Singular limit for reactive diffusive transport through an array of thin channels in case of critical diffusivity

Abstract

We consider a nonlinear 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 $\epsilon$, and the equation inside the layer depends on the parameter $\epsilon$. We consider the critical scaling of the diffusion coefficients in the channels and nonlinear Neumann-boundary condition on the channels' lateral boundaries. We derive effective models in the limit $\epsilon \to 0 $, when the channel-domain is replaced by an interface $\Sigma$ between the two bulk-domains. Due to the critical size of the diffusion coefficients, we obtain jumps for the solution and its normal fluxes across $\Sigma$, involving the solutions of local cell problems on the reference channel in every point of the interface $\Sigma$.