Differential and Integral Equations

Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity

Juncheng Wei and Matthias Winter

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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.

Article information

Differential Integral Equations, Volume 16, Number 10 (2003), 1153-1180.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B25: Singular perturbations 35B45: A priori estimates 37L10: Normal forms, center manifold theory, bifurcation theory 92C15: Developmental biology, pattern formation


Wei, Juncheng; Winter, Matthias. Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity. Differential Integral Equations 16 (2003), no. 10, 1153--1180. https://projecteuclid.org/euclid.die/1356060543

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