Ukrainian Mathematical Journal

, Volume 67, Issue 9, pp 1349–1366 | Cite as

Homogenized Model of Diffusion in Porous Media with Nonlinear Absorption on the Boundary

  • M. V. Goncharenko
  • L. A. Khil’kova

We consider a boundary-value problem used to describe the process of stationary diffusion in a porous medium with nonlinear absorption on the boundary. We study the asymptotic behavior of the solution when the medium becomes more and more porous and denser located in a bounded domain Ω. A homogenized equation for the description of the main term of the asymptotic expansion is constructed.


Porous Medium Generalize Solution Elliptic Problem Diffusion Equation Homogenize Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. V. Berlyand and M. V. Goncharenko, “Homogenization of the diffusion equation in a medium with weak absorption,” Teor. Funkts. Funkts. Anal. Prilozhen., 52, 113–122 (1989).MathSciNetzbMATHGoogle Scholar
  2. 2.
    C. Conca, J. Diaz, and C. Timofte, “Effective chemical processes in porous media,” Math. Models. Methods Appl. Sci., 13, No. 10, 1437–1462 (2003).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    C. Conca, J. Diaz, A. Linan, and C. Timofte, “Homogenization in chemical reactive floes,” Electron. J. Different. Equat., No. 40, 1–22 (2004).Google Scholar
  4. 4.
    C. Conca, J. Diaz, A. Linan, and C. Timofte, “Homogenization results for chemical reactive flows through porous media,” New Trends Contin. Mech., 6, 99–107 (2005).MathSciNetzbMATHGoogle Scholar
  5. 5.
    D. Cioranescu, P. Donato, and R. Zaki, “Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions,” Asymptot. Anal., 53, 209–235 (2007).MathSciNetzbMATHGoogle Scholar
  6. 6.
    C. Timofte, “Homogenization in nonlinear chemical reactive flows,” in: Proc. of the 9 th WSEAS Internat. Conf. on Applied Mathematics (Istanbul, Turkey, May 27–29, 2006), pp. 250–255.Google Scholar
  7. 7.
    T. A. Mel’nyk and O. A. Sivak, “Asymptotic analysis of a boundary-value problem with nonlinear multiphase boundary interactions in a perforated domain,” Ukr. Mat. Zh., 61, No. 4, 494–512 (2009); English translation: Ukr. Math. J., 61, No. 4, 592–612 (2009).Google Scholar
  8. 8.
    T. A. Mel’nyk and O. A. Sivak, “Asymptotic expansion for the solution of an elliptic problem with boundary multiphase interactions of the Dirichlet and Neumann types in a perforated domain,” Visn. Kyiv. Univ., Ser. Fiz.-Mat. Nauk., 3, 63–67 (2010).zbMATHGoogle Scholar
  9. 9.
    T. A. Mel’nyk and O. A. Sivak, “Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains,” J. Math. Sci., 177, No. 1, 50–70 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    T. A. Mel’nyk and O. A. Sivak, “Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains,” Asymptot. Anal., 75, 79–92 (2011).MathSciNetzbMATHGoogle Scholar
  11. 11.
    V. A. Marchenko and E. Ya. Khruslov, Averaged Models of Microinhomogeneous Media [in Russian], Naukova Dumka, Kiev (2005).Google Scholar
  12. 12.
    V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, and K.-T. Ngoan, “Averaging and G-convergence of differential operators,” Usp. Mat. Nauk, 34, Issue 5, 65–133 (1979).MathSciNetzbMATHGoogle Scholar
  13. 13.
    G. Dal Maso, An introduction to Γ-Convergence, Birkhäuser, Boston (1993).Google Scholar
  14. 14.
    A. Braides, Γ-Convergence for Beginners, Oxford Univ. Press, Oxford (2002).Google Scholar
  15. 15.
    E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer, New York (1980).zbMATHGoogle Scholar
  16. 16.
    N. S. Bakhvalov and G. P. Panasenko, Averaging of Processes in Periodic Systems. Mathematical Problems of Mechanics of Composite Materials [in Russian], Nauka, Moscow (1984).zbMATHGoogle Scholar
  17. 17.
    L. Tartar, “Compensated compactness and applications to partial differential equations in non-linear analysis and mechanics,” in: R. S. Knops (editor), Heriot-Watt Symposium, Vol. IV, Pitman, London (1979).Google Scholar
  18. 18.
    G. Nguetseng, “Asymptotic analysis for a stiff variational problem arising in mechanics,” SIAM J. Math. Anal., 21, No. 6, 1394–1414 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    D. Cioranescu, A. Damlamian, and G. Griso, “The periodic unfolding method in homogenization,” SIAM J. Math. Anal., 40, No. 4, 1585–1620 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    B. Cabarrubias and P. Donato, “Existence and uniqueness for a quasilinear elliptic problem with nonlinear robin condition,” Carpath. J. Math., 27, No. 2, 173–184 (2011).MathSciNetzbMATHGoogle Scholar
  21. 21.
    E. Ya. Khruslov, “Asymptotic behavior of solutions of the second boundary-value problem in the case of refinement of the boundary of domain,” Mat. Sb., 106(148), No. 4(8), 604–621 (1978).Google Scholar
  22. 22.
    O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).zbMATHGoogle Scholar
  23. 23.
    V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad University, Leningrad (1985).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • M. V. Goncharenko
    • 1
  • L. A. Khil’kova
    • 2
  1. 1.Institute for Low Temperature Physics and EngineeringUkrainian National Academy of SciencesKharkovUkraine
  2. 2.Institute of Chemical TechnologiesDal’ East-Ukrainian National UniversityRubezhnoeUkraine

Personalised recommendations