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Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence

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Abstract

We consider a quasilinear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. The existence and uniqueness results are proved by using Rothe’s method combined with the technique of two-scale convergence.

Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution.

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References

  1. G. Allaire: Two-scale convergence and homogenization of periodic structures. School on Homogenization, ICTP, Trieste, September 6–17, 1993.

    Google Scholar 

  2. G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482–1518. zbl

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Almqvist, E.K. Essel, L.-E. Persson, P. Wall: Homogenization of the unstationary incompressible Reynolds equation. Tribol. Int. 40 (2007), 1344–1350.

    Article  Google Scholar 

  4. A. Bensoussan, J.-L. Lions, G. Papanicolaou: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam, 1978.

    MATH  Google Scholar 

  5. S. Fučík, A. Kufner: Nonlinear Differential Equations. Elsevier Scientific Publishing Company, Amsterdam-Oxford-New York, 1980. zbl

    MATH  Google Scholar 

  6. J. Kačur: Method of Rothe and nonlinear parabolic boundary value problems of arbitrary order. Czech. Math. J. 28 (1978), 507–524. zbl

    Google Scholar 

  7. J. Kačur: Method of Rothe in Evolution Equations. B.G. Teubner Verlagsgesellschaft, Leipzig, 1985. zbl

    MATH  Google Scholar 

  8. K. Kuliev: Parabolic problems on non-cylindrical domains. The method of Rothe. PhD. Thesis. Faculty of Applied Sciences, University of West Bohemia, Pilsen, 2007.

    Google Scholar 

  9. K. Kuliev, L.-E. Persson: An extension of Rothe’s method to non-cylindrical domains. Appl. Math. 52 (2007), 365–389. zbl

    Article  MathSciNet  MATH  Google Scholar 

  10. J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris, 1969. (In French.) zbl

    MATH  Google Scholar 

  11. J.-L. Lions, D. Lukkassen, L.-E. Persson, P. Wall: Reiterated homogenization of monotone operators. C. R. Acad. Sci. Paris 330 (2000), 675–680. zbl

    MathSciNet  MATH  Google Scholar 

  12. J.-L. Lions, D. Lukkassen, L.-E. Persson, P. Wall: Reiterated homogenization of nonlinear monotone operators. Chin. Ann. Math., Ser. B 22 (2001), 1–12. zbl

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Lukkassen, G. Nguetseng, P. Wall: Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002), 35–86. zbl

    MathSciNet  MATH  Google Scholar 

  14. G. Nguetseng: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608–623. zbl

    Article  MathSciNet  MATH  Google Scholar 

  15. B.G. Pachpatte: Inequalities for Differential and Integral Equations. Academic Press, San Diego, 1998. zbl

    MATH  Google Scholar 

  16. L.-E. Persson, L. Persson, N. Svanstedt, J. Wyller: The Homogenization Method. An Introduction. Studentlitteratur, Lund, 1993. zbl

    MATH  Google Scholar 

  17. K. Rektorys: The Method of Discretization in Time and Partial Differential Equations. D. Reidel Publishing Company, London, and SNTL, Prague, 1982. zbl

    MATH  Google Scholar 

  18. J. Vala: The method of Rothe and two-scale convergence in nonlinear problems. Appl. Math. 48 (2003), 587–606. zbl

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Wall: Homogenization of Reynolds equation by two-scale convergence. Chin. Ann. Math. Ser. B 28 (2007), 363–374.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Luleå, Sweden

    Emmanuel Kwame Essel & Lars-Erik Persson

  2. Cape Coast, Ghana

    Emmanuel Kwame Essel

  3. Plzeň, Czech Republic

    Komil Kuliev & Gulchehra Kulieva

  4. Department of Mathematics, Luleå University of Technology, SE-971 87, Luleå, Sweden

    Emmanuel Kwame Essel & Lars-Erik Persson

  5. Department of Mathematics and Statistics, University of Cape Coast, Cape Coast, Ghana

    Emmanuel Kwame Essel & Komil Kuliev

  6. Department of Mathematics, University of West Bohemia, Univerzitní 22, 306 14, Pilsen, Czech Republic

    Gulchehra Kulieva

Authors
  1. Emmanuel Kwame Essel
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  2. Komil Kuliev
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  3. Gulchehra Kulieva
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  4. Lars-Erik Persson
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Corresponding author

Correspondence to Emmanuel Kwame Essel.

Additional information

The first author’s research was financed by The Ghana Government Scholarship Secretariat and I.S.P. of Uppsala University, Sweden.

The research of the second and third authors were supported by the Research Plan MSM4977751301 of the Ministry of Education, Youth and Sports of the Czech Republic.

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Essel, E.K., Kuliev, K., Kulieva, G. et al. Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence. Appl Math 55, 305–327 (2010). https://doi.org/10.1007/s10492-010-0023-7

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  • DOI: https://doi.org/10.1007/s10492-010-0023-7

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