### Abstract

We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on R
^{n}
with n≥2 and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow V
_{γ}
=κ
_{γ}
+c with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen–Cahn equation plays a key role.

Original language | English (US) |
---|---|

Pages (from-to) | 585-626 |

Number of pages | 42 |

Journal | Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2019 |

### Fingerprint

### Keywords

- Anisotropic Allen–Cahn equation
- Anisotropic mean curvature flow
- Spreading front
- Stability
- Wulff shape

### Cite this

**
Asymptotic behavior of spreading fronts in the anisotropic Allen–Cahn equation on R
^{n}
.** / Matano, Hiroshi; Mori, Yoichiro; Nara, Mitsunori.

Research output: Contribution to journal › Article

^{n}',

*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*, vol. 36, no. 3, pp. 585-626. https://doi.org/10.1016/j.anihpc.2018.07.003

}

TY - JOUR

T1 - Asymptotic behavior of spreading fronts in the anisotropic Allen–Cahn equation on R n

AU - Matano, Hiroshi

AU - Mori, Yoichiro

AU - Nara, Mitsunori

PY - 2019/5/1

Y1 - 2019/5/1

N2 - We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on R n with n≥2 and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow V γ =κ γ +c with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen–Cahn equation plays a key role.

AB - We consider the Cauchy problem for the anisotropic (unbalanced) Allen–Cahn equation on R n with n≥2 and study the large time behavior of the solutions with spreading fronts. We show, under very mild assumptions on the initial data, that the solution develops a well-formed front whose position is closely approximated by the expanding Wulff shape for all large times. Such behavior can naturally be expected on a formal level and there are also some rigorous studies in the literature on related problems, but we will establish approximation results that are more refined than what has been known before. More precisely, the Hausdorff distance between the level set of the solution and the expanding Wulff shape remains uniformly bounded for all large times. Furthermore, each level set becomes a smooth hypersurface in finite time no matter how irregular the initial configuration may be, and the motion of this hypersurface is approximately subject to the anisotropic mean curvature flow V γ =κ γ +c with a small error margin. We also prove the eventual rigidity of the solution profile at the front, meaning that it converges locally to the traveling wave profile everywhere near the front as time goes to infinity. In proving this last result as well as the smoothness of the level surfaces, an anisotropic extension of the Liouville type theorem of Berestycki and Hamel (2007) for entire solutions of the Allen–Cahn equation plays a key role.

KW - Anisotropic Allen–Cahn equation

KW - Anisotropic mean curvature flow

KW - Spreading front

KW - Stability

KW - Wulff shape

UR - http://www.scopus.com/inward/record.url?scp=85061129106&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061129106&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2018.07.003

DO - 10.1016/j.anihpc.2018.07.003

M3 - Article

AN - SCOPUS:85061129106

VL - 36

SP - 585

EP - 626

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

SN - 0294-1449

IS - 3

ER -