Concentration phenomena on Y-shaped metric graph for the Gierer-Meinhardt model with heterogeneity

In this paper, we consider the Gierer-Meinhardt model with the heterogeneity in both the activator and the inhibitor on the Y-shaped compact metric graph. Using the Lyapunov-Schmidt reduction method, we construct a one-peak stationary solution, which concentrates at a suitable point. In particular, we reveal that the location of concentration point is determined by the interaction of the heterogeneity function for the activator with the geometry of the domain, represented by the associated Green's function. Moreover, based on our main result, we determine the precise location of concentration point for non-heterogeneity case. Furthermore, we also present the effect of heterogeneity by using a concrete example. (C) 2020 Elsevier Ltd. All rights reserved.

Automated summary

This paper investigates the Gierer-Meinhardt model with heterogeneity in both activator and inhibitor on a Y-shaped compact metric graph. A one-peak stationary solution, concentrated at a suitable point, is constructed using the Lyapunov-Schmidt reduction method. It is shown that the location of concentration point is determined by the interaction of the activator's heterogeneity function with the geometry of the domain represented by the associated Green's function. Additionally, the precise location of concentration point for the non-heterogeneity case is determined, and the effect of heterogeneity is demonstrated using a concrete example.