Please use this identifier to cite or link to this item: http://bura.brunel.ac.uk/handle/2438/569
 Title: Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity Authors: Winter, MWei, 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},\, 00,\\ \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 1s+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. URI: http://www.elsevier.com/wps/find/journaldescription.cws_home/622868/description#descriptionhttp://bura.brunel.ac.uk/handle/2438/569 Appears in Collections: Dept of Mathematics Research PapersMathematical Sciences

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