KPP equation with random, time-space depedent, rate
We consider the reaction-diffusion equation (du)(dt) = Du + k u(u-1) for D the Laplacian, t > 0, x ? Rd and u(0, .) (close to) the indicator function of the unit ball. The rate of reaction k > 0 is of the form k=k(t, x; w), it is stationary and ergodic, and, in fact, has a specific form. We prove existence of a limiting speed c, meaning that the solution at time t looks like the indicator function of a ball of radius ct + o(t). We also study the dependence of c on the fluctuations of k via a "disorder" parameter; a phase transition takes place, similar to the localization transition for polymers.