KPP fronts in shear flows with cutoff reaction rates

Abstract

We consider the effect of a shear flow which has, without loss of generality, a zero mean flow rate, on a Kolmogorov–Petrovskii–Piscounov (KPP)‐type model in the presence of a discontinuous cutoff at concentration . In the long‐time limit, a permanent‐form traveling wave solution is established which, for fixed , is unique. Its structure and speed of propagation depends on (the strength of the flow relative to the propagation speed in the absence of advection) and (the square of the front thickness relative to the channel width). We use matched asymptotic expansions to approximate the propagation speed in the three natural cases , , and , with particular associated orderings on , while remains fixed. In all the cases that we consider, the shear flow enhances the speed of propagation in a manner that is similar to the case without cutoff (). We illustrate the theory by evaluating expressions (either directly or through numerical integration) for the particular cases of the plane Couette and Poiseuille flows.