Ancient shrinking spherical interfaces in the Allen-Cahn flow

We consider the parabolic Allen-Cahn equation in Rn, n≥2,ut=Δu+(1−u2)u in Rn×(−∞,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→−∞. These interfaces are resemble at main order copies of the shrinking sphere ancient solution to mean the flow by mean curvature of surfaces: |x|=−2(n−1)t. More precisely, if w(s) denotes the heteroclinic 1-dimensional solution of w″+(1−w2)w=0w(±∞)=±1 given by w(s)=tanh⁡(s2) we haveu(x,t)≈∑j=1k(−1)j−1w(|x|−ρj(t))−12(1+(−1)k) as t→−∞ whereρj(t)=−2(n−1)t+12(j−k+12)log⁡(|t|log⁡|t|)+O(1),j=1,…,k.