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

We consider the shadow system of the Gierer-Meinhardt system in a smooth bounded domain Omega subset R(N),A(t)=epsilon(2)DeltaA-A+A(p)/xi(q),x is element of Omega, t>0, tau/Omega/xi(t)=-/Omega/xi+1/xi(s) integral(Omega)A(r)dx, t>0 with the Robin boundary condition epsilon partial differentialA/partial differentialnu+a(A)A=0, x is element of partial differentialOmega, where a(A)>0, the reaction rates (p,q,r,s) satisfy 1<p<(N+2/N-2)(+), q>0, r>0, s>or=0, 1<qr/(s+1)(p-1)<+infinity, the diffusion constant is chosen such that epsilon<1, and the time relaxation constant is such that tau>or=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<infinity, then for a(A)>1 and tau sufficiently small the interior spike is stable. (ii) For N=1 if r=2 and 1<p<or=3 or if r=p+1 and 1<p<infinity, then for 0<a(A)<1 the near-boundary spike is stable. (iii) For N=1 if 3<p<5 and r=2, then there exist a(0) is element of (0,1) and mu(0)>1 such that for a is element of (a(0),1) and mu=2q/(s+1)(p-1) is element of (1,mu(0)) 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 epsilon-->0.