### Abstract

This paper studies traveling fronts to the Allen-Cahn equation in ℝ^{N} for N ≥ 3. Let (N - 2)-dimensional smooth surfaces be the boundaries of compact sets in ℝ^{N - 1} and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation.

Original language | English |
---|---|

Pages (from-to) | 455-476 |

Number of pages | 22 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 47 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- Allen-Cahn equation
- Nonsymmetric
- Traveling front

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Computational Mathematics

### Cite this

**An (N - 1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen-Cahn equation.** / Taniguchi, Masaharu.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - An (N - 1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen-Cahn equation

AU - Taniguchi, Masaharu

PY - 2015

Y1 - 2015

N2 - This paper studies traveling fronts to the Allen-Cahn equation in ℝN for N ≥ 3. Let (N - 2)-dimensional smooth surfaces be the boundaries of compact sets in ℝN - 1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation.

AB - This paper studies traveling fronts to the Allen-Cahn equation in ℝN for N ≥ 3. Let (N - 2)-dimensional smooth surfaces be the boundaries of compact sets in ℝN - 1 and assume that all principal curvatures are positive everywhere. We define an equivalence relation between them and prove that there exists a traveling front associated with a given surface and that it is asymptotically stable for given initial perturbation. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation.

KW - Allen-Cahn equation

KW - Nonsymmetric

KW - Traveling front

UR - http://www.scopus.com/inward/record.url?scp=84924014527&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84924014527&partnerID=8YFLogxK

U2 - 10.1137/130945041

DO - 10.1137/130945041

M3 - Article

AN - SCOPUS:84924014527

VL - 47

SP - 455

EP - 476

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 1

ER -