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Title: Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity
Authors: Winter, M
Wei, J
Keywords: Stability;Hopf bifurcations; Finite diffusivities;;Gierer-Meinhardt system
Issue Date: 2003
Publisher: Elsevier
Citation: Diff Int Equations 16: 1153-1180
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 \ep < \frac{\sqrt{p-1}}{\pi}. In this paper, we show that for any \ep < \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.
Appears in Collections:Dept of Mathematics Research Papers
Mathematical Sciences

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