{
  "abstract": "We investigate the free boundary propagation of a non-linear degenerate parabolic problem with both fourth and second order non-linearities, which may be used to model viscous thin-film flow in the regime of weak slippage with an additional contributions from a singular disjoining pressure brought forth by inter-molecular forces. By carefully treating the individual scaling behaviors of the non-linearities, we obtain asymptotically optimal upper and lower bounds on the rate of forward support propagation. These rates show that the fourth-order operator dominates the propagation rate on short time scales, but the long-term rate of propagation is controlled by the second-order operator. We estimate further the time at which this trade-off in propagation rate occurs. Lastly, we examine sufficient conditions to ensure that the free boundary does not immediately advance in a given direction, leading to a detailed picture of how the competition between the second/fourth-order operators influence the free boundary dynamics of solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23022",
  "author_total": 1,
  "authors": [
    {
      "name": "Joshua Utley"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5hqrfh3tq05nk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23022",
  "fulltext_url": null,
  "id": "5hqrfh3tq05nk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23022",
  "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/2609.23022",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5hqrfh3tq05nk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23022",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23022"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Free Boundary Propagation of a Thin-Film Equation with Nonlinear Diffusion: Scale-Consistent Methods for Optimal Results",
  "title_text": "Free Boundary Propagation of a Thin-Film Equation with Nonlinear Diffusion: Scale-Consistent Methods for Optimal Results",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/5hqrfh3tq05nk",
  "venue": "arXiv",
  "versions": []
}