{
  "abstract": "We study the Neumann problem $u_t-\\varepsilonΔu=e^u-1-au\\ (a>1)$ on a smooth bounded domain $Ω\\subset\\mathbb{R}^2$. For the spatially homogeneous problem, $0$ is stable, the positive equilibrium $ξ_a$ is unstable, and solutions starting above $ξ_a$ blow up in finite time. Although finite-time blow-up persists at every diffusivity, we show that sufficiently large diffusion recovers this scalar trichotomy uniformly on every bounded $H^1$ ball, and that blow-up occurs precisely when the spatial mean crosses $ξ_a$. For initial data with $\\|u_0\\|_{H^1}\\le R$ and spatial mean at most $ξ_a-δ$, let $\\varepsilon_{\\mathrm{unif}}(R,δ)$ denote the uniform diffusion threshold above which all such solutions are global and converge to $0$. We prove $\\log \\varepsilon_{\\mathrm{unif}}(R,δ)=R^2/(8π)+O(\\log R)$ as $R\\to\\infty$. The domain-independent coefficient $1/(8π)$ arises from the sharp mean-zero Moser--Trudinger inequality. A matching lower bound is obtained from boundary-concentrating Moser profiles via a localized Kaplan argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28061",
  "author_total": 1,
  "authors": [
    {
      "name": "Juneyoung Seo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2s4p3ttvhtxhs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28061",
  "fulltext_url": null,
  "id": "2s4p3ttvhtxhs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28061",
  "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": "W7204813212",
  "pdf_url": "https://arxiv.org/pdf/2608.28061",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2s4p3ttvhtxhs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28061",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28061"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Large-diffusion dynamics for a planar Neumann heat equation with exponential nonlinearity",
  "title_text": "Large-diffusion dynamics for a planar Neumann heat equation with exponential nonlinearity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/2s4p3ttvhtxhs",
  "venue": "arXiv",
  "versions": []
}