Tools for stability analysis of fractional reaction diffusion systems

TL;DR

This paper extends the linearization principle to fractional reaction-diffusion systems with time derivatives of order less than one, providing tools to analyze their stability and phenomena like Turing instability.

Contribution

It proves the linearization principle for fractional reaction-diffusion equations and applies it to analyze stability and Turing patterns in these systems.

Findings

Established the linearization principle for fractional systems.

Derived conditions for stability and instability in fractional reaction-diffusion models.

Extended classical Turing instability analysis to fractional derivatives.

Abstract

The linearization principle states that the stability (or instability) of solutions to a suitable linearization of a nonlinear problem implies the stability (or instability) of solutions to the original nonlinear problem. In this work, we prove this principle for solutions of abstract fractional reaction-diffusion equations with a fractional derivative in time of order $\alpha\in (0,1)$. Then, we apply these results to particular fractional reaction-diffusion equations, obtaining, for example, the counterpart of the classical Turing instability in the case of fractional equations.