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

Abstract

We consider the shadow system of the Gierer-Meinhardt system in a smooth bounded domain $\Omega\subset R^N$: \[ \left\{ \begin{array}{ll} A_t=\ep^2 \Delta A- A+\frac{A^p}{\xi^q}, &x\in\Om,\, t>0,\\[2mm] \tau |\Om|\xi_t= -|\Om|\xi+ \frac{1}{\xi^s}\int_{\Om} A^r\,dx, & t>0\\ % A^{'} (-1)= A^{'} (1)= H^{'} (-1) = H^{'} (1) =0, \end{array} \right. \] with Robin boundary condition \[\ep \frac{\partial A}{\partial \nu}+a_A A=0, \quad x\in\partial\Om, \] where $a_A>0$, the reaction rates $(p, q, r, s)$ satisfy \[1 <p < \left(\frac{N+2}{N-2}\right)_+,\quad q> 0,\quad r>0,\quad s\geq 0,\quad 1< \frac{ qr}{(s+1)( p-1)} < +\infty,\] the diffusion constant is chosen such that $\ep<<1$ and the time relaxation constant such that $\tau\geq 0$. 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 $1<p<\infty$ then for $a_A>1$ and $\tau$ sufficiently small the interior spike is stable. (ii) For $N=1$ if $r=2$ and $1<p\leq 3$ or if $r=p+1$ and $1<p<\infty$ 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 $a_0\in (0,1)$ and $\mu_0>1$ such that for $a\in(a_0,1)$ and $ \mu= \frac{2q}{(s+1)(p-1)}\in(1,\mu_0)$ the near-boundary spike solution is unstable. This instability is not present for the Neumann boundary condition but only arises for Robin boundary condition. Further we show that the corresponding eigenvalue is of order $O(1)$ as $\ep\to0$.