{
  "abstract": "We study energy methods for the time-fractional equation $\\partial_t^α(u-u_0)-(a(t,x)u_x)_x+(F(t,x,u))_x=Q(t,x,u)$ on the real line, where $0<α<1$, the coefficient $a$ is measurable and bounded from above and from below by positive constants, $σ\\partial_xF\\ge0$, $σQ\\le0$, and the data are in $L^1\\cap L^\\infty$. Using Yosida approximations of the kernel, an iteration on the $L^{2^m}$ norms gives the parabolic decay $\\|u(t)\\|_{L^\\infty}\\le C\\|u_0\\|_{L^1}t^{-α/2}$. In $L^2$ it gives the rate $t^{-α/4}$, which, already for the linear equation, improves the rate $t^{-α/5}$ obtained before by the classical energy method. For the convection--diffusion equation $\\partial_t^α(u-u_0)-u_{xx}+(|u|^{q-1}u)_x=0$, $q>1$, the parabolic rate is optimal for $q\\ge2$, and for $1<q<2$ the convection improves it. An $L^1$ time-regularity estimate of Bénilan--Crandall type gives the pointwise flux bound $|u|^{q-1}u-u_x=O(\\|u_0\\|_{L^1}t^{-α})$. This bound takes the place of the Ole\\uınik estimate and yields the optimal rate $\\|u(t)\\|_{L^\\infty}\\lesssim t^{-α/q}$ for $1<q<2$. For nonnegative data, matching lower bounds hold for the $L^p$ norms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07185",
  "author_total": 1,
  "authors": [
    {
      "name": "Liviu I. Ignat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/80f2b52mmky8a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07185",
  "fulltext_url": null,
  "id": "80f2b52mmky8a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07185",
  "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/2611.07185",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/80f2b52mmky8a.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07185",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07185"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Energy methods and optimal decay for time-fractional equations",
  "title_text": "Energy methods and optimal decay for time-fractional equations",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:47Z",
  "url": "https://research.draconitic.com/paper/80f2b52mmky8a",
  "venue": "arXiv",
  "versions": []
}