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}

For spreading solutions, it also remains to determine whether $W_s(t)$ tends to infinity and whether its growth rate can be estimated.

— An Upper Bound for the Mean Speed of Transition Fronts and Unbounded Front Widths for Fisher KPP Equations in Almost Periodic Media  (2609.30760 - Liang et al., 25 Sep 2026) in Section 7, “Further remarks” (Section \ref{sec:scope})

The main mechanism that differentiates the mixed operator from the purely local or purely fractional laplacian is the combination of different scales, which rises the interesting question of whether one of the scales prevail or both coexist in the end.

— Interior Regularity of Mixed Local-Nonlocal Parabolic Semilinear Equations  (2610.06537 - Junquera et al., 5 Oct 2026) in Section 1, Introduction