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 $u=u_c$. In the long-time limit, a permanent-form traveling wave solution is established which, for fixed $u_c>0$, is unique. Its structure and speed of propagation depends on $A$ (the strength of the flow relative to the propagation speed in the absence of advection) and $B$ (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 $A\to \infty$, $A\to 0$, and $A=O\left(1\right)$, with particular associated orderings on $B$, while $u_c\in (0,1)$ 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 ($u_c=0$). We illustrate the theory by evaluating expressions (either directly or through numerical integration) for the particular cases of the plane Couette and Poiseuille flows.