Nonlocal Fisher--KPP propagation with heavy tails

Yuanyang Hu

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arXiv · 2610.30986

Nonlocal Fisher--KPP propagation with heavy tails

Yuanyang Hu

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Abstract

We study a nonlocal Fisher--KPP equation on the real line with compactly supported initial data. The reaction satisfies $f(s)\le f^{\prime}(0)s$ for $0\le s\le1$, with a power bound on $f^{\prime}(0)s-f(s)$ 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 $t\log t$ 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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Yuanyang Hu. 2026-10-22. Nonlocal Fisher--KPP propagation with heavy tails. https://arxiv.org/abs/2610.30986

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