Applications of Mathematics

, Volume 52, Issue 5, pp 431–446 | Cite as

Homogenization of some parabolic operators with several time scales

  • Liselott Flodén
  • Marianne Olsson


The main focus in this paper is on homogenization of the parabolic problem ∂ t uɛ − ∇ · (a(x/ɛ,t/ɛ,t r )∇u ɛ ) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.


homogenization G-convergence multiscale convergence parabolic asymptotic expansion 


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  1. [1]
    G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482–1518.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    G. Allaire, M. Briane: Multiscale convergence and reiterated homogenization. Proc. R. Soc. Edinburgh, Sect. A 126 (1996), 297–342.zbMATHMathSciNetGoogle Scholar
  3. [3]
    S. Brahim-Otsmane, G.A. Francfort, and F. Murat: Correctors for the homogenization of the heat and wave equations. J. Math. Pures Appl. 71 (1992), 197–231.zbMATHMathSciNetGoogle Scholar
  4. [4]
    A. Bensoussan, J.-L. Lions, and G. Papanicolaou: Asymptotic Analysis for Periodic Structures. Stud. Math. Appl. North-Holland, Amsterdam-New York-Oxford, 1978.Google Scholar
  5. [5]
    D. Cioranescu, P. Donato: An Introduction to Homogenization. Oxford Lecture Ser. Math. Appl. Oxford University Press, Oxford, 1999.Google Scholar
  6. [6]
    F. Colombini, S. Spagnolo: Sur la convergence de solutions d’équations paraboliques. J. Math. Pures Appl. 56 (1977), 263–305. (In French.) zblzbMATHMathSciNetGoogle Scholar
  7. [7]
    A. Dall’Aglio, F. Murat: A corrector result for H-converging parabolic problems with time-dependent coefficients. Ann. Sc. Norm. Super. Pisa Cl. Sci., IV. Ser. 25 (1997), 329–373.zbMATHMathSciNetGoogle Scholar
  8. [8]
    A. Holmbom: Homogenization of parabolic equations. An alternative approach and some corrector-type results. Appl. Math. 42 (1997), 321–343.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    A. Holmbom, N. Svanstedt, N. Wellander: Multiscale convergence and reiterated homogenization for parabolic problems. Appl. Math. 50 (2005), 131–151.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    D. Lukkassen, G. Nguetseng, P. Wall: Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002), 35–86.zbMATHMathSciNetGoogle Scholar
  11. [11]
    G. Nguetseng: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608–623.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    A. Pankov: G-convergence and Homogenization of Nonlinear Partial Differential Operators. Mathematics and its Applications. Kluwer, Dordrecht, 1997.Google Scholar
  13. [13]
    A. Profeti, B. Terreni: Uniformità per una convergenza di operatori parabolichi nel caso dell’omogenizzazione. Boll. Unione Math. Ital. Ser. B 16 (1979), 826–841. (In Italian.)zbMATHMathSciNetGoogle Scholar
  14. [14]
    S. Spagnolo: Convergence of parabolic equations. Boll. Unione Math. Ital. Ser. B 14 (1977), 547–568.zbMATHMathSciNetGoogle Scholar
  15. [15]
    N. Svanstedt: G-convergence of parabolic operators. Nonlinear Anal. Theory Methods Appl. 36 (1999), 807–843.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    N. Svanstedt: Correctors for the homogenization of monotone parabolic operators. J. Nonlinear Math. Phys. 7 (2000), 268–283.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2007

Authors and Affiliations

  • Liselott Flodén
    • 1
  • Marianne Olsson
    • 1
  1. 1.Department of Engineering, Physics and MathematicsMid Sweden UniversityÖstersundSweden

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