Cut-offs in a degenerate advection–reaction–diffusion equation — a case study

We investigate the effect of a Heaviside cut-off on the front propagation dynamics of a degenerate advection–reaction–diffusion equation. In particular, we consider two formulations of the equation, one with the cut-off function multiplying the reaction kinetics alone and one in which the cut-off is also applied to the advection term. We prove the existence and uniqueness of a “critical” front solution in both cases, and we derive the leading-order correction to the front propagation speed in dependence on the advection strength and the cut-off parameter. We show that, while the asymptotics of the correction in the cut-off parameter remains unchanged to leading order when the advection term is cut off, the corresponding coefficient is different. Finally, we consider a generalised family of advection–reaction–diffusion equations, and we identify scenarios in which the application of a cut-off to the advection term substantially affects the front propagation speed. Our analysis relies on geometric techniques from dynamical systems theory and, specifically, on geometric desingularisation, also known as “blow-up”.