Abstract
We consider a reaction–diffusion equation on a 3D thin porous media of thickness \(\varepsilon \) which is perforated by periodically distributed cylinders of size \(\varepsilon \). On the boundary of the cylinders, we prescribe a dynamical boundary condition of pure-reactive type. As \(\varepsilon \rightarrow 0\), in the 2D limit the resulting reaction–diffusion equation has a source term coming from the dynamical-type boundary conditions imposed on boundaries of the original 3D domain.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anguiano, M.: Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media. Mediterr. J. Math. 17, 18 (2020)
Anguiano, M.: Homogenization of parabolic problems with dynamical boundary conditions of reactive–diffusive type in perforated media. Z. Angew. Math. Mech. 100(10), e202000088 (2020)
Anguiano, M., Suárez-Grau, F.J.: Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Netw. Heterog. Media 14(2), 289–316 (2019)
Cioranescu, D., Donato, P.: Homogénéisation du problème de Neumann non homogène dans des ouverts perforés. Asymptot. Anal. 1, 115–138 (1988)
Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford University Press, Oxford (1999)
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)
Cioranescu, D., Paulin, J.Saint Jean: Homogenization in open sets with holes. J. Math. Anal. Appl. 71, 590–607 (1979)
Conca, C., Díaz, J.I., Timofte, C.: Effective chemical processes in porous media. Math. Models Methods Appl. Sci. 13(10), 1437–1462 (2003)
Conca, C., Donato, P.: Non homogeneous Neumann problems in domains with small holes. RAIRO Modél. Math. Anal. Numér 22, 561–607 (1988)
Goldstein, G.R.: Derivation and physical interpretation of general boundary conditions. Adv. Differ. Equ. 11(4), 457–480 (2006)
Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non linèaires. Dunod, Paris (1969)
Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)
Timofte, C.: Parabolic problems with dynamical boundary conditions in perforated media. Math. Model. Anal. 8(4), 337–350 (2003)
Tartar, L.: Problèmes d’homogénéisation dans les équations aux dérivées partielles. In: Cours Peccot, Collège de France (1977)
Vanninathan, M.: Homogenization of eigenvalues problems in perforated domains. Proc. Indian Acad. Sci. 90, 239–271 (1981)
Acknowledgements
The author would like to thank the anonymous referees for several useful remarks and comments that led to the improvement of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Yong Zhou.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Anguiano, M. Reaction–Diffusion Equation on Thin Porous Media. Bull. Malays. Math. Sci. Soc. 44, 3089–3110 (2021). https://doi.org/10.1007/s40840-021-01103-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-021-01103-0