The Mathematical Institute, University of Oxford, Eprints Archive

Stability of spikes in the shadow Gierer-Meinhardt system with Robin boundary conditions

Maini, P. K. and Wei, J. and Winter, M. (2007) Stability of spikes in the shadow Gierer-Meinhardt system with Robin boundary conditions. Chaos, 17 (3). 037106-1-037106-16. ISSN n/a

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Abstract

We consider the shadow system of the Gierer-Meinhardt system in a smooth bounded domain RN,At=2A−A+,x, t>0, ||t=−||+Ardx, t>0 with the Robin boundary condition +aAA=0, x, where aA>0, the reaction rates (p,q,r,s) satisfy 1<p<()+, q>0, r>0, s0, 1<<+, the diffusion constant is chosen such that 1, and the time relaxation constant is such that 0. We rigorously prove the following results on the stability of one-spike solutions: (i) If r=2 and 1<p<1+4/N or if r=p+1 and 1<p<, then for aA>1 and sufficiently small the interior spike is stable. (ii) For N=1 if r=2 and 1<p3 or if r=p+1 and 1<p<, then for 0<aA<1 the near-boundary spike is stable. (iii) For N=1 if 3<p<5 and r=2, then there exist a0(0,1) and µ0>1 such that for a(a0,1) and µ=2q/(s+1)(p−1)(1,µ0) the near-boundary spike solution is unstable. This instability is not present for the Neumann boundary condition but only arises for the Robin boundary condition. Furthermore, we show that the corresponding eigenvalue is of order O(1) as 0. ©2007 American Institute of Physics

Item Type:Article
Additional Information:n/a
Uncontrolled Keywords:n/a
Subjects:A - C > Biology and other natural sciences
Research Groups:Centre for Mathematical Biology
ID Code:661
Deposited By:Philip Maini
Deposited On:02 Oct 2007
Last Modified:20 Jul 2009 14:23

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