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Terrace solutions for non-Lipschitz multistable nonlinearities

Traveling wave solutions of reaction-diffusion equations are well-studied for Lipschitz continuous monostable and bistable reaction functions. These special solutions play a key role in mathematical biology and in particular in the study of ecological invasions. However, if there are more than two stable steady states, the invasion phenomenon may become more intricate and involve intermediate steps, which leads one to consider not a single but a collection of traveling waves with ordered speeds. In this paper we show that, if the reaction function is discontinuous at the stable steady states, then such a collection of traveling waves exists and even provides a special solution which we call a terrace solution. More precisely, we will address both the existence and uniqueness of the terrace solution.