Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations

Masaharu Taniguchi

doi:10.3934/dcds.2020126

Article Contents

2020, Volume 40, Issue 6: 3981-3995. Doi: 10.3934/dcds.2020126 open access

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Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations

Masaharu Taniguchi

Research Institute for Interdisciplinary Science, Okayama University, 3-1-1, Tsushimanaka, Kita-ku, Okayama City, 700-8530, Japan

Received: April 2019

Revised: September 2019

Published: March 2020

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Abstract

For a balanced bistable reaction-diffusion equation, the existence of axisymmetric traveling fronts has been studied by Chen, Guo, Ninomiya, Hamel and Roquejoffre [4]. This paper gives another proof of the existence of axisymmetric traveling fronts. Our method is as follows. We use pyramidal traveling fronts for unbalanced reaction-diffusion equations, and take the balanced limit. Then we obtain axisymmetric traveling fronts in a balanced bistable reaction-diffusion equation. Since pyramidal traveling fronts have been studied in many equations or systems, our method might be applicable to study axisymmetric traveling fronts in these equations or systems.

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

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Figure 1. A level set $ \left\{ {U(\mathit{\boldsymbol{x'}}, {x_n}) = {\theta _0}} \right\}$ of $ U $

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Figure 2. A level set $ \{V_{k}(\mathit{\boldsymbol{x}}', x_{n}) = \theta_{0}\} $ of a square pyramidal traveling front $ V_{k} $

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