  We investigate the uniform boundedness of the fronts of the solutions to the
randomized Fisher-KPP equation and to its linearization, the parabolic Anderson
model. It has been known that for the standard (i.e. deterministic) Fisher-KPP
equation, as well as for the special case of a randomized Fisher-KPP equation
with so-called ignition type nonlinearity, one has a uniformly bounded (in
time) transition front. Here, we show that this property of having a uniformly
bounded transition front fails to hold for the general randomized Fisher-KPP
equation. Nevertheless, we establish that this property does hold true for the
parabolic Anderson model.
