*Forma,* Vol. 4 (No. 1), pp. 65-73, 2001

*Original Paper*

## Phase Separation in an Activator-Inhibitor Medium

Masayasu Mimura, Shin-ichiro Ei and Masataka Kuwamura

Department of Mathematics, Hiroshima University, Hiroshima 730, Japan

(Received September 9, 1989; Accepted October 9, 1989)

**Keywords: **
Phase Separation, Activator- Inhibitor System, Singular Perturbation

**Abstract. **
Evolutional process of solutions is studied for some reaction-diffusion systems with activator-inhibitor kinetics

*u*_{t} = *e*^{2}*u*_{xx} + (1 - *u*^{2})(*u* - *z*), *z*_{t} = _{I}udx - *gz*

in some interval *I* with Neumann boundary conditions. When *e* is very small, it is shown by using multi-time scale arguments that the dynamics of solutions is divided into three stages; The first is the occurrence of internal layers, the second is the propagation of layers with the velocity *O*(*e*) and the final is the very slow propagation of layers with the velocity *O*(e^{-A/e}) for some constant *A*>0.