Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices
Abstract
Let P denote normalized nearest-neighbor averaging on ℤd and set A=I−P. We consider ∂tu+Asu=f(u),t>0, x∈ℤd, 0<s<1. For a Fisher-KPP reaction with a=f'(0)>0, compactly supported data have logarithmic propagation rate a/(d+2s). In the logistic case, each fixed level set lies between two constant multiples of exp(at/(d+2s)); this localization remains valid for data with the critical algebraic tail. Nondecreasing front-like data in one dimension instead propagate at rate a/(2s), and the resulting acceleration rules out planar traveling fronts of finite speed in every dimension. We also analyze the singular limit s↑1. If τs=−log(1−s), then throughout at<τs compactly supported solutions follow the Wulff shape of the local nearest-neighbor equation. After a constant multiple of τs, their propagation is exponential. Half-space data exhibit the corresponding transition from the directional local speed to the exponent a/(2s). The analysis rests on sharp bounds for the subordinated kernel, algebraic barriers, a compactness--Liouville argument, and a decomposition of the fractional walk into its nearest-neighbor and rare long-jump parts.
Related
- Propagation Dynamics for Multidimensional Nonlocal Diffusion Equations: A General Freidlin-Gärtner Formula
- Nonlocal Boundary Conditions for Truncated-Fractional Differential Equations Posed on Bounded Domains
- Nonlocal Fisher--KPP propagation with heavy tails
- Fractional gradients and nonlocal PDEs in variable exponent spaces
- Symmetry of solutions to nonlocal Neumann problems with the fractional Laplacian in half spaces
BibTeX
@misc{hu2026sharp,
title = {{Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices}},
author = {Yuanyang Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05124},
eprint = {2609.05124},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05124},
}
RIS
TY - UNPB
TI - Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices
AU - Yuanyang Hu
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05124
UR - https://arxiv.org/abs/2609.05124
AB - Let P denote normalized nearest-neighbor averaging on ℤ^d and set A=I−P. We consider ∂_tu+A^su=f(u),t>0, x∈ℤ^d, 0<s<1. For a Fisher-KPP reaction with a=f'(0)>0, compactly supported data have logarithmic propagation rate a/(d+2s). In the logistic case, each fixed level set lies between two constant multiples of exp(at/(d+2s)); this localization remains valid for data with the critical algebraic tail. Nondecreasing front-like data in one dimension instead propagate at rate a/(2s), and the resulting acceleration rules out planar traveling fronts of finite speed in every dimension. We also analyze the singular limit s↑1. If τ_s=−log(1−s), then throughout at<τ_s compactly supported solutions follow the Wulff shape of the local nearest-neighbor equation. After a constant multiple of τ_s, their propagation is exponential. Half-space data exhibit the corresponding transition from the directional local speed to the exponent a/(2s). The analysis rests on sharp bounds for the subordinated kernel, algebraic barriers, a compactness--Liouville argument, and a decomposition of the fractional walk into its nearest-neighbor and rare long-jump parts.
ER -