Level-set geometry in the intermediate transition window

Determine the level-set geometry of solutions to the fractional Fisher–KPP equation on the integer lattice in the intermediate regime between the early local-propagation range $at<\tau_s$ and the late exponential-propagation range $t\geq C\tau_s$ as $s\uparrow1$, including the matching of the local large-deviation profile with the emerging algebraic tail and the subsequent approach to the asymptotic expanding-barrier rate.

Background

The paper establishes two asymptotic regimes for the fractional Fisher–KPP equation on the integer lattice as the fractional order approaches one. Before the transition, when at<τsat<\tau_s with τs=−log⁡(1−s)\tau_s=-\log(1-s), localized solutions follow the Wulff shape of the nearest-neighbor local equation, while after a constant multiple of τs\tau_s they exhibit exponential propagation governed by the fractional tail.

The authors explicitly state that their proof does not determine the level-set geometry in the interval between these regimes. Resolving this gap would require identifying how the local large-deviation profile matches the increasingly significant algebraic long-jump tail near at=τsat=\tau_s, and then tracking the resulting level sets until the uniform expanding barriers attain their asymptotic propagation rate.

References

We do not obtain a bounded-width localization of the full transition set ${x\ind:\varepsilon<u(t,x)<1-\varepsilon}$ that is uniform over all directions, for $0<\varepsilon<1/2$. Such a conclusion requires lower bounds uniform in $e$, with a specified lattice approximation to each ray.

— Bramson correction and convergence to the critical wave for Fisher-KPP equations on $\mathbb Z^d$  (2609.10143 - Hu, 9 Sep 2026) in Section 7, Section "Weighted estimates and other directions"

The proof separates the ranges $at<\tau_s$ and $t\geq C\tau_s$. It does not determine the level-set geometry between these two time scales. Near $at=\tau_s$, the amplified coefficient of the long-jump tail is no longer small in operator norm, although its spatial decay is still relevant on the ballistic scale. Determining the level-set geometry in the remaining interval would require matching the local large-deviation profile with the emerging algebraic tail and then following the latter until the uniform expanding barrier reaches its asymptotic rate.

— Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices  (2609.05124 - Hu, 4 Sep 2026) in Remark \ref{tr:rem:intermediate-window}, Section \ref{tr:sec:level-set-formulation-and-scale-separation}