Skip to main content

On Attractors of Reaction–Diffusion Equations in a Porous Orthotropic Medium

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, access via your institution.

Fig. 1.

Notes

  1. From open sources.

REFERENCES

  1. V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems in Domains with Fine-Grained Boundary (Naukova Dumka, Kiev, 1974) [in Russian].

    MATH  Google Scholar 

  2. O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization (North-Holland, Amsterdam, 1992).

    MATH  Google Scholar 

  3. V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Nauka, Moscow, 1993; Springer-Verlag, Berlin, 1994).

  4. 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.

    MATH  Google Scholar 

  5. A. G. Belyaev, A. L. Pyatnitskii, and G. A. Chechkin, Sb. Math. 192 (7), 933–949 (2001).

    MathSciNet  Article  Google Scholar 

  6. A. V. Babin and M. I. Vishik, Attractors of Evolution Equations (Nauka, Moscow, 1989; North-Holland, Amsterdam, 1992).

  7. V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics (Am. Math. Soc., Providence, R.I., 2002).

    MATH  Google Scholar 

  8. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Springer-Verlag, New York, 1988).

    Book  Google Scholar 

  9. K. A. Bekmaganbetov, G. A. Chechkin, and V. V. Chepyzhov, Chaos Solitons Fractals 140, 110208 (2020).

    MathSciNet  Article  Google Scholar 

  10. M. I. Vishik and V. V. Chepyzhov, Sb. Math. 192 (1), 11–47 (2001).

    MathSciNet  Article  Google Scholar 

Download references

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

Authors

Corresponding authors

Correspondence to K. A. Bekmaganbetov, V. V. Chepyzhov or G. A. Chechkin.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562421030030

Keywords:

  • attractors
  • homogenization
  • reaction–diffusion equation
  • nonlinear equations
  • weak convergence
  • perforated domain
  • rapidly oscillating terms
  • strange term