{
  "abstract": "We establish sharp temporal lower bounds for the full spatial $L^2$-norm of energy solutions to $\\partial_tu-Δu=vu$ on $\\mathbb{R}^n$, with $n\\geq3$. For global solutions, nonvanishing at a single time implies the lower bound $ce^{-Ct}$ when $v$ is small in the scale-invariant space $L^\\infty_tL^{n/2}_x$, and $ce^{-Ct^2}$ throughout the subcritical range $v\\in L^\\infty_tL^p_x$, with $p>n/2$, without smallness. The linear time exponent is optimal already for the free heat equation, while we show that the quadratic exponent is optimal for bounded complex potentials on $\\mathbb{R}^3$. No additional regularity of the potential is required. The proof uses a convexified mixed Carleman estimate whose norms are designed together with the final time truncation. Integration over spatial centers removes the heat-kernel weight and yields quantitative comparisons of the full solution norms. To prove quadratic sharpness in the Euclidean energy class, we construct a time-dependent function $g$ that localizes Meshkov's periodic parabolic example while keeping the induced potential bounded across the zeros of the periodic profile.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30630",
  "author_total": 2,
  "authors": [
    {
      "name": "Igor Kukavica"
    },
    {
      "name": "Qi Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/90k21wjtmndbh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30630",
  "fulltext_url": null,
  "id": "90k21wjtmndbh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30630",
  "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",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7214609517",
  "pdf_url": "https://arxiv.org/pdf/2609.30630",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/90k21wjtmndbh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30630",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30630"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Quantitative Decay for Linear Parabolic Equations",
  "title_text": "Quantitative Decay for Linear Parabolic Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/90k21wjtmndbh",
  "venue": "arXiv",
  "versions": []
}