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

TL;DR

This paper examines the boundedness of transition fronts in the randomized Fisher-KPP and parabolic Anderson models, revealing that boundedness holds in the latter but not in the general former, highlighting differences in their long-term behaviors.

Contribution

It demonstrates that the uniform boundedness of transition fronts, known for certain Fisher-KPP cases, does not extend to the general randomized Fisher-KPP equation, but does hold for the parabolic Anderson model.

Findings

Bounded transition fronts do not hold for the general randomized Fisher-KPP equation.

Bounded transition fronts are confirmed for the parabolic Anderson model.

Highlights differences in front behavior between models under randomness.

Abstract

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.