HENRI BERESTYCKI, EHESS, Paris
Reaction-diffusion equations and propagation phenomena

Reaction diffusion equations and systems play a central role in reactive flows, chemical waves, combustion theory, systems undergoing phase changes and in modelling a variety of biological and ecological systems, in particular in describing different kinds of invasions. The important features of these equations are the propagation of fronts and the spreading or quenching of confined initial data. In this series of lectures, I will first review the classical results in the framework of the classical homogeneous reaction-diffusion equations. I will describe some of the models and recall the basic results of Kolmogorov, Petrovsky and Piskunov, Aronson and Weinberger and of Fife and McLeod. The main thrust of the course will then be to present the most recent developments regarding non homogeneous media. First, I will show several cases where there still are travelling waves but which are non planar. Then, I will discuss in detail the case of periodic media. There, the notion of pulsating travelling wave extends that of travelling wave. Also, one can derive formulas for the asymptotic speed of spreading. I will present properties dealing with the qualitative influence of various factors on the speed of propagation. Lastly, I will present some results on propagation and generalized fronts in very general non homogeneous media. These lectures build on several recent joint works with François Hamel, Nikolai Nadirashvili and Hiroshi Matano.