On positive solutions generated by semi-strong saturation effect for the Gierer-Meinhardt system

Abstract

In this paper, we study the existence and the asymptotic behavior of a positive solution to the one-dimensional stationary shadow system of the Gierer-Meinhardt system with saturation. We equip a reaction term of activator with saturation effect $\kappa_0 \varepsilon^{2\alpha}$ for $\alpha\in (0,1)$ (semi-strong saturation effect). Here, $\varepsilon$ > 0 stands for the diffusion constant of activator. For sufficiently small $\varepsilon$, we show the existence of a new type of solutions which has the following properties:

(a) the solution has an internal transition-layer of $O(\varepsilon)$ in width,

(b) the transition-layer is located in the position of $O(\varepsilon^\alpha)$ from the boundary $x=0$,

(c) the solution concentrates at $x=0$ with the amplitude of the order of $O(\varepsilon^{-\alpha})$ when $\varepsilon \ll 1$.