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We consider a two-species competition-diffusion system involving a smallparameter $\varepsilon>0$ and discuss the validity of formal asymptoticexpansions of solutions near the sharp interface limit $\varepsilon\approx0$.We assume that the corresponding ODE system has two stable equilibria. As inthe scalar Allen--Cahn equation, it is known that the motion of the sharpinterfaces of such systems is governed by the mean curvature flow with adriving force. The formal expansion also suggests that the profile of thetransition layers converges to that of a traveling wave solution as$\varepsilon\rightarrow0$. In this paper, we rigorously verify this latteransatz for a large class of initial data.  The proof relies on a rescaling argument, the super--subsolution method and aLiouville type theorem for eternal solutions of parabolic systems. Roughlyspeaking, the Liouville type theorem states that any eternal solution that liesbetween two traveling waves is itself a traveling wave. The same Liouville typetheorem was established for the scalar Allen--Cahn equation by Berestycki andHamel. In view of their importance, we prove the Liouville type theorems in arather general framework, not only for two-species competition-diffusionsystems but also for $m$-species cooperation-diffusion systems possibly withtime periodic or spatially periodic coefficients.

Validity of formal expansions for singularly perturbed competition-diffusion systems