Please use this identifier to cite or link to this item: http://bura.brunel.ac.uk/handle/2438/1224
 Title: Stability of spikes in the shadow Gierer-Meinhardt system with Robin boundary conditions Authors: Winter, MMaini, PKWei, J Keywords: Gierer-Meinhardt system;Robin boundary condition;Spike;Stability Issue Date: 2007 Publisher: American Institute of Physics Citation: Chaos 17: 037106, Sep 2007 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 $1 0,\quad r>0,\quad s\geq 0,\quad 1< \frac{ qr}{(s+1)( p-1)} < +\infty,$ 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 $11$ and $\tau$ sufficiently small the interior spike is stable. (ii) For $N=1$ if $r=2$ and $11$ 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 $\ep\to0$. URI: http://chaos.aip.org/http://bura.brunel.ac.uk/handle/2438/1224 Appears in Collections: Dept of Mathematics Research PapersMathematical Sciences

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