Ancient Solutions to the Allen–Cahn Equation in Catenoids

Abstract

Let $N\ge 2$ and $F:{\mathbb{R}}^N\to \mathbb{R}$ be the unique increasing radially symmetric function satisfying the minimal surface equation for graphs with the initial conditions $F(1)=0$ and $\underset{r\to 1}{lim}{F}_r(r)=\mathrm{\infty };$ $r=|x|.$ We construct an ancient solution to Allen–Cahn equation ${\tilde{u}}_t={\mathrm{\Delta }}_M\tilde{u}+(1-{\tilde{u}}^2)\tilde{u}$ in $M\times (-\mathrm{\infty },0),$ where $M=\{(x,\pm F(|x|)):x\in {\mathbb{R}}^N,|x|\ge 1\}$ is a $N$ -dimensional catenoid in ${\mathbb{R}}^{N+1}$ and ${\mathrm{\Delta }}_M$ is the Laplace Beltrami operator of $M.$ In particular, we construct a solution of the form $u(t,r,F(r))=u(t,r,-F(r))$ such that $$u(t,r,F(r))\approx \sum _{j=1}^k(-1{)}^{j-1}w(r-{\rho }_j(t))-\frac{1}{2}(1+(-1{)}^k)\text{ as }t\to -\mathrm{\infty },$$ where $w(s)$ is a solution of $w^{″}+(1-w^2)w=0$ with $w(\pm \mathrm{\infty })=\pm 1,$ given by $w(s)=\tanh (s/\sqrt{2}),$ and $${\rho }_j(t)=\sqrt{-2(N-1)t}+\frac{1}{\sqrt{2}}(j-\frac{k+1}{2})\log \left(\frac{|t|}{\log |t|}\right)+O(1),j=1,\dots ,k.$$