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On the shape of the stable patterns for activator-inhibitor systems in two-dimensional domains

Author(s): Yasuhito Miyamoto
Journal: Quart. Appl. Math. 65 (2007), 357-374.
MSC (2000): Primary 35B35, 35K57; Secondary 35J60, 35P15
Posted: March 5, 2007
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Abstract: We continue to study the shape of the stable steady states of the so-called shadow limit of activator-inhibitor systems in two-dimensional domains

$\displaystyle u_t=D_u\Delta u+f(u,\xi)\;$ in $\displaystyle \; \Omega\times\mathbb{R}_+$    and    $\displaystyle \tau\xi_t=\frac{1}{\vert\Omega\vert}\iint_{\Omega}g(u,\xi)dxdy \;$ in $\displaystyle \;\mathbb{R}_+,$    
$\displaystyle \partial_{\nu}u=0\;$ on $\displaystyle \; \partial\Omega\times\mathbb{R}_+,$    

where $ f$ and $ g$ satisfy the following: $ g_{\xi}<0$, and there is a function $ k(\xi)\in C^0$ such that $ f_{\xi}(u,\xi)=k(\xi)g_{u}(u,\xi)$. This class of reaction-diffusion systems includes the FitzHugh-Nagumo system and a special case of the Gierer-Meinhardt system. In the author's previous paper ``An instability criterion for activator-inhibitor systems in a two-dimensional ball'' (J. Diff. Eq. 229 (2006), 494-508), we obtain a necessary condition about the profile of $ u$ on the boundary of the domain for a steady state $ (u,\xi)$ to be stable when the domain is a two-dimensional ball. In this paper, we give a necessary condition about the profile of $ u$ in the domain when the domain is a two-dimensional ball, annulus or rectangle. Roughly speaking, we show that if $ (u,\xi)$ is stable for some $ \tau >0$, then the shape of $ u$ is like a boundary one-spike layer even if $ D_u$ is not small.

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Additional Information:

Yasuhito Miyamoto
Affiliation: Research Institute for Mathematical Sciences, Kyoto Univ., Kyoto, 606-8502, Japan
Email: miyayan@sepia.ocn.ne.jp

PII: S0033-569X-07-01038-2
Keywords: Activator-inhibitor system, shadow system, reaction-diffusion system, stability, nodal curve, nodal domain
Received by editor(s): May 23, 2006
Posted: March 5, 2007
Copyright of article: Copyright 2007, Brown University
The copyright for this article reverts to public domain after 28 years from publication.


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