Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity
Year: 2003
Type: article
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.
Authors: Juncheng Wei, Matthias Winter
Institutions: Chinese University of Hong Kong, University of Stuttgart
Language: English
Subfield: Computer Networks and Communications
Field: Computer Science
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1571293882