Effective transmission conditions for reaction–diffusion processes in domains separated by thin channels

ABSTRACT

We consider a reaction–diffusion equation in a domain consisting of two bulk regions connected via small channels periodically distributed within a thin layer. The height and the thickness of the channels are of order ε, and the equation inside the layer depends on the parameter ε and an additional parameter $\gamma \in [-1,1)$, which describes the size of the diffusion in the layer. We derive effective models for the limit $ϵ\to 0$, when the channel-domain is replaced by an interface between the two bulk-domains.

KEYWORDS:

• Array of channels
• homogenization
• two-scale convergence
• reaction–diffusion equation
• effective transmission conditions

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35B27
• 35K57
• 35Q92

Acknowledgments

A.B. acknowledges the support by the RTG 2339 IntComSin of the German Science Foundation (DFG). M.G. was supported by the SCIDATOS project, funded by the Klaus Tschira Foundation (Grant Number 00.0277.2015), and the Odysseus program of the Research Foundation - Flanders FWO (Project-Nr. G0G1316N). Parts of the paper have been created during a research stay of AB at the Hasselt University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

AB acknowledges the support by the RTG 2339 IntComSin of the Deutsche Forschungsgesellschaft (DFG). MG was supported by the SCIDATOS project, funded by the Klaus Tschira Foundation [grant number 00.0277.2015], and the Odysseus program of the Research Foundation - Flanders FWO (Fonds Wetenschappelijk Onderzoek) (Project-Nr. G0G1316N). Parts of the paper have been created during a research stay of AB at the Hasselt University.