Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Volume 33, Issue 7, Pages 2885-2900Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcds.2013.33.2885
Keywords
Reaction-diffusion equation; Gierer-Meinhardt system; Turing pattern; Hamilton structure; asymptotic behavior of the solution
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Funding
- Marie Curie International Research Staff Exchange Scheme (IRSES) with the 7th European Community Framework Programme [PIRSESGA-2009-247486]
- project Archimedes Center for Modeling, Analysis and Computation [FP7-REGPOT-2009-1]
- JST project, CREST (Alliance for Breakthrough Between Mathematics and Sciences)
- Grants-in-Aid for Scientific Research [24540220] Funding Source: KAKEN
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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 = 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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