Existence, Uniqueness and Homogenization of Nonlinear Parabolic Problems with Dynamical Boundary Conditions in Perforated Media

Abstract

We consider a nonlinear parabolic problem with nonlinear dynamical boundary conditions of pure-reactive type in a media perforated by periodically distributed holes of size \(\varepsilon \). The novelty of our work is to consider a nonlinear model where the nonlinearity also appears in the boundary. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when \(\varepsilon \) goes to zero, a new nonlinear parabolic problem defined on a unified domain without holes with zero Dirichlet boundary condition and with extra terms coming from the influence of the nonlinear dynamical boundary conditions is rigorously derived.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Cioranescu, D., Donato, P., Zaki, R.: Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions. Asymptot. Anal. 53, 209–235 (2007)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Cabarrubias, B., Donato, P.: Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions. Appl. Anal. 91(6), 1111–1127 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chourabi, I., Donato, P.: Homogenization and correctors of a class of elliptic problems in perforated domains. Asymptot. Anal. 92, 1–43 (2015)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chourabi, I., Donato, P.: Homogenization of elliptic problems with quadratic growth and nonhomogenous Robin conditions in perforated domains. Chin. Ann. Math. 37B(6), 833–852 (2016)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Donato, P., Monsurrò, S., Raimondi, F.: Homogenization of a class of singular elliptic problems in perforated domains. Nonlinear Anal. 173, 180–208 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Timofte, C.: Parabolic problems with dynamical boundary conditions in perforated media. Math. Model. Anal. 8(4), 337–350 (2003)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Wang, W., Duan, J.: it Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions. Commun. Math. Phys. 275(1), 163–186 (2007)

    Article  Google Scholar 

  8. 8.

    Tartar, L.: Problèmes d’homogénéisation dans les équations aux dérivées partielles. Cours Peccot Collège de France (1977)

  9. 9.

    Cioranescu, D., Donato, P.: Homogénéisation du problème du Neumann non homogéne dans des ouverst perforés. Asymptot. Anal. 1, 115–138 (1988)

    Article  Google Scholar 

  10. 10.

    Vanninathan, M.: Homogenization of eigenvalues problems in perforated domains. Proc. Indian Acad. Sci. 90, 239–271 (1981)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non linèaires. Dunod, Paris (1969)

    Google Scholar 

  12. 12.

    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence (1991)

  13. 13.

    Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  14. 14.

    Cioranescu, D., Saint Jean Paulin, J.: Homogenization in open sets with holes. J. Math. Anal. Appl. 71, 590–607 (1979)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lectures Series in Mathematics and its Applications, 17, New York (1999)

  16. 16.

    Conca, C., Díaz, J.I., Liñán, A., Timofte, C.: Homogenization in chemical reactive flows. Electron. J. Diff. Equ. 2004(40), 1–22 (2004)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Cioranescu, D., Donato, P., Ene, H.: Homogenization of the Stokes problem with non homogeneous slip boundary conditions. Math. Methods Appl. Sci. 19, 857–881 (1996)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to María Anguiano.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was completed with the support of Junta de Andalucía (Spain), Proyecto de Excelencia P12-FQM-2466.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Anguiano, M. Existence, Uniqueness and Homogenization of Nonlinear Parabolic Problems with Dynamical Boundary Conditions in Perforated Media. Mediterr. J. Math. 17, 18 (2020). https://doi.org/10.1007/s00009-019-1459-y

Download citation

Keywords

  • Homogenization
  • perforated media
  • dynamical boundary conditions

Mathematics Subject Classification

  • Primary 35B27
  • Secondary 35K57