Traveling waves for discrete reaction-diffusion equations in the general monostable case

Abstract

We consider general fully nonlinear discrete reaction-diffusion equations ut=F[u], described by some function F. In the positively monostable case, we study monotone traveling waves of velocity c, connecting the unstable state 0 to a stable state 1. Under Lipschitz regularity of F, we show that there is a minimal velocity cF+ such that there is a branch of traveling waves with velocities c≥cF+ and no traveling waves for c<cF+. We also show that the map F↦cF+ is not continuous for the L∞ norm on F. Assuming more regularity of F close to the unstable state 0, we show that cF+≥cF⁎, where the velocity cF⁎ can be computed from the linearization of the equation around the unstable state 0. In addition, we show that the inequality can be strict for certain nonlinearities F. On the contrary, under a KPP condition on F, we show the equality cF+=cF⁎. Finally, we provide an example in which cF+ is negative.