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.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Cioranescu, D., Donato, P., Zaki, R.: Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions. Asymptot. Anal. 53, 209–235 (2007)
Cabarrubias, B., Donato, P.: Homogenization of a quasilinear elliptic problem with nonlinear Robin boundary conditions. Appl. Anal. 91(6), 1111–1127 (2012)
Chourabi, I., Donato, P.: Homogenization and correctors of a class of elliptic problems in perforated domains. Asymptot. Anal. 92, 1–43 (2015)
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)
Donato, P., Monsurrò, S., Raimondi, F.: Homogenization of a class of singular elliptic problems in perforated domains. Nonlinear Anal. 173, 180–208 (2018)
Timofte, C.: Parabolic problems with dynamical boundary conditions in perforated media. Math. Model. Anal. 8(4), 337–350 (2003)
Wang, W., Duan, J.: it Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions. Commun. Math. Phys. 275(1), 163–186 (2007)
Tartar, L.: Problèmes d’homogénéisation dans les équations aux dérivées partielles. Cours Peccot Collège de France (1977)
Cioranescu, D., Donato, P.: Homogénéisation du problème du Neumann non homogéne dans des ouverst perforés. Asymptot. Anal. 1, 115–138 (1988)
Vanninathan, M.: Homogenization of eigenvalues problems in perforated domains. Proc. Indian Acad. Sci. 90, 239–271 (1981)
Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non linèaires. Dunod, Paris (1969)
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence (1991)
Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)
Cioranescu, D., Saint Jean Paulin, J.: Homogenization in open sets with holes. J. Math. Anal. Appl. 71, 590–607 (1979)
Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lectures Series in Mathematics and its Applications, 17, New York (1999)
Conca, C., Díaz, J.I., Liñán, A., Timofte, C.: Homogenization in chemical reactive flows. Electron. J. Diff. Equ. 2004(40), 1–22 (2004)
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)
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.
About this article
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
- perforated media
- dynamical boundary conditions
Mathematics Subject Classification
- Primary 35B27
- Secondary 35K57