{
  "abstract": "We consider on a Riemannian manifold $M$ the Leibenson equation \\begin{equation*}\\label{eqabs}\\partial _{t}u=Δ_{p}u^{q},\\end{equation*} where $p>1$ and $q>0$. When $q(p-1)\\geq 1$, we prove conservation of mass for solutions of Leibenson's equation assuming only the volume bound $V(x_0, r)\\leq \\exp\\left(C r^{\\frac{p}{p-1}}\\right)$ for some $x_{0}\\in M$ and all large enough $r>0$. When $q(p-1)< 1$, we prove this property assuming $V(x_0, r)\\leq Cr^{N}$ and $p>N[1-q(p-1)]$, which matches the threshold in $\\mathbb{R}^{n}$ with $N=n$. We also show that solutions on the hyperbolic space $\\mathbb{H}^{n}$ have a finite extinction time in the case $q(p-1)< 1$, which implies that the conservation of mass property does not hold. Using the conservation of mass result in the case $q(p-1)=1$, we also prove a $L^{p-1}$- Liouville property, which partially answers a conjecture stated by I. Holopainen \\cite{holopainen2000sharp}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02732",
  "author_total": 1,
  "authors": [
    {
      "name": "Philipp Sürig"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0rdeed1g31b2q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02732",
  "fulltext_url": null,
  "id": "0rdeed1g31b2q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02732",
  "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"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7207603434",
  "pdf_url": "https://arxiv.org/pdf/2609.02732",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0rdeed1g31b2q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02732",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02732"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Conservation of mass for solutions of Leibenson's equation on Riemannian Manifolds",
  "title_text": "Conservation of mass for solutions of Leibenson's equation on Riemannian Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/0rdeed1g31b2q",
  "venue": "arXiv",
  "versions": []
}