Boundary spikes in the Gierer-Meinhardt system (Q1866583)

The authors consider the following system of reaction-diffusion equations \[ \begin{cases} a_t-\varepsilon^2\Delta a+a-a^2/h=0 &\text{in }\Omega\times (0,\infty),\\ h_t-D\Delta h+h-a^2=0 &\text{in }\Omega\times (0,\infty),\\ \partial_na=\partial_nh=0 &\text{on }\partial\Omega\times (0,\infty),\\ a(\cdot,0)= a_0>0, \;h(\cdot,0)=h_0>0, &{}\end{cases} \tag{1} \] where \((a,h)\) represents the concentration of the activator and inhibitor respectively, and the parameters \(\varepsilon^2\), \(D>0\) are their diffusion rates, \(\Omega\subset\mathbb R^N\), \(N\leq 3\) is a bounded domain with smooth boundary, \(\partial_n\) denotes the exterior normal derivative. The main goal of this paper is to show the existence of steady state solutions to (1) for the range of \(\varepsilon\) and \(D\).