{
  "abstract": "In this paper, we study the following Gierer--Meinhardt system \\begin{equation*} \\begin{cases} u_t=d_1Δu-a_1u+u^p/v^q+δ_1(x),\\quad x\\in Ω, t>0,\\\\ v_t=d_2Δv-a_2v+u^r/v^s+δ_2(x), \\quad x\\in Ω, t>0,\\\\ \\partial_νu=\\partial_νv=0,\\qquad\\qquad \\qquad\\qquad\\quad x\\in \\partial Ω, t>0,\\\\ u(x,0)=u_0(x),\\quad v(x,0)=v_0(x),\\quad x\\in Ω, \\end{cases} \\end{equation*} where $Ω\\subset \\mathbb{R}^N$ is a bounded domain, $ν$ is the outer norm vector with respect to the smooth boundary $\\partial Ω$, $d_1,d_2, p,q,r>0$, $s>-1$, $p-1<r$, $qr=(p-1)(s+1)$, $δ_1(x)$, $δ_2(x)$, $u_0(x)$ and $v_0(x)$ are continuous functions. For every pair $d_1,d_2>0$, we respectively establish the global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent $qr=(p-1)(s+1)$ under certain conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01968",
  "author_total": 2,
  "authors": [
    {
      "name": "Xianfa Song"
    },
    {
      "name": "Zeen Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jatmwyr1tybe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01968",
  "fulltext_url": null,
  "id": "6jatmwyr1tybe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01968",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7216035098",
  "pdf_url": "https://arxiv.org/pdf/2610.01968",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jatmwyr1tybe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01968",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01968"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent $qr=(p-1)(s+1)$",
  "title_text": "Global existence, uniform boundedness and asymptotic behavior for Gierer--Meinhardt system with critical exponent qr=(p−1)(s+1)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/6jatmwyr1tybe",
  "venue": "arXiv",
  "versions": []
}