Citation: G. V. Sandrakov, “Multiphase homogenized diffusion models for problems with several parameters”, Izv. RAN. Ser. Mat., 71:6 (2007), 119–182; Izv. Math., 71:6 (2007), 1193–1252

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv711
• https://doi.org/10.4213/im711
• http://mi.mathnet.ru/eng/izv/v71/i6/p119

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This publication is cited in the following articles:
1. Bellieud M., “Torsion effects in elastic composites with high contrast”, SIAM J. Math. Anal., 41:6 (2010), 2514–2553
2. V. V. Vlasov, N. A. Rautian, A. S. Shamaev, “Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics”, Journal of Mathematical Sciences, 190:1 (2013), 34–65
3. Vlasov V.V., Rautian N.A., “On the Asymptotic Behavior of Solutions of Integro-Differential Equations in a Hilbert Space”, Differ. Equ., 49:6 (2013), 718–730
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