(Un-)bounded transition fronts for the parabolic Anderson model and the randomized F-KPP equation

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, the transition front is uniformly bounded (in time). Here, we show that this property of having a uniformly bounded transition front fails to hold for the general randomized Fisher-KPP equation. In contrast, for the parabolic Anderson model we do establish this property under some assumptions.