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In an earlier paper, [1] , the authors treated a family of Allen–Cahn model problems for which 0 and 1 are solutions and further solutions were found that are near 1 on a prescribed set, T + Ω , where T ⊂ Z n , and near 0 on ( Z n ∖ T ) + Ω . Here Ω ⊂ ( 0 , 1 ) n . In this paper, a more general class of potentials is treated for which the pair, { 0 , 1 } , is replaced by Z and the existence of a far richer structure of shadowing solutions, including unbounded ones, is established. [ABSTRACT FROM AUTHOR]

Recent papers have studied the existence of phase transition solutions for Allen-Cahn type equations. These solutions are either single or multi-transition spatial heteroclinics or homo- clinics between simpler equilibrium states. A sufficient condition for 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. [ABSTRACT FROM AUTHOR]