The critical wave speed for the Fisher-Kolmogorov-Petrowskii-Piscounov equation with cut-off

Abstract

The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than epsilon = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold e, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c(crit)(epsilon) = 2 - pi(2)/(ln epsilon)(2) + O((ln epsilon)(-3)), as epsilon -> 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of ccrit on e as well as for the universality of the corresponding leading-order coefficient (pi(2)).