An $(N-1)$-Dimensional Convex Compact Set Gives an $N$-Dimensional Traveling Front in the Allen--Cahn Equation

Source abstract

This paper studies traveling fronts to the Allen--Cahn equation in $\mathbb{R}^{N}$ for $N\geq 3$. Let $(N-2)$-dimensional smooth surfaces be the boundaries of compact sets in $\mathbb{R}^{N-1}$ 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.