Article ID Journal Published Year Pages File Type
4610020 Journal of Differential Equations 2015 34 Pages PDF
Abstract

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⁎.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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