Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Hu, Yuanyang
Institution: Henan University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223342382