Pulsating fronts for nonlocal dispersion and KPP nonlinearity

In this paper we are interested in propagation phenomena for nonlocal reaction–diffusion equations of the type:∂u∂t=J⁎u−u+f(x,u)t∈R,x∈RN, where J is a probability density and f is a KPP nonlinearity periodic in the x variables. Under suitable assumptions we establish the existence of pulsating fronts describing the invasion of the 0 state by a heterogeneous state. We also give a variational characterization of the minimal speed of such pulsating fronts and exponential bounds on the asymptotic behavior of the solution.