Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 728–731 | Cite as

Singularly Perturbed System of Parabolic Equations in the Critical Case

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Abstract

We examine a system of singularly perturbed parabolic equations in the case where the small parameter is involved as a coefficient of both time and spatial derivatives and the spectrum of the limit operator has a multiple zero point. In such problems, corner boundary layers appear, which can be described by products of exponential and parabolic boundary-layer functions. Under the assumption that the limit operator is a simple-structure operator, we construct a regularized asymptotics of a solution, which, in addition to corner boundary-layer functions, contains exponential and parabolic boudary-layer functions. The construction of the asymptotics is based on the regularization method for singularly perturbed problems developed by S. A. Lomov and adapted to singularly perturbed parabolic equations with two viscous boundaries by A. S. Omuraliev.

Keywords and phrases

singularly perturbed parabolic equation parabolic boundary layer regularized asymptotics exponential boundary layer 

AMS Subject Classification

35K51 35B25 

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References

  1. 1.
    V. F. Butuzov and L. V. Kalachev, “Asymptotic approximation of solutions of a boundary-value problem for a singularly perturbed parabolic equation in the critical case,” Mat. Zametki, 89, No. 6, 819–830 (1986).MathSciNetMATHGoogle Scholar
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    S. A. Lomov, Introduction to the General Theory of Singular Perturbations [in Russian], Nauka, Moscow (1981).MATHGoogle Scholar
  3. 3.
    A. S. Omuraliev, “Regularization of a two-dimensional singularly perturbed parabolic problem,” Zh. Vychisl. mat. mat. Fiz., 46, No. 8, 1423–1432 (2006).MathSciNetGoogle Scholar
  4. 4.
    A. D. Polyanin, Reference Book on Linear Equations of Mathematical Physics [in Russian], Moscow (2001).Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Kyrgyz Turkish Manas UniversityBishkekKyrgyz Republic
  2. 2.Naryn State UniversityNarynKyrgyz Republic

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