Ancient shrinking spherical interfaces in the Allen-Cahn flow

Abstract

We consider the parabolic Allen-Cahn equation in ${\mathbb{R}}^n$, $n\ge 2$, $$u_t=\Delta u+(1-u^2)uin{\mathbb{R}}^n\times (-\infty ,0].$$ 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 {\em 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 $w^{\prime \prime }+(1-w^2)w=0$ $w(\pm \infty )=\pm 1$ given by $w(s)=\tanh \left(\frac{s}{\sqrt{2}}\right)$ we have $$u(x,t)\approx \sum _{j=1}^k(-1{)}^{j-1}w(|x|-{\rho }_j(t))-\frac{1}{2}(1+(-1{)}^k)ast\to -\infty$$ where $${\rho }_j(t)=\sqrt{-2(n-1)t}+\frac{1}{\sqrt{2}}\left(j-\frac{k+1}{2}\right)\log \left(\frac{|t|}{\log |t|}\right)+O(1),j=1,\dots ,k.$$