{
  "abstract": "We consider the radially symmetric version of the reaction-diffusion equation $u_t-dΔu=f(u)$ with a monostable nonlinearity $f$, viewed as a model for the spreading of a species with population range $r<h(t)$ and density $u(t,r)$ ($r=|x|$), where the free boundary $r=h(t)$ is governed by $u(t,h(t))=δ>0$ and $h'(t)=-d u_r(t,h(t))/δ$. For the one-dimensional case ($N=1$), Du \\cite{DN} proved that when $δ\\in(0,1)$, spreading occurs: $u\\to1$ locally uniformly in $\\mathbb{R}$, $h(t)\\to\\infty$, and $\\lim_{t\\to\\infty}[h(t)-c_*t]=\\tilde{h}\\in\\mathbb{R}$ with no logarithmic shift. In the present paper we consider $N\\ge2$ and establish a complete trichotomy: spreading for $δ\\in(0,1)$; transition for $δ=1$, where $u\\to1$ uniformly on $[0,h(t)]$ and $h(t)\\to h_\\infty\\in(0,\\infty)$; and vanishing for $δ>1$, where $h(t)\\to0$ and $u\\toδ$ uniformly on $[0,h(t)]$. For the spreading regime, by constructing sharp upper and lower solutions, we prove that the solution converges globally to the semi-wave profile and reveal a logarithmic shift of the form $ \\lim_{t\\to\\infty}\\big[h(t)-c_*t+c_N(δ)\\log t\\big]=\\hat{h}\\in\\mathbb{R}$, with the coefficient $c_N(δ)>0$ satisfying $ \\lim_{δ\\to0}c_N(δ)=d(N-1)/c_0$, where $d(N-1)/c_0$ is the shift coefficient for the high-dimensional radial pushed-case Cauchy problem. These results reveal the connection to the spreading behavior modeled by the corresponding Cauchy problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.25672",
  "author_total": 4,
  "authors": [
    {
      "name": "Hongkai Cao"
    },
    {
      "name": "Jingyi Cui"
    },
    {
      "name": "Chengzhe Tang"
    },
    {
      "name": "Xiaoyan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8zfzd3jf6vz5d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.25672",
  "fulltext_url": null,
  "id": "8zfzd3jf6vz5d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.25672",
  "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",
      "2739|Mathematical and Theoretical Epidemiology and Ecology Models"
    ]
  ],
  "openalex": "W7204488741",
  "pdf_url": "https://arxiv.org/pdf/2608.25672",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8zfzd3jf6vz5d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2608.25672",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.25672"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The high dimensional monostable reaction-diffusion equation with free boundary and radial symmetry",
  "title_text": "The high dimensional monostable reaction-diffusion equation with free boundary and radial symmetry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/8zfzd3jf6vz5d",
  "venue": "arXiv",
  "versions": []
}