We study the reaction-diffusion equation $u_t=\Delta u + f(x, u)$ with $f(0, u) = f(1, u) = 0$. Then $u = 1$ and $u = 0$ are respectively stable and unstable equilibriums. The equation is frequently used in modeling invasions of one equilibrium state of a physical process by another. For instance, forest fires spreading.
I am going to focus on ignition reactions. I will discuss the previous results including the existence of fronts, bounded transition zone and homogenization. I will introduce the ongoing work (with my mentor Andrej Zlatos) on stochastic homogenization and several open questions along the way.