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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 (kurokaw3@is.titech.ac.jp; masaharu.taniguchi@is.titech.ac.jp)
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 (kurokaw3@is.titech.ac.jp; masaharu.taniguchi@is.titech.ac.jp)
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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

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