### 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−1}w(|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 |
---|---|

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

### Cite this

*Annales De L Institut Henri Poincare: Analyse Non Linéaire*,

*35*(1), 187-215. https://doi.org/10.1016/j.anihpc.2017.03.005