Precise propagation rates of the fractional Fisher-KPP equation with free boundaries
Abstract
In this paper, we determine the propagation rates of the Fisher-KPP equation with free boundaries and fractional diffusion (−Δ)s (0<s<1), which was considered in \cite{CDN26} as a model for species spreading, where the density function u(t,x) satisfies the fractional Fisher-KPP equation ut+(−Δ)su=f(u) over a one-dimensional spatial domain $(g(t), h(t))$, representing the expanding population range, with x=g(t) and x=h(t) the free boundaries. It was shown in \cite{CDN26} that the model is well-posed, and its long-time dynamics exhibits a spreading-vanishing dichotomy. We show in this paper that, when spreading occurs, \[ \begin{cases} \lim_{t\to\infty}\frac{h(t)}{t^{1/(2s)}} =\lim_{t\to\infty}\frac{-g(t)}{t^{1/(2s)}} =\left(\frac{2^{1-2s}μ}{1-2s}\right)^{1/(2s)} & \mbox{ if s∈(0,1/2),}\2mm] limt→∞h(t)/(tln t) =limt→∞(−g(t))/(tln t) =μ& if $s=1/2$, [2mm] limt→∞h(t)/t =limt→∞(−g(t))/t =c0(s) & if $s∈(1/2,1)$, where c0(s)∈ (0,∞) is uniquely determined by the associated semi-wave problem. The level sets of u(t,x) propagate with the same rates as the free boundaries. The proof uses delicate upper and lower solutions constructed from a smooth profile function in the case s∈ (0, 1/2], and from the semi-wave profile when s∈ (1/2, 1).
Related
- The fractional Fisher-KPP equation with free boundaries
- Sharp Propagation and the Local-to-Nonlocal Transition for Fractional Fisher-KPP Equations on Integer Lattices
- Nonlocal Fisher--KPP propagation with heavy tails
- Directional delays and singular strong-barrier limits for Fisher--KPP propagation across a semipermeable interface
- Slow Beats Fast at the Kesten-Stigum Threshold: Minimax, Fisher-Information and Belief-Propagation Characterizations of the Information-Computation Gap in Sparse Stochastic Block Models
BibTeX
@misc{chen2026precise,
title = {{Precise propagation rates of the fractional Fisher-KPP equation with free boundaries}},
author = {Huyuan Chen and Yihong Du and Wenjie Ni},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12838},
eprint = {2610.12838},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12838},
}
RIS
TY - UNPB
TI - Precise propagation rates of the fractional Fisher-KPP equation with free boundaries
AU - Huyuan Chen
AU - Yihong Du
AU - Wenjie Ni
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12838
UR - https://arxiv.org/abs/2610.12838
AB - In this paper, we determine the propagation rates of the Fisher-KPP equation with free boundaries and fractional diffusion (−Δ)^s (0<s<1), which was considered in \cite{CDN26} as a model for species spreading, where the density function u(t,x) satisfies the fractional Fisher-KPP equation u_t+(−Δ)^su=f(u) over a one-dimensional spatial domain $(g(t), h(t))$, representing the expanding population range, with x=g(t) and x=h(t) the free boundaries. It was shown in \cite{CDN26} that the model is well-posed, and its long-time dynamics exhibits a spreading-vanishing dichotomy. We show in this paper that, when spreading occurs, \[ \begin{cases} \lim_{t\to\infty}\frac{h(t)}{t^{1/(2s)}} =\lim_{t\to\infty}\frac{-g(t)}{t^{1/(2s)}} =\left(\frac{2^{1-2s}μ}{1-2s}\right)^{1/(2s)} & \mbox{ if s∈(0,1/2),}\2mm] lim_t→∞h(t)/(tln t) =lim_t→∞(−g(t))/(tln t) =μ& if $s=1/2$, [2mm] lim_t→∞h(t)/t =lim_t→∞(−g(t))/t =c_0(s) & if $s∈(1/2,1)$, where c_0(s)∈ (0,∞) is uniquely determined by the associated semi-wave problem. The level sets of u(t,x) propagate with the same rates as the free boundaries. The proof uses delicate upper and lower solutions constructed from a smooth profile function in the case s∈ (0, 1/2], and from the semi-wave profile when s∈ (1/2, 1).
ER -