Existence of transition fronts at or above the upper speed endpoint

Determine whether Fisher–KPP equations in one-dimensional almost periodic media with finite upper speed endpoint \(\overline w=\lambda_1/L(\lambda_1)\) admit transition fronts with global mean speed \(w\geq\overline w\), particularly at the endpoint \(w=\overline w\).

Background

For the continuous and lattice Fisher–KPP equations in almost periodic media, prior results establish existence of transition fronts for every global mean speed in $[w^*,\overline w)$ and nonexistence below $w^*$. The paper proves the upper bound $w\leq\lambda_1/L(\lambda_1)=\overline w$ when $L(\lambda_1)>0$. Consequently, the existence question at the upper endpoint, and above it in the formulation of the preceding results, is not resolved by the known theory. The paper’s strict upper-bound argument does not settle the endpoint case.