Reaction-Cattaneo equation

As a second approach, we use reaction-Cattaneo equations, see Sect. 2.2.2, to study the effect of inertia on Turing instabilities [206]. The uniform steady state of the two-species reaction-Cattaneo system. 

As for HRDEs, the Turing condition C4 = 0 for the reaction-Cattaneo system leads to exactly the same conditions as for the standard reaction-diffusion equation, namely (10.42) and (10.40) the Turing condition is independent of r and ty for reaction-Cattaneo equations. As for HRDEs, the spatial Hopf bifurcation cannot occur in the regime of small inertia, if (10.23) is satisfied. 

Reaction-Cattaneo Systems and Reaction-Telegraph Equations... 

We call the hyperbolic system (2.17) and (2.18) a reaction-Cattaneo system. Eu and Al-Ghoul have derived such systems from generalized hydrodynamic theory [9, 7, 8, 6]. Reaction-Cattaneo systems can also be obtained from extended irreversible thermodynamics [223], see for example [282]. If we differentiate (2.17) with respect to t and (2.18) with respect to x and eliminate mixed second derivatives, we obtain the so-called reaction-telegraph equation. 

This relation between the relaxation time t of the flux and the time scale l/F (p) of the reaction appears to be a purely mathematical requirement. The following mesoscopic approach will shed light on the foundational problems of the reaction-Cattaneo system (2.17) and (2.18) and the reaction-telegraph equation (2.19) hinted at by points (i) and (ii). 

For n-variable systems, the reaction-Cattaneo systems and reaction-telegraph equations read, i = 1,... 

The appropriate spatial modes for the fluxes 8J and 8Jy are sin jr), and we obtain the characteristic equation for the reaction-Cattaneo system, det J = 0, where... 

Remark 10.2 These equations do not constitute a reaction-Cattaneo system because of the contribution of the intrinsic death rates f (p , Py), / = 1, 2, to the decay rate of the flows. Further, no reaction-telegraph equations can be derived for the total densities, unless the death rates /i (p , Py) are constants. 

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See also in sourсe #XX -- [ ]

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