Ancient shrinking spherical interfaces in the Allen-Cahn flow

Abstract

We consider the parabolic Allen-Cahn equation in $\mathbb{R}^n$, $n\ge 2$, $$u_t= \Delta u + (1-u^2)u \quad \hbox{ in } \mathbb{R}^n \times (-\infty, 0].$$ 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 $w'' + (1-w^2)w=0$ $w(\pm \infty)= \pm 1$ given by $w(s) = \tanh \left(\frac s{\sqrt{2}} \right) $ we have $$ u(x,t) \approx \sum_{j=1}^k (-1)^{j-1}w(|x|-\rho_j(t)) - \frac 12 (1+ (-1)^{k}) \quad \hbox{ as } t\to -\infty $$ where $$\rho_j(t)=\sqrt{-2(n-1)t}+\frac{1}{\sqrt{2}}\left(j-\frac{k+1}{2}\right)\log\left(\frac {|t|}{\log |t| }\right)+ O(1),\quad j=1,\ldots ,k.$$