{
  "abstract": "We consider the Neumann initial boundary value problem associated to the chemotaxis system \\begin{align}\\label{prob:abstract}\\tag{$\\star$} \\begin{cases} u_t = \\big((u+1)^{m-1} u_x - u(u+1)^m v_x\\big)_x & \\text{in $(0, 1) \\times (0, \\infty)$}, \\\\ v_t = v_{xx} - v + u, &\\text{in $(0, 1) \\times (0, \\infty)$}, \\end{cases} \\end{align} where $m \\in \\mathbb R$ is a given parameter. The relation between diffusion and taxis sensitivity is critical since the ratio $u(u+1)^m/(u+1)^{m-1}$ grows like $u^{2/n}$ for large $u$ with $n = \\dim((0, 1)) = 1$. Nonetheless, we show that there is no critical mass phenomenon if $m \\le -1$; that is, in that case all solutions emanating from suitably regular initial data are globally bounded. For certain parabolic-elliptic simplifications of \\eqref{prob:abstract}, we obtain the same conclusion for all $m \\in (-\\infty, -1] \\cup (0, \\infty)$ and even for all $m \\in \\mathbb R$ if the initial datum is additionally assumed to be monotone. This stands in contrast to critical mass phenomena known to occur for critical quasilinear Keller-Segel systems considered in higher-dimensional domains. Accordingly, we make use of several special features of the one-dimensional setting such as the boundedness of the energy functional from below, the embedding $W^{1, n} \\hookrightarrow L^\\infty$, and the fact that the mass accumulation function solves a spatially non-degenerate parabolic equation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.18006",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinru Cao"
    },
    {
      "name": "Mario Fuest"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7v2t4a1z7xnqf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.18006",
  "fulltext_url": null,
  "id": "7v2t4a1z7xnqf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.18006",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2606.18006",
  "pmcid": null,
  "pmid": null,
  "published": "2026-06-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7v2t4a1z7xnqf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2606.18006",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.18006"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Absence of critical mass phenomena in one-dimensional critical quasilinear Keller-Segel systems",
  "title_text": "Absence of critical mass phenomena in one-dimensional critical quasilinear Keller-Segel systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/7v2t4a1z7xnqf",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1016/j.jde.2026.114753",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}