Fast solutions and asymptotic behavior in a reaction–diffusion equation

In this paper we deal with the existence of traveling waves solutions (t.w.s.) for the reaction–diffusion equationut=uxx+f(u),ut=uxx+f(u), in a very general setting of reaction terms f with two distinguished stationary states, say 0 and 1. We link the existence of some type of solutions of the second order ODEu″+cu′+f(u)=0,u″+cu′+f(u)=0, with the existence of fast t.w.s. By defining fast solutions for this ODE, we find a value cM>0cM>0 related to the existence of global fast solutions and determine cMcM through a variational formula. Our results allow us particularly to show that any solution u(x,t)u(x,t) of the reaction–diffusion equation with compactly supported initial data and 0≤u(x,0)≤1,x∈R, satisfieslimt→+∞⁡u(x+ct,t)=0, uniformly on compact sets, for all c  , |c|>cM|c|>cM.Finally, we connect cMcM with the minimum speed of propagation of all t.w.s. c⁎c⁎.