HOW TRAVELLING WAVES ATTRACT THE SOLUTIONS OF KPP-TYPE EQUATIONS

We consider in this paper a general reaction-diffusion equation of the KPP (Kohnogorov, Petrovskii, Piskunov) type, posed on an infinite cylinder. Such a model will have a family of pulsating waves of constant speed, larger than a critical speed c(*). The family of all supercritical waves attracts a large class of initial data, and we try to understand how. We describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity: the solution will converge to a whole set of translates of the same wave, and we identify the convergence dynamics as that of an effective drift, around which an effective diffusion process occurs. In several nontrivial particular cases, we are able to describe the dynamics by an effective equation.