{
  "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 $u_t+(-Δ)^su=f(u)$ 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 $s\\in(0,1/2)$,}\\\\[2mm] \\lim_{t\\to\\infty}\\frac{h(t)}{t\\ln t} =\\lim_{t\\to\\infty}\\frac{-g(t)}{t\\ln t} =μ& \\mbox{ if $s=1/2$,}\\\\[2mm] \\lim_{t\\to\\infty}\\frac{h(t)}{t} =\\lim_{t\\to\\infty}\\frac{-g(t)}{t} =c_0(s) & \\mbox{ if $s\\in(1/2,1)$}, \\end{cases} \\] where $c_0(s)\\in (0,\\infty)$ 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 $s\\in (0, 1/2]$, and from the semi-wave profile when $s\\in (1/2, 1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12838",
  "author_total": 3,
  "authors": [
    {
      "name": "Huyuan Chen"
    },
    {
      "name": "Yihong Du"
    },
    {
      "name": "Wenjie Ni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/594ptqnqw6cjc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12838",
  "fulltext_url": null,
  "id": "594ptqnqw6cjc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12838",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7221505385",
  "pdf_url": "https://arxiv.org/pdf/2610.12838",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/594ptqnqw6cjc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12838",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12838"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Precise propagation rates of the fractional Fisher-KPP equation with free boundaries",
  "title_text": "Precise propagation rates of the fractional Fisher-KPP equation with free boundaries",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/594ptqnqw6cjc",
  "venue": "arXiv",
  "versions": []
}