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 PDE 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¿8 and then take the homogenisation limit e¿0 that is behind the derivation of the two-scale model? Here e is the width of a thin two-dimensional strip. Using the method of asymptotic expansions, we show that it does not matter whether we first take e¿0 and then a¿8, or vice-versa. Finally, we illustrate numerically the solution behaviour of the two-scale model in case of a fast reaction.