Virtual linearity for KPP reaction-diffusion equations

We show that long time solution dynamic for general reaction-advection-diffusion equations with KPP reactions is virtually linear in the following sense. Its leading order depends on the non-linear reaction only through its linearization at $u=0$, and it can also be recovered for general initial data by instead solving the PDE for restrictions of the initial condition to unit cubes on $R^d$ (the latter means that non-linear interaction of these restricted solutions has only lower order effects on the overall solution dynamic). The result holds under a uniform bound on the advection coefficient, which we show to be sharp. We also extend it to models with non-local diffusion and KPP reactions.