Hostname: page-component-f554764f5-nt87m Total loading time: 0 Render date: 2025-04-13T20:24:15.597Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  26 September 2011

Yu Kurokawa  and
Masaharu Taniguchi
Yu Kurokawa
Affiliation:
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-38 Ookayama, Meguro-ku, Tokyo 152-8552, Japan ([email protected]; [email protected])
Masaharu Taniguchi
Affiliation:
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-38 Ookayama, Meguro-ku, Tokyo 152-8552, Japan ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 141 , Issue 5 , October 2011 , pp. 1031 - 1054
Copyright
Copyright © Royal Society of Edinburgh 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable