## Differential and Integral Equations

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

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 21 December 2012