Memory Effects in RD Equation

To account for memory effects in the transport, one is tempted to replace the standard Pick s first law form of the constitutive equation by a nonlocal phenomenological relation between the flux and the density gradient  

Combining the constitutive equation for the flux (5.40) with the continuity equation for the particle density, we obtain the nonlocal RD equation 

If one considers the exponential memory kernel in (5.41) and takes the temporal derivative, the reaction-telegraph equation (2.19) is recovered. 

We assume that the reaction term is of the KPP type, F(p) = rp l — p). Our main goal is to find the front velocity for a general memory kernel in terms of its Laplace transform, K, and to show that fronts travel with a finite velocity in the fast reaction limit only if the initial value of the memory kernel is positive definite. This general result will be applied to some specific memory kernels. 

We proceed as in the previous sections and find that the Hamilton-Jacobi equation for (5.41) can be written as 

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