Ancient shrinking spherical interfaces in the Allen-Cahn flow

Abstract

We consider the parabolic Allen Cahn equation in R-n, n >= 2,

u(t) = Delta u + (1 - u(2))u in R-n x (-infinity, 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 vertical bar t vertical bar) one to each other as t -> infinity. These interfaces are resemble at main order copies of the shrinking sphere ancient solution to mean the flow by mean curvature of surfaces: vertical bar x vertical bar = root-2(n -1)t. More precisely, if w(s) denotes the heteroclinic 1-dimensional solution of w '' + (1 w(2))w = 0 w(+/-infinity) = +/- 1 given by w(s) = tanh (s/root 2) we have

u(x, t) approximate to Sigma(k)(j=1)(-1)(j-1)w(vertical bar x vertical bar - rho(j) (t)) - 1/2(1+(-1)(k)) as t -> -infinity

where

rho(j)(t) = root-2(n - 1)t + 1/root 2 (j - k+1/2) log(vertical bar t vertical bar/log vertical bar t vertical bar) + O(1), j=1,...,k. (C) 2017 Elsevier Masson SAS. All rights reserved.