Spatial homogenization and internal layers in a reaction-diffusion system

Abstract

For a system of reaction-diusion equations of activator-inhibitor type, we show that solutions undergo at least three stages of dynamical behaviour when the activator diuses slowly and reacts fast, and the inhibitor diuses fast. In the first stage, the inhibitor quickly decays to its spatial average (spatial homogenization of the inhibitor). In the second stage, the activator develops internal layers (formation of internal layers). In the third stage, the layers move according to a certain motion law (motion of interfaces) which is described by a system of ordinary dierential equations on finite time intervals. Asymptotic behaviour of the solutions of the interface equation is also analyzed. To describe the behaviour of the solutions of the reaction-diusion equations after the last interface equation becomes powerless, another type of interface equation is proposed.