CONVERGENCE TO SELF-SIMILAR PROFILES IN REACTION-DIFFUSION SYSTEMS

We study a reaction-diffusion system on the real line, where the reactions of the species are given by one reversible reaction pair alpha X-1 reversible arrow beta X-2 satisfying the mass-action law. Under prescribed (different) positive limits at x -> -infinity and x -> +infinity we investigate the long-time behavior of solutions. Rescaling space and time according to the parabolic scaling with tau = log(1 + t) and y = x/root 1+t, we show that solutions converge exponentially for tau -> infinity to a self-similar profile. In the original variables, these profiles correspond to asymptotically self-similar behavior describing the phenomenon of diffusive mixing of the different states at infinity. Our method provides global exponential convergence for all initial states with finite entropy relative to the self-similar profile. For the case alpha = beta >= 1 we can allow for profiles with arbitrary limiting states at +/-infinity, while for alpha > beta >= 1 we need to assume that the two states at infinity are sufficiently close such that the profile is flat enough.