Henry, B. I.; Wearne, S. L. Existence of Turing instabilities in a two-species fractional reaction-diffusion system. (English) Zbl 1103.35047 SIAM J. Appl. Math. 62, No. 3, 870-887 (2002). Summary: We introduce a two-species fractional reaction-diffusion system to model activator-inhibitor dynamics with anomalous diffusion such as occurs in spatially inhomogeneous media. Conditions are derived for Turing-instability induced pattern formation in these fractional activator-inhibitor systems whereby the homogeneous steady state solution is stable in the absence of diffusion but becomes unstable over a range of wave-numbers when fractional diffusion is present. The conditions are applied to a variant of the Gierer–Meinhardt reaction kinetics which has been generalized to incorporate anomalous diffusion in one or both of the activator and inhibitor variables. The anomalous diffusion extends the range of diffusion coefficients over which Turing patterns can occur. An intriguing possibility suggested by this analysis, which can arise when the diffusion of the activator is anomalous but the diffusion of the inhibitor is regular, is that Turing instabilities can exist even when the diffusion coefficient of the activator exceeds that of the inhibitor. Cited in 92 Documents MSC: 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) 92E20 Classical flows, reactions, etc. in chemistry Keywords:reaction-diffusion; Turing pattern; anomalous diffusion; inhomogeneous media PDFBibTeX XMLCite \textit{B. I. Henry} and \textit{S. L. Wearne}, SIAM J. Appl. Math. 62, No. 3, 870--887 (2002; Zbl 1103.35047) Full Text: DOI