{
  "abstract": "Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption $$ \\partial_tu=Δu^m-|x|^σu^p, \\quad (x,t)\\in\\real^N\\times(0,\\infty), \\quad N\\geq1, $$ with exponents $1<p<m$ and $σ>0$, are classified. Looking for solutions in the form $$ u(x,t)=t^{-α}f(|x|t^β), \\quad α=\\frac{σ+2}{σ(m-1)+2(p-1)}, \\quad β=\\frac{m-p}{σ(m-1)+2(p-1)}, $$ it is shown that all their profiles satisfy the behavior at infinity given by $$ \\lim\\limits_{ξ\\to\\infty}ξ^{σ/(p-1)}f(ξ)=\\left(\\frac{1}{p-1}\\right)^{1/(p-1)}, $$ but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with $f(0)>0$, $f'(0)=0$, another unique solution such that $f$ presents a \\emph{dead-core}; that is, $f\\equiv0$ for $ξ\\in[0,ξ_0]$ for some $ξ_0>0$, and, finally, there exists $K^*\\in(0,\\infty)$ such that, for any $K\\in(0,K^*)$, there is at least a solution such that $$ \\lim\\limits_{ξ\\to0}ξ^{-(σ+2)/(m-p)}f(ξ)=K. $$ The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20397",
  "author_total": 2,
  "authors": [
    {
      "name": "Razvan Gabriel Iagar"
    },
    {
      "name": "Diana-Rodica Munteanu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f1c2kh41c38q7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20397",
  "fulltext_url": null,
  "id": "f1c2kh41c38q7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20397",
  "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.DS"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213602916",
  "pdf_url": "https://arxiv.org/pdf/2609.20397",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f1c2kh41c38q7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20397",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20397"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "A porous medium equation with dominating weighted absorption: three types of self-similar solutions",
  "title_text": "A porous medium equation with dominating weighted absorption: three types of self-similar solutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/f1c2kh41c38q7",
  "venue": "arXiv",
  "versions": []
}