{
  "abstract": "We investigate forward and backward smoothing effects in Lebesgue spaces $L^p$ and $\\mathcal{M}^p:=L^{p,\\infty}$ for the Cauchy problem associated to the nonlinear and nonlocal fractional diffusion equation $\\partial_t u+(-Δ)^{\\frac\\sigma2}|u|^{m-1}u=0$ in $\\mathbb{R}^N$, $0<σ<2$, in the very fast range $0<m\\le m_c:=\\frac{N-σ}{N}$. We prove that very weak solutions have an $\\mathcal{M}^p$--$L^\\infty$ smoothing effect if $p>p^*:=\\frac{N}σ(1-m)$, and we construct counterexamples showing the failure of any $L^p$--$\\mathcal{M}^q$ forward ($q>p$) smoothing effect if $1< p< p^*$, $m<m_c$ or $L^1$--$\\mathcal{M}^q$ ($q>1$) if $m=m_c$. We also prove a backward $\\mathcal{M}^p$--$L^1$ smoothing effect whenever $1\\le p<p^*$, $m<m_c$, and we construct counterexamples showing that there is no $L^p$--$\\mathcal{M}^q$ backward ($q<p$) smoothing effect if $p> p^*$. Regarding the threshold value $p=p^*$, we prove that all solutions starting in $\\mathcal{M}^{p^*}$ become extinct in finite time, and show the failure of any $\\mathcal{M}^{p^*}$--$\\mathcal{M}^q$ smoothing before extinction for any $q\\neq p^*$. The construction of counterexamples is based on new uniqueness and comparison results for very weak solutions, combined with the existence of self-similar solutions with suitable properties. The same approach yields new counterexamples for both forward and backward smoothing effects also in the local case $σ=2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12703",
  "author_total": 3,
  "authors": [
    {
      "name": "Mohammed-El-Mahdi Boudaoud"
    },
    {
      "name": "Arturo de Pablo"
    },
    {
      "name": "Fernando Quirós"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5g62tyt2swjkh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12703",
  "fulltext_url": null,
  "id": "5g62tyt2swjkh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12703",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213172422",
  "pdf_url": "https://arxiv.org/pdf/2609.12703",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5g62tyt2swjkh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12703",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12703"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fractional very fast diffusion equations in Lebesgue spaces: uniqueness and smoothing effects",
  "title_text": "Fractional very fast diffusion equations in Lebesgue spaces: uniqueness and smoothing effects",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/5g62tyt2swjkh",
  "venue": "arXiv",
  "versions": []
}