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Nonlocal Fisher--KPP propagation with heavy tails
Abstract
We study a nonlocal Fisher--KPP equation on the real line with compactly supported initial data. The reaction satisfies f(s)≤ f′(0)s for 0≤ s≤1, with a power bound on f′(0)s−f(s) near zero. For continuous even probability kernels J comparable to |x|−γ, γ>1, we prove a global two-sided estimate and locate every fixed level set with a bounded error in log|x|. The propagation scale is exp(f'(0)t/γ) for every γ>1. This differs sharply from the free-boundary propagation studied by Du and Ni \cite{DuNiJEMS}. For polynomial kernel tails, spreading in their model has linear, algebraic or tlog t growth. We also obtain arrival-time and weighted-integral estimates and identify a spatial tail coefficient when |x|γJ(x) has a positive limit.
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BibTeX
@misc{hu2026nonlocal,
title = {{Nonlocal Fisher--KPP propagation with heavy tails}},
author = {Yuanyang Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30986},
eprint = {2610.30986},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30986},
}
RIS
TY - UNPB
TI - Nonlocal Fisher--KPP propagation with heavy tails
AU - Yuanyang Hu
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30986
UR - https://arxiv.org/abs/2610.30986
AB - We study a nonlocal Fisher--KPP equation on the real line with compactly supported initial data. The reaction satisfies f(s)≤ f^′(0)s for 0≤ s≤1, with a power bound on f^′(0)s−f(s) near zero. For continuous even probability kernels J comparable to |x|^−γ, γ>1, we prove a global two-sided estimate and locate every fixed level set with a bounded error in log|x|. The propagation scale is exp(f'(0)t/γ) for every γ>1. This differs sharply from the free-boundary propagation studied by Du and Ni \cite{DuNiJEMS}. For polynomial kernel tails, spreading in their model has linear, algebraic or tlog t growth. We also obtain arrival-time and weighted-integral estimates and identify a spatial tail coefficient when |x|^γJ(x) has a positive limit.
ER -