Maini, P. K. and Wei, J. and Winter, M. (2007) Stability of spikes in the shadow GiererMeinhardt system with Robin boundary conditions. Chaos, 17 (3). 037106103710616. ISSN n/a

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Abstract
We consider the shadow system of the GiererMeinhardt 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 onespike 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 nearboundary 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 nearboundary 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:  29 May 2015 18:26 
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