| Summary: | 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,="" and="" chosen="" constant="" diffusion="" epsilon<<1,="" is="" relaxation="" such="" tau="" that="" the="" time="">or=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<infinity,="" a(a)="" and="" for="" if="" n="" or="" r="p+1" then="">1 and tau sufficiently small the interior spike is stable. (ii) For N=1 if r=2 and 1<p<or=3 (0,1)="" (iii)="" 0<a(a)<1="" 1<p<infinity,="" 3<p<5="" a(0)="" and="" element="" exist="" for="" if="" is="" mu(0)="" n="1" near-boundary="" of="" or="" r="2," spike="" stable.="" the="" then="" there="">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.</p<or=3></p<1+4></qr></p<(n+2>
|
|---|