Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations

Yu Kurokawa, Masaharu Taniguchi

Research output: Contribution to journalArticlepeer-review

38 Citations (Scopus)


We study travelling-front solutions of pyramidal shapes in the Allen-Cahn equation in RN with N 3. It is well known that two-dimensional V-form travelling fronts and three-dimensional pyramidal travelling fronts exist and are stable. The aim of this paper is to show that for N 4 there exist N-dimensional pyramidal travelling fronts. We construct a supersolution and a subsolution, and find a pyramidal travelling-front solution between them. For the construction of a supersolution we use a multi-scale method. ©

Original languageEnglish
Pages (from-to)1031-1054
Number of pages24
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Issue number5
Publication statusPublished - Oct 2011
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)


Dive into the research topics of 'Multi-dimensional pyramidal travelling fronts in the Allen-Cahn equations'. Together they form a unique fingerprint.

Cite this