Validity of formal expansions for singularly perturbed competition-diffusion systems

We consider a two-species competition-diffusion system involving a small parameter $\epsilon >0$ and discuss the validity of formal asymptotic expansions of solutions near the sharp interface limit $\epsilon \approx 0$. We assume that the corresponding ODE system has two stable equilibria. As in the scalar Allen–Cahn equation, it is known that the motion of the sharp interfaces of such systems is governed by the mean curvature flow with a driving force. The formal expansion also suggests that the profile of the transition layers converges to that of a traveling wave solution as $\epsilon \to 0$. In this paper, we rigorously verify this latter ansatz for a large class of initial data. The proof relies on a rescaling argument, the super–subsolution method and a Liouville type theorem for eternal solutions of parabolic systems. Roughly speaking, the Liouville type theorem states that any eternal solution that lies between two traveling waves is itself a traveling wave. The same Liouville type theorem was established for the scalar Allen–Cahn equation by Berestycki and Hamel. In view of their importance, we prove the Liouville type theorems in a rather general framework, not only for two-species competition-diffusion systems but also for $m$-species cooperation-diffusion systems possibly with time periodic or spatially periodic coefficients.