Point-condensation phenomena and saturation effect for the one-dimensional Gierer–Meinhardt system

Abstract

In this paper, we are concerned with peak solutions to the following one-dimensional Gierer–Meinhardt system with saturation:

where $\epsilon ,D>0$, $\kappa ⩾0$, $\sigma ⩾0$. The saturation effect of the activator is given by the parameter $\kappa$. We will give a sufficient condition of $\kappa$ for which point-condensation phenomena emerge. More precisely, for fixed $D>0$, we will show that the Gierer–Meinhardt system admits a peak solution when $\epsilon$ is sufficiently small under the assumption: $\kappa$ depends on $\epsilon$, namely, $\kappa =\kappa (\epsilon )$, and there exists a limit ${\lim }_{\epsilon \to 0}\kappa {\epsilon }^{-2}={\kappa }_0$ for certain ${\kappa }_0\in [0,\infty )$.