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

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 generalized 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.