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 κ₀ ε^2α for α ∈ (0,1) (semi-strong saturation effect). Here, ε > 0 stands for the diffusion constant of activator. For sufficiently small ε, 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(ε) in width,
(b) the transition-layer is located in the position of O(ε^α) from the boundary x = 0,
(c) the solution concentrates at x = 0 with the amplitude of the order of O(ε^−α) when ε ≪ 1.