Recent papers have studied the existence of phase transition solutions for model equations of Allen-Cahn type equations. These solutions are either single or multi-transition spatially heteroclinics or homoclinics between simpler equilibrium states. A sufficient condition of the construction of the multi-transition solutions is that there are gaps in the ordered set of single transition solutions. In this paper we explore the necessity of these gap conditions.
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Hiroshi Matano, Paul H. Rabinowitz, On the necessity of gaps. J. Eur. Math. Soc. 8 (2006), no. 2, pp. 355–373DOI 10.4171/JEMS/57