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

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...

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Bibliographic Details
Published inIMA journal of applied mathematics Vol. 86; no. 2; pp. 247 - 293
Main Authors Yang, Jichen, Rademacher, Jens D M
Format Journal Article
LanguageEnglish
Published Oxford University Press 01.04.2021
Subjects
reaction–subdiffusion
fractional PDEs
Turing instability
spectrum
stability
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Summary: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 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.
ISSN:0272-4960
1464-3634
DOI:10.1093/imamat/hxaa044