Society for Industrial and Applied MathematicsActa Medica Okayama0036-14104712015An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen--Cahn equation455476ENMasaharuTaniguchiThis paper studies traveling fronts to the Allen–Cahn equation in RN for N ≥ 3. Let
(N |2)-dimensional smooth surfaces be the boundaries of compact sets in RN|1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation.No potential conflict of interest relevant to this article was reported.