Spike Solutions to the Supercritical Fractional Gierer–Meinhardt System

Abstract

Localized solutions are known to arise in a variety of singularly perturbed reaction–diffusion systems. The Gierer–Meinhardt (GM) system is one such example and has been the focus of numerous rigorous and formal studies. A more recent focus has been the study of localized solutions in systems exhibiting anomalous diffusion, particularly with Lévy flights. In this paper, we investigate localized solutions to a one-dimensional fractional GM system for which the inhibitor’s fractional order is supercritical. Specifically, we assume the fractional orders of the activator and inhibitor are, respectively, in the ranges \(s_1\in (1/4,1)\) and \(s_2\in (0,1/2)\). Using the method of matched asymptotic expansions, we reduce the construction of multi-spike solutions to solving a nonlinear algebraic system. The linear stability of the resulting multi-spike solutions is then addressed by studying a globally coupled eigenvalue problem. In addition to these formal results, we also rigorously establish the existence and stability of ground state solutions when the inhibitor’s fractional order is nearly critical. The fractional Green’s function, for which we present a rapidly converging series expansion, is prominently featured throughout both the formal and rigorous analysis in this paper. Moreover, we emphasize that the striking similarities between the one-dimensional supercritical GM system and the classical three-dimensional GM system can be attributed to the leading-order singular behaviour of the fractional Green’s function.