2017/03/26 by Manuel del Pino, del Pino, Manuel, Konstantinos T. Gkikas +1
Computer Science · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Solidification and crystal growth phenomena #math.AP
paper · pdf · doi:10.48550/arxiv.1703.08797
arxiv created 2017/03/26 · openalex publication_date 2017/03/26 · arxiv updated 2017/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the parabolic Allen-Cahn equation in ℝn, n≥ 2, ut= Δu + (1-u2)u \hbox 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 \em 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=0 w(± ∞)= ± 1 given by w(s) = \tanh (\frac s√(2) ) we have u(x,t) ≈ ∑j=1k (-1)j-1w(|x|-ρj(t)) - \frac 12 (1+ (-1)k) \hbox as t→ -∞ where ρj(t)=√(-2(n-1)t)+(1)/(√(2))(j-(k+1)/(2))log(\frac |t|log |t| )+ O(1), j=1,… ,k.