{
  "abstract": "We establish local Aronson-Bénilan estimates for positive solutions to the porous medium and fast diffusion equations on complete noncompact Riemannian manifolds with a small scale-invariant integral bound on the negative part of the Ricci curvature in $L^p$, where $p>n/2$. No uniform lower bound on the volume of unit balls is assumed, and hence the results apply to manifolds that may collapse. For the fast diffusion equation, we obtain the estimate in the range $1-2/n<m<1$. The auxiliary function method, which was designed for the heat equation under integral curvature conditions, is not directly applicable to those two nonlinear equations. The pressure operator in both cases contains a solution-dependent coefficient, which produces a residual term involving $\\partial_t J$ in the evolution inequality. Its treatment requires a positive lower bound for $J$, and moreover, control of $-\\partial_t\\log J$. We obtain these bounds by linearizing the auxiliary equation, applying Duhamel's formula, and combining local Dirichlet heat kernel estimates with a singular Volterra inequality. Additionally, in the fast diffusion case, the negativity of $v$ changes the sign structure of the evolution inequality, so a separate coercive estimate is required to close the argument. Local Aronson-Bénilan estimates follow for both diffusion regimes and integrating them along space-time curves gives the corresponding Harnack inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.09962",
  "author_total": 2,
  "authors": [
    {
      "name": "Jinpeng Miao"
    },
    {
      "name": "Bin Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a2xv5sdmpg1ta.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.09962",
  "fulltext_url": null,
  "id": "a2xv5sdmpg1ta",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.09962",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.09962",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a2xv5sdmpg1ta.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:33Z",
      "native_id": "2611.09962",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.09962"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Local Aronson-Bénilan estimates for the porous medium and fast diffusion equations on collapsing manifolds",
  "title_text": "Local Aronson-Bénilan estimates for the porous medium and fast diffusion equations on collapsing manifolds",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:33Z",
  "url": "https://research.draconitic.com/paper/a2xv5sdmpg1ta",
  "venue": "arXiv",
  "versions": []
}