{
  "abstract": "We study the master equation $ (\\partial_t -Δ)^{s} u(x,t) = R(|x|)f(u(x,t))\\quad\\mbox{in}\\ \\mathbb{R}^n\\times\\mathbb{R}, $ where $s\\in(0,1)$, $R$ is positive and strictly decreasing in the radial variable, and $f$ is positive, locally Lipschitz, and satisfies $f'(0)<0$. Existing symmetry results for this operator require $f(0)=0$ with $f'(0)\\ge 0$ (or, more generally, that $f$ be non-decreasing near the origin), because the sign of the moving-plane comparison inequality is then controlled by the nonlocal diffusion term. When $f'(0)<0$, the linear term in the comparison inequality appears at the same order as the nonlocal diffusion but with the opposite sign, destroying the comparison principle underlying the standard cut-off perturbation argument. We prove that, despite this obstruction, every positive bounded classical solution with uniform spatial decay is radially symmetric and strictly radially decreasing in $x$ for each $t$. The proof reconciles the direct method of moving planes with this adverse sign by exploiting the interplay between the spatial decay of $u$ and the strict radial monotonicity of $R$. This appears to be the first symmetry result for the master operator that accommodates a local damping mechanism at the origin.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20476",
  "author_total": 1,
  "authors": [
    {
      "name": "Yahui Niu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8c01kcb7dd3r5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20476",
  "fulltext_url": null,
  "id": "8c01kcb7dd3r5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20476",
  "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": "W7213610720",
  "pdf_url": "https://arxiv.org/pdf/2609.20476",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8c01kcb7dd3r5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20476",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20476"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Radial Symmetry and Strict Radial Decrease for Master Equations with Decreasing Radial Potentials",
  "title_text": "Radial Symmetry and Strict Radial Decrease for Master Equations with Decreasing Radial Potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/8c01kcb7dd3r5",
  "venue": "arXiv",
  "versions": []
}