Boundary spikes in the Gierer-Meinhardt system
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 \(D\).
scientific article; zbMATH DE number 1894767