# Spike Solutions to the Supercritical Fractional Gierer-Meinhardt System

## TL;DR

This paper investigates localized spike solutions in a one-dimensional supercritical fractional Gierer-Meinhardt system, employing asymptotic methods and stability analysis, and establishes rigorous existence results related to fractional Green's functions.

## Contribution

It introduces a novel analysis of multi-spike solutions in a supercritical fractional GM system, combining formal asymptotics with rigorous existence and stability proofs.

## Key findings

- Construction of multi-spike solutions via nonlinear algebraic systems
- Stability analysis through eigenvalue problems
- Rigorous existence of ground-state solutions near critical fractional order

## 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 L\'evy flights. In this paper we investigate localized solutions to a one-dimensional fractional GM system for which the inhibitor's fractional order is supercritical. 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.