Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems

Yan-Yu Chen; Yoshihito Kohsaka; Hirokazu Ninomiya

doi:10.3934/dcdsb.2014.19.697

Article Contents

2014, Volume 19, Issue 3: 697-714. Doi: 10.3934/dcdsb.2014.19.697

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Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems

1.
Department of Mathematics, Tamkang University, No. 151, Yingzhuan Rd., Tamsui Dist., New Taipei City 25137
2.
Division of System Engineering for Mathematics, Muroran Institute of Technology, 27-1 Mizumoto-cho, Muroran 050-8585
3.
School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo, 164-8525

Received: August 31, 2013

Revised: November 30, 2013

Published: April 2014

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Abstract

In this study, we consider the traveling spots that were observed in the photosensitive Belousov-Zhabotinsky reaction experiment conducted by Mihailuk et al. in 2001. First, we introduce the interface equation by the singular limit analysis of a FitzHugh--Nagumo-type reaction-diffusion system. Then, we obtain the profile of the support of the solution. Next, we prove the uniqueness of the traveling spot by studying ordinary differential equations that describe its front and back. In addition, we provide an upper bound for the width of the spot. Furthermore, we compare the singular limit problem with the wave front interaction model proposed by Zykov and Showalter in 2005 and obtain traveling fingers.

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Mathematics Subject Classification: Primary: 35K57, 35C07; Secondary: 35R35.

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