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2003 Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity
Juncheng Wei, Matthias Winter
Differential Integral Equations 16(10): 1153-1180 (2003).

Abstract

We consider the following shadow system of the Gierer-Meinhardt model: \[ \left\{\begin{array}{l} A_t= \epsilon^2 A_{xx} -A +\frac{A^p}{\xi^q},\, 0 <x <1,\, t>0,\\ \tau \xi_t= -\xi + \xi^{-s} \int_0^1 A^2 \,dx,\\ A>0,\, A_x (0,t)= A_x(1, t)=0, \end{array} \right. \] where $1 <p <+\infty,\, \frac{2q}{p-1} >s+1,\, s\geq 0$, and $\tau >0$. It is known that a nontrivial monotone steady-state solution exists if and only if $ \epsilon < \frac{\sqrt{p-1}}{\pi}.$ In this paper, we show that for any $\epsilon < \frac{\sqrt{p-1}}{\pi}$, and $p=2$ or $p=3$, there exists a unique $\tau_c>0$ such that for $\tau <\tau_c$ this steady state is linearly stable, while for $\tau>\tau_c$ it is linearly unstable. (This result is optimal.) The transversality of this Hopf bifurcation is proven. Other cases for the exponents as well as extensions to higher dimensions are also considered. Our proof makes use of functional analysis and the properties of Weierstrass functions and elliptic integrals.

Citation

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Juncheng Wei. Matthias Winter. "Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity." Differential Integral Equations 16 (10) 1153 - 1180, 2003.

Information

Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1074.35017
MathSciNet: MR2014805

Subjects:
Primary: 35K57
Secondary: 35B25 , 35B45 , 37L10 , 92C15

Rights: Copyright © 2003 Khayyam Publishing, Inc.

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Vol.16 • No. 10 • 2003
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