Multi-dimensional traveling fronts in bistable reaction-diffusion equations

Masaharu Taniguchi

doi:10.3934/dcds.2012.32.1011

Masaharu Taniguchi

Open Access

https://doi.org/10.3934/dcds.2012.32.1011

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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 $\min_{0\leq \theta\leq 2\pi}g(\theta)=0$.

Highlights

In this paper we consider the following equation ∂u = ∆u + f (u)∂t u|t=0 = u0 in R3, t > 0, in R3 .A given function u0 belongs to BU (R3 )
We write a typical example: f (u) = −(u+1)(u+a)(u−1) with a ∈ (0, 1). This equation is called the Allen-Cahn equation, the Nagumo equation or the scalar Ginzburg-Landau equation. It appears in various fields, say, in population genetics [10], ecology [35, 34], bistable transmission in electronic circuits [25], phase transitions in metallurgy, van der Waals theory and Landau theory [1, 21]
Generalized pyramidal traveling fronts or traveling fronts of various kinds of smooth shapes are constructed in this paper for a bistable or multistable reaction-diffusion with

Summary

Introduction

A given function u0 belongs to BU (R3 ). BU (R3 ) is the space of bounded uniformly continuous functions from R3 to R with the supremum norm. We write a typical example: f (u) = −(u+1)(u+a)(u−1) with a ∈ (0, 1) This equation is called the Allen-Cahn equation, the Nagumo equation or the scalar Ginzburg-Landau equation. It appears in various fields, say, in population genetics [10], ecology [35, 34], bistable transmission in electronic circuits [25], phase transitions in metallurgy, van der Waals theory and Landau theory [1, 21]

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Conclusion