Ancient shrinking spherical interfaces in the Allen-Cahn flow

Manuel del Pino, Konstantinos T. Gkikas

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We consider the parabolic Allen–Cahn equation in R n, n≥2, u t=Δu+(1−u 2)u in R n×(−∞,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−w 2)w=0 w(±∞)=±1 given by w(s)=tanh⁡([Formula presented]) we have u(x,t)≈∑j=1k(−1) j−1w(|x|−ρ j(t))−[Formula presented](1+(−1) k) as t→−∞ where ρ j(t)=−2(n−1)t+[Formula presented](j−[Formula presented])log⁡([Formula presented])+O(1),j=1,…,k.

Original languageEnglish
Pages (from-to)187-215
Number of pages29
JournalAnnales De L Institut Henri Poincare: Analyse Non Linéaire
Volume35
Issue number1
DOIs
Publication statusPublished - 1 Jan 2018

Keywords

  • Allen–Cahn equation
  • Ancient solutions
  • Nonlinear parabolic equation

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics

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