{
  "abstract": "In this paper, we prove the Liouville-type theorem for equation $u_t-Δu=u^p+M|\\nabla u|^q$ via the integral Bernstein method, where $q=2p/(p+1)$ is critical and $M<0$. As an application, we can establish a universal priori estimate for this equation by using the doubling lemma. These results generalize the previous work of Liang and Zhang. We can also prove the Liouville-type theorem for equation $u_t-Δu=u^p+M|\\nabla u|^2$, which corresponds to the supercritical case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28481",
  "author_total": 1,
  "authors": [
    {
      "name": "Xue Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/381m6vy32v8gc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28481",
  "fulltext_url": null,
  "id": "381m6vy32v8gc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28481",
  "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": "W7223123048",
  "pdf_url": "https://arxiv.org/pdf/2610.28481",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/381m6vy32v8gc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28481",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28481"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term",
  "title_text": "Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/381m6vy32v8gc",
  "venue": "arXiv",
  "versions": []
}