Stability of spikes in the shadow Gierer-Meinhardt system with Robin boundary conditions - Maini, P; Wei, J; Winter, M (2007)
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 1<p<,="" aa="" and="" for="" if="" n="" or="" r="p+1" then="">1 and sufficiently small the interior spike is stable. (ii) For N=1 if r=2 and 1<p3 (iii)="" 0<aa<1="" 1<p<,="" 3<p<5="" a0(0,1)="" and="" exist="" for="" if="" is="" n="1" near-boundary="" or="" r="2," spike="" stable.="" the="" then="" there="" 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</p3></p<1+4></p<()+,>