Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Table of Contents - October 2011 - Volume 141, Issue 05  

Research Article

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

Yu Kurokawaa1 and Masaharu Taniguchia1

a1 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-38 Ookayama, Meguro-ku, Tokyo 152-8552, Japan (kurokaw3@is.titech.ac.jp; masaharu.taniguchi@is.titech.ac.jp)

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.

(Received July 30 2010)

(Accepted December 09 2010)

(Online publication September 26 2011)