  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 $\bbR^d$ (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.
