Nonlocal Fisher--KPP propagation with heavy tails

We study a nonlocal Fisher–KPP equation on the real line with compactly supported initial data. The reaction satisfies $f(s)\le f^{\mathrm{\prime }}(0)s$ for $0\le s\le 1$, with a power bound on $f^{\mathrm{\prime }}(0)s-f(s)$ near zero. For continuous even probability kernels $J$ comparable to $\mathrm{\mid }x{\mathrm{\mid }}^{-\gamma }$, $\gamma >1$, we prove a global two-sided estimate and locate every fixed level set with a bounded error in $\log \mathrm{\mid }x\mathrm{\mid }$. The propagation scale is $\exp (f^{\mathrm{\prime }}(0)t\mathrm{/}\gamma )$ for every $\gamma >1$. 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 $\mathrm{\mid }x{\mathrm{\mid }}^{\gamma }J(x)$ has a positive limit.