Global-in-time behavior of the solution to a Gierer-Meinhardt system

Georgia Karali; Takashi Suzuki; Yoshio Yamada

doi:10.3934/dcds.2013.33.2885

Article Contents

2013, Volume 33, Issue 7: 2885-2900. Doi: 10.3934/dcds.2013.33.2885

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Global-in-time behavior of the solution to a Gierer-Meinhardt system

1.
Department of Applied Mathematics, University Crete, P.O. Box 2208, 71409, Heraklion, Crete
2.
Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikane-yama, Toyonaka, Osaka, 560-8531
3.
Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku 169-8555, Tokyo

Received: March 31, 2012

Revised: August 31, 2012

Published: June 2013

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Abstract

Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is expected in the presence of local self-enhancement and long-range inhibition. The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes periodic-in-time solutions, that is, $\tau=\frac{s+1}{p-1}$. Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually disappear if those parameters are in a region without local self-enhancement or long-range inhibition.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35Q92.

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