Existence and stability of symmetric and asymmetric patterns for the half-Laplacian Gierer–Meinhardt system in one-dimensional domain

Abstract

In this paper, we study the existence and stability of multiple spikes pattern to the fractional Gierer–Meinhardt model with periodic boundary conditions and the fractional power [Formula: see text]. Specifically, we rigorously establish the existence of symmetric multiple spikes and asymmetric two-spikes solutions by the classical Lyapunov–Schmidt reduction method. We also investigate the stability of the constructed solution by studying its associated large and small eigenvalue problems, where we need to consider two nonlocal eigenvalue problems in their fractional versions. In the study of the large eigenvalue problem, the quantity [Formula: see text] is the critical threshold which determines the stability of [Formula: see text]-peaked solutions. For the symmetric two-spikes pattern we obtain the asymptotic expansion for the critical threshold [Formula: see text] up to the second order. Moreover, we provide some elementary properties of the Green’s function, including the first and second derivatives, they are linked to the location of the spikes and the stability. Among these properties on the Green’s function, we find out that the polygamma function [Formula: see text] plays a crucial role.