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Title:  Stability of monotone solutions for the shadow GiererMeinhardt system with finite diffusivity 
Authors:  Winter, M Wei, J 
Keywords:  Stability Hopf bifurcations; Finite diffusivities; GiererMeinhardt system 
Publication Date:  2003 
Publisher:  Elsevier 
Citation:  Diff Int Equations 16: 11531180 
Abstract:  We consider the following shadow system of the GiererMeinhardt 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}{p1} >s+1,\, s\geq 0, and \tau >0.
It is known that a nontrivial monotone steadystate solution exists if and only if
\ep < \frac{\sqrt{p1}}{\pi}.
In this paper, we show that for any \ep < \frac{\sqrt{p1}}{\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. 
URI:  http://www.elsevier.com/wps/find/journaldescription.cws_home/622868/description#description http://bura.brunel.ac.uk/handle/2438/569 
Appears in Collections:  Mathematical Science Dept of Mathematics Research Papers

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