Reaction-subdiffusion systems and memory: spectra, Turing instability and decay estimates

TL;DR

This paper investigates reaction-subdiffusion systems with memory effects, analyzing spectra, Turing instability, and decay behavior, revealing conditions for algebraic versus exponential decay in different classes of such systems.

Contribution

It introduces generalized spectra via dispersion relations to study Turing instability and decay estimates in reaction-subdiffusion equations, providing new insights into their qualitative behavior.

Findings

Identification of conditions for Turing instability in subdiffusion systems

Derivation of algebraic decay for certain classes of stable spectra

Proof of exponential decay in another class of subdiffusion equations

Abstract

The modelling of linear and nonlinear reaction-subdiffusion processes is more subtle than normal diffusion and causes different phenomena. The resulting equations feature a spatial Laplacian with a temporal memory term through a time fractional derivative. It is known that the precise form depends on the interaction of dispersal and reaction, and leads to qualitative differences. We refine these results by defining generalised spectra through dispersion relations, which allows us to examine the onset of instability and in particular inspect Turing type instabilities. These results are numerically illustrated. Moreover, we prove expansions that imply for one class of subdiffusion reaction equations algebraic decay for stable spectrum, whereas for another class this is exponential.