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

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 $γ\in [-1,1)$, 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.