Multi-dimensional traveling fronts in bistable reaction-diffusion equations

Masaharu Taniguchi

doi:10.3934/dcds.2012.32.1011

Article Contents

2012, Volume 32, Issue 3: 1011-1046. Doi: 10.3934/dcds.2012.32.1011

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Multi-dimensional traveling fronts in bistable reaction-diffusion equations

Masaharu Taniguchi^1,

1.
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, O-okayama 2-12-1-W8-38, Tokyo 152-8552

Received: October 2010

Revised: September 2011

Published: March 2012

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Abstract

This paper studies traveling front solutions of convex polyhedral shapes in bistable reaction-diffusion equations including the Allen-Cahn equations or the Nagumo equations. By taking the limits of such solutions as the lateral faces go to infinity, we construct a three-dimensional traveling front solution for any given $g\in C^{\infty}(S^{1})$ with $\min_{0\leq \theta\leq 2\pi}g(\theta)=0$.

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Mathematics Subject Classification: Primary: 35C07, 35B20; Secondary: 35K57.

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