Ancient shrinking spherical interfaces in the Allen–Cahn flow

Abstract

We consider the parabolic Allen–Cahn equation in ${\mathbb{R}}^n$, $n\ge 2$,

We construct an ancient radially symmetric solution $u(x,t)$ with any given number k of transition layers between −1 and +1. At main order they consist of k time-traveling copies of w with spherical interfaces distant $O(\log |t|)$ one to each other as $t\to -\infty$. These interfaces are resemble at main order copies of the shrinking sphere ancient solution to mean the flow by mean curvature of surfaces: $|x|=\sqrt{-2(n-1)t}$. More precisely, if $w(s)$ denotes the heteroclinic 1-dimensional solution of $w^{″}+(1-w^2)w=0$ $w(\pm \infty )=\pm 1$ given by $w(s)=\tanh \left(\frac{s}{\sqrt{2}}\right)$ we have

where