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Ukrainian Mathematical Journal

, Volume 45, Issue 4, pp 564–580 | Cite as

G-convergence of periodic parabolic operators with a small parameter by the time derivative

  • N. R. Sidenko
Article
  • 23 Downloads

Abstract

In this paper, we consider a sequenceP k of divergent parabolic operators of the second order, which are periodic in time with periodT=const, and a sequenceP Ψ k of shifts of these operators by an arbitrary periodic vector function Ψ εX=L2((0,T) × Ω)n where Ω is a bounded Lipschitz domain in the space ℝn. The compactness of the family {P Ψ k ¦ Ψ εX, k ε ℕ ink with respect to strongG-convergence, the convergence of arbitrary solutions of the equations with the operatorP Ψ k , and the local character of the strongG-convergence in Ω are proved under the assumptions that the matrix of coefficients ofL2 is uniformly elliptic and bounded and that their time derivatives are uniformly bounded in the space L2(Ω;L2(0,T)).

Keywords

Time Derivative Small Parameter Vector Function Lipschitz Domain Local Character 
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References

  1. 1.
    V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, “On theG-convergence of parabolic operator,”Usp. Mat. Nauk.,36, No. 1, 11–58 (1981).Google Scholar
  2. 2.
    V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, “Averaging of parabolic operators,”Tr. Most Mat. Ob-va,45, 182–236 (1982).Google Scholar
  3. 3.
    P. M. Kun'ch, and O. A. Pankov, “G-convergence of monotone parabolic operators,”Dokl. Akad. Nauk Ukr. SSR, Ser. A., No. 8, 8–10 (1986).Google Scholar
  4. 4.
    P. M. Kun'ch,G-Convergence of Nonlinear Parabolic Operators [in Russian], Author's Review of the Candidate Degree Thesis (Physics and Mathematics), L'vov (1988).Google Scholar
  5. 5.
    N. R. Sidenko, “G-convergence and averaging of periodic parabolic operators with a small parameter by the time derivative,” in:Nonlinear Boundary-Value Problems in Mathematical Physics and Their Applications [in Russian], Institute of Mathematics, Ukra-inian Academy of Sciences, Kiev (1990), pp. 107–110.Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • N. R. Sidenko
    • 1
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev

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