KPP fronts in shear flows with cut-off reaction rates

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 cut-off at concentration $u = u_c$. 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 $A=O(1)$, with particular associated orderings on $B$, whilst $u_c\in(0,1)$ 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 cut-off ($u_c=0$). We illustrate the theory by evaluating expressions (either directly or through numerical integration) for the particular cases of the plane Couette and Poiseuille flows.