  We investigate in this paper propagation phenomena for the heterogeneous
reaction-diffusion equation $\partial_t u -\Delta u = f(t,u)$, $x\in R^N$,
$t\in\R$, where f=f(t,u) is a KPP monostable nonlinearity which depends in a
general way on t. A typical f which satisfies our hypotheses is f(t,u)=m(t)
u(1-u), with m bounded and having positive infimum. We first prove the
existence of generalized transition waves (recently defined by Berestycki and
Hamel, Shen) 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. Lastly, we prove some spreading properties
for the solution of the Cauchy problem.

Analysis of PDEs