Layer Oscillations in Reaction-Diffusion Systems

Abstract

This paper considers a two-component system of reaction-diffusion equations involving two parameters $\varepsilon $ and $\tau $: \[ \varepsilon \tau u_1 = \varepsilon ^2 u_{xx} + f( u,v ),\qquad v_1 = v_{xx} + g( u,v ). \]. The stability of stationary internal layer-solutions when $\varepsilon $ is sufficiently small is investigated. It is shown that when $\tau $ becomes smaller than some critical value, such solutions are destabilized and there appear layer-oscillating solutions that behave as does “breathing motion.” This is due to the instability of internal layer-solutions via Hopf-bifurcation.