Abstract:

In this seminar, I will first review some known results about the propagation phenomena of spatially periodic reaction-diffusion equations with bistable nonlinearities. Then I will discuss the existence and some qualitative properties of transition fronts of such equations. The notion of transition fronts, introduced by Berestycki-Hamel (2012), generalizes the standard notion of traveling waves. In this talk, under the a priori existence of a non-stationary traveling wave, I will show that transition fronts are reduced to this traveling wave, and they are unique up to shift in time. On the other hand, when no non-stationary traveling waves exist, I will provide an example of the existence of a new type of transition fronts which are not standard traveling waves. This talk is based on a joint work with François Hamel and Xiao-Qiang Zhao (CVPDE, 2017).