Propagation phenomena for time heterogeneous KPP reaction-diffusion equations

We investigate in this paper propagation phenomena for the heterogeneous reaction-diffusion equation. ∂tu-δu=f(t,u), x∈RN, t∈R, where f=f(t, u) is a KPP monostable nonlinearity which depends in a general way on t∈R. A typical f which satisfies our hypotheses is f(t, u) = μ(t) u(1 - u), with μ∈L∞(R) such that essinft∈Rμ(t)>0. We first prove the existence of generalized transition waves (recently defined in Berestycki and Hamel (2007) [4]) for a given class of speeds. As an application of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t∈R. Lastly, we prove some spreading properties for the solution of the Cauchy problem. © 2012 Elsevier Masson SAS.