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 ε, D > 0, κ 0, σ 0. The saturation effect of the activator is given by the parameter κ. We will give a sufficient condition of κ 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 ε is sufficiently small under the assumption: κ depends on ε, namely, κ = κ(ε), and there exists a limit lim ε→0 κε −2 = κ 0 for certain κ 0 ∈ [0, ∞).