Multi-peak solutions for the Schnakenberg model with heterogeneity on star shaped graphs

Abstract

In this paper, the Schnakenberg model with heterogeneity function is considered on star shaped metric graphs. We establish the existence and the linear stability of N-peak stationary solutions. In particular, we reveal that the location, amplitude, and stability are decided by the effects of the heterogeneity function and the geometry of the graph, represented by the associated Green’s function. The existence theorem is shown by using Lyapunov–Schmidt reduction method, and the stability is analyzed by investigating the associated linearized eigenvalue problem. Also, by considering several concrete examples, we describe how the location and the stability are decided by the interaction of the geometry of the graph with the heterogeneity function in detail. Moreover, we give the classification of the lengths of edges for the case of one spike per one edge. The case of a one-dimensional interval case without heterogeneity function case was studied by Iron et al. (2004). Although the proof of our main results is based on their strategy, we present all key estimates needed the analysis for the case of a star graph with heterogeneity function in detail.