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On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system
Kavallaris, Nikos I. Suzuki, Takashi
Kavallaris, Nikos I.
Suzuki, Takashi
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2017-03-21
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Abstract
The purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.
Citation
Kavallaris, N. I., & Suzuki, T. (2017). On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system. Nonlinearity, 30(5), 1734. https://doi.org/10.1088/1361-6544/aa64b2
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London Mathematical Society
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Nonlinearity
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Article
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en
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This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://iopscience.iop.org/article/10.1088/1361-6544/aa64b2/meta
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0951-7715
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1361-6544