Resumen

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 wavenumbers 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.

RESUMEN

Brailovsky and Sivashinsky have proposed a model for pressure-driven combustion in the form of a degenerate parabolic system for temperature, concentration, and pressure. It is shown that the existence and uniqueness problem for traveling front solutions can be completely solved by exploiting the existing invariants and by phase plane methods. The approach yields exact propagation speeds which are noticeably larger than the approximations obtained so far.