On the Gierer-Meinhardt System with Saturation

Abstract

We consider the following shadow Gierer-Meinhardt system with saturation: \left\{\begin{array}{l} A_t=\epsilon^2 \Delta A -A + \frac{A^2}{ \xi (1+k A^2)} \ \ \mbox{in} \ \Omega \times (0, \infty),\\ \tau \xi_t= -\xi +\frac{1}{|\Omega|} \int_\Om A^2\,dx \ \ \mbox{in} \ (0, +\infty), \frac{\partial A}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega\times(0,\infty), \end{array} \right. where \ep>0 is a small parameter, $\tau \geq 0,\, k>0 and \Omega \subset R^n is smooth bounded domain. The case k=0 has been studied by many authors in recent years. Here we give some sufficient conditions on $k$ for the existence and stability of stable spiky solutions. In the one-dimensional case we have a complete answer to the stability behavior. Central to our study are a parameterized ground-state equation and the associated nonlocal eigenvalue problem (NLEP) which is solved by functional analysis arguments and the continuation method.