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.
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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).
C. Conca, J. Diaz, and C. Timofte, “Effective chemical processes in porous media,” Math. Models. Methods Appl. Sci., 13, No. 10, 1437–1462 (2003).
C. Conca, J. Diaz, A. Linan, and C. Timofte, “Homogenization in chemical reactive floes,” Electron. J. Different. Equat., No. 40, 1–22 (2004).
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).
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).
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.
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).
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).
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).
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).
V. A. Marchenko and E. Ya. Khruslov, Averaged Models of Microinhomogeneous Media [in Russian], Naukova Dumka, Kiev (2005).
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).
G. Dal Maso, An introduction to Γ-Convergence, Birkhäuser, Boston (1993).
A. Braides, Γ-Convergence for Beginners, Oxford Univ. Press, Oxford (2002).
E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Springer, New York (1980).
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).
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).
G. Nguetseng, “Asymptotic analysis for a stiff variational problem arising in mechanics,” SIAM J. Math. Anal., 21, No. 6, 1394–1414 (1990).
D. Cioranescu, A. Damlamian, and G. Griso, “The periodic unfolding method in homogenization,” SIAM J. Math. Anal., 40, No. 4, 1585–1620 (2008).
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).
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).
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1973).
V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad University, Leningrad (1985).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 9, pp. 1201–1216, September, 2015.
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Goncharenko, M.V., Khil’kova, L.A. Homogenized Model of Diffusion in Porous Media with Nonlinear Absorption on the Boundary. Ukr Math J 67, 1349–1366 (2016). https://doi.org/10.1007/s11253-016-1158-9
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DOI: https://doi.org/10.1007/s11253-016-1158-9