Weakly pushed nature of "pulled" fronts with a cutoff.

The concept of pulled fronts with a cutoff epsilon has been introduced to model the effects of the discrete nature of the constituent particles on the asymptotic front speed in models with continuum variables (pulled fronts are the fronts that propagate into an unstable state, and have an asymptotic front speed equal to the linear spreading speed v* of small linear perturbations around the unstable state). In this paper, we demonstrate that the introduction of a cutoff actually makes such pulled fronts weakly pushed. For the nonlinear diffusion equation with a cutoff, we show that the longest relaxation times tau(m) that govern the convergence to the asymptotic front speed and profile, are given by tau(-1)(m) approximately equal to [(m+1)(2)-1]pi(2)/ln(2)epsilon, for m=1,2,....