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 language | English |
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Pages (from-to) | 187-215 |
Number of pages | 29 |
Journal | Annales De L Institut Henri Poincare: Analyse Non Linéaire |
Volume | 35 |
Issue number | 1 |
DOIs | |
Publication status | Published - 1 Jan 2018 |
Keywords
- Allen–Cahn equation
- Ancient solutions
- Nonlinear parabolic equation
ASJC Scopus subject areas
- Analysis
- Mathematical Physics