Abstract
A system of reaction–diffusion equations in a perforated domain with rapidly oscillating terms in the equation and in the boundary conditions is studied. A nonlinear function in the equations may not satisfy the Lipschitz condition and, hence, the uniqueness theorem for the corresponding initial–boundary value problem for the considered system of reaction–diffusion equations may not be satisfied. It is proved that the trajectory attractors of this system weakly converge in the corresponding topology to the trajectory attractors of the homogenized reaction–diffusion system with a “strange term” (potential).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
From open sources.
REFERENCES
V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems in Domains with Fine-Grained Boundary (Naukova Dumka, Kiev, 1974) [in Russian].
O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization (North-Holland, Amsterdam, 1992).
V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Nauka, Moscow, 1993; Springer-Verlag, Berlin, 1994).
D. Cioranescu and F. Murat, “Un terme étrange venu d’ailleurs I & II,” in Nonlinear Partial Differential Equations and Their Applications: Collège de France Seminar, Ed. by H. Berzis and J. L. Lions (Pitman, London, 1982), Vol. 2, pp. 98–138; Vol. 3, pp. 154–178.
A. G. Belyaev, A. L. Pyatnitskii, and G. A. Chechkin, Sb. Math. 192 (7), 933–949 (2001).
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (Nauka, Moscow, 1989; North-Holland, Amsterdam, 1992).
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics (Am. Math. Soc., Providence, R.I., 2002).
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Springer-Verlag, New York, 1988).
K. A. Bekmaganbetov, G. A. Chechkin, and V. V. Chepyzhov, Chaos Solitons Fractals 140, 110208 (2020).
M. I. Vishik and V. V. Chepyzhov, Sb. Math. 192 (1), 11–47 (2001).
Funding
The first author’s research was supported by the Ministry of Education and Science of the Republic of Kazakhstan, grant no. AR08855579. The second author’s study (presented in Section 1) was supported by the Russian Foundation for Basic Research, project no. 20-01-00469. The third author’s work (in Section 2) was supported by the Russian Science Foundation, project no. 20-11-20272.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Bekmaganbetov, K.A., Chepyzhov, V.V. & Chechkin, G.A. On Attractors of Reaction–Diffusion Equations in a Porous Orthotropic Medium. Dokl. Math. 103, 103–107 (2021). https://doi.org/10.1134/S1064562421030030
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562421030030