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This paper deals with the homogenization of the reaction-diffusion equations in a domain containing periodically distributed holes of size ε, with a dynamical boundary condition of reactive-diffusive type, i.e., we consider the following nonlinear boundary condition on the surface of the holeswhere ∆Γ denotes the Laplace-Beltrami operator on the surface of the holes, ν is the outward normal to the boundary, δ > 0 plays the role of a surface diffusion coefficient and g is the nonlinear term. We generalize our previous results (see [3]) established in the case of a dynamical boundary condition of pure-reactive type, i.e., with δ = 0. We prove the convergence of the homogenization process to a nonlinear reaction-diffusion equation whose diffusion matrix takes into account the reactive-diffusive condition on the surface of the holes.

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We consider a non‐stationary Stokes system in a thin porous medium of thickness ε that is perforated by periodically distributed solid cylinders of size ε, and containing a fissure of width ηε. Passing to the limit when ε goes to zero, we find a critical size ηε≈ε23 in which the flow is described by a 2D quasi‐stationary Darcy law coupled with a 1D quasi‐stationary Reynolds problem. Copyright © 2017 John Wiley & Sons, Ltd.