Stability of spikes in the shadow Gierer-Meinhardt system with Robin boundary conditions.
Stability of spikes in the shadow Gierer-Meinhardt system with Robin boundary conditions.
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DOI：
10.1063/1.2768156
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发表时间：
2007-09
期刊：
影响因子：
2.9
通讯作者：
P. Maini;Juncheng Wei;M. Winter
中科院分区：
文献类型：
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作者：
P. Maini;Juncheng Wei;M. Winter
考虑光滑有界区域欧米茄子集R(N)中Gierer-Meinhardt系统的影子系统，A(T)=epsilon(2)DeltaA-A+A(P)/Xi(Q)，x是Omega，t>；0的元素，tau/Omega/Xi(T)=-/Omega/Xi+1/Xi(S)积分(Omega)A(R)dx，t>；0，其中a(A)>；0是偏微分Omega的元素.0，反应速率(p，q，r，S)满足10，r>；0，S；或=0，1或=0。我们严格地证明了以下关于单峰波解稳定性的结果：(I)当r=2和11且tau足够小时，内部波峰是稳定的。(Ii)对于N=1，如果r=2且11，使得对于(a(0)，1)的is元素且u=2q/(S+1)(p-1)是(1，u(0))的元素，则近边界尖峰解是不稳定的。这种不稳定性在Neumann边界条件下不存在，而只在Robin边界条件下出现。此外，我们还证明了相应的本征值是O(1)阶为-->；0。
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 10, r>0, s>or=0, 1or=0. We rigorously prove the following results on the stability of one-spike solutions: (i) If r=2 and 11 and tau sufficiently small the interior spike is stable. (ii) For N=1 if r=2 and 11 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.