A two-scale reaction-diffusion system with micro-cell reaction concentrated on a free boundary

We discuss the fast-reaction limit of a two-scale reaction-diffusion model. We point out that if the reaction constant a explodes to infinity, then a two-scale PDEs system with free boundary at the micro cell is obtained. The aim of this note is to answer the question: Can the same two-scale free-boundary problem be obtained if we first pass to the fast-reaction limit a → ∞ and then take the homogenization limit ε → 0 that is behind the derivation of the two-scale model? Here ε is the typical height of a representative rectangular micro-cell. Using the method of asymptotic expansions, we show that it does not matter whether we first take ε → 0 and then a → ∞, or vice-versa. Finally, we illustrate numerically the solution behavior of the two-scale model in case of a fast reaction.