EXISTENCE OF STABLE SPATIALLY PERIODIC STEADY-STATES IN COUPLED REACTION-DIFFUSION EQUATIONS

A new approach to the study of steady states with periodic pattern in coupled reaction-diffusion systems (RD-systems) is presented. The discussion is focused on a given constant steady state with diffusion coefficients as bifurcation parameters. We apply the theorems of reversible systems to the steady-state system of RD-systems, an idea recently used by Kazarinoff and Yan in the study of a chemical model. Here we generalize this idea to a class of RD-systems. As a result, an existence criterion is derived for coupled equations of the form ut = d1uxx - f(hook)1(u, v, k), vt= d2vxx - f(hook)2(u, v, k), x ∈ [0,2L], k ∈ Rm, ux(0) = vx(0) = ux,(2L) = vx(2L) = 0.The temporal stability of these bifurcating periodic steady states is also analyzed. According to a discussion by Segel and Jackson, a stability condition is obtained. The spatial structure of the bifurcating periodic steady states is approximated by the spatial structure of the solutions of the linearized system. This approximation reveals some characteristic features of the solutions and explains the essential pattern of the periodic steady states in each system. To illustrate the applications and provide a comparison with previous works, we apply our results to six models which have been studied by others. Predicted by the existence criterion and stability condition, numerical approximations of spatially periodic steady states are obtained in these models. Their spatial structures are analyzed according to our characterization, which are also compared to and verified by numerical results. © 1993 Academic Press, Inc.