(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.

Automated summary

This paper investigates the uniform boundedness property of the fronts of solutions for the randomized Fisher-KPP equation and its linearization model, the parabolic Anderson model. It is known that this property holds for the deterministic Fisher-KPP equation and a specific case of the randomized Fisher-KPP equation with ignition type nonlinearity. However, we find that this property fails to hold for the general randomized Fisher-KPP equation. In contrast, we establish this property for the parabolic Anderson model under certain assumptions.