Front speeds, cut-offs, and the blow-up method in scalar reaction-diffusion equations

The study of front propagation in reaction-diffusion systems is a central topic in non-equilibrium physics. Recently, the effects of a cut-off in the reaction terms on the front propagation speed have received much attention. Cut-offs were introduced by Brunet and Derrida in the classical FKPP equation to model phenomena in which concentrations cannot fall below a small threshold; they typically induce a shift in the propagation speed that is largely independent of the choice of cut-off function.

We define a dynamical systems framework for the study of front propagation in cut-off scalar reaction-diffusion equations that is based on geometric singular perturbation theory and geometric desingularization (blow-up). We study the asymptotics of the shift in propagation speed in terms of the cut-off parameter, and we show that this asymptotics is determined by a certain normal form system in blown-up phase space; in particular, we argue that the occurrence of logarithmic (switchback) terms in the asymptotics is caused by resonances in this normal form. Finally, we discuss two examples, the FKPP equation analyzed by Brunet and Derrida and an equation with density-dependent diffusivity, and we indicate how our results can be extended to more general families of scalar reaction-diffusion equations with cut-off.