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Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains

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We consider quasilinear and linear parabolic problems with rapidly oscillating coefficients in a domain Ω ε that is ε-periodically perforated by small holes of order

. The holes are divided into three ε-periodical sets depending on boundary conditions. The homogeneous Dirichlet boundary conditions are imposed for holes of one set, whereas, for holes in the remaining sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For a solution to the quasilinear problem we find the leading terms of the asymptotic expansion and prove asymptotic estimates that show the influence of perturbation parameters. In the linear case, we construct and justify a complete asymptotic expansion of the solution by using the two-scale asymptotic expansion method. Bibliography: 25 titles. Illustrations: 1 figure.

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Authors and Affiliations

  1. National Taras Shevchenko University of Kyiv, 64, Volodymyrska St., Kyiv, 01033, Ukraine

    T. A. Mel’nik

  2. Swansea University, Swansea, SA2 8PP, UK

    O. A. Sivak

Authors
  1. T. A. Mel’nik
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  2. O. A. Sivak
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Correspondence to T. A. Mel’nik.

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Dedicated to Professor V. V. Zhikov on the occasion of his 70th birthday

Translated from Problems in Mathematical Analysis 59, July 2011, pp. 43–62.

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Mel’nik, T.A., Sivak, O.A. Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains. J Math Sci 177, 50–70 (2011). https://doi.org/10.1007/s10958-011-0447-y

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