{
  "abstract": "We explore the inverse problem of determining the reaction coefficient $q(x',t)$ in the time-fractional diffusion equation ${}^CD_t u-Lu+q(x',t)u=f$ in $\\mathbb{R}^n\\times(0,T]$, $x=(x',x_n)$, $0<α<1$, from the trace of the solution on the hyperplane $\\{x_n=0\\}$. Here ${}^CD_t$ is the Gerasimov--Caputo derivative and $L$ is a uniformly elliptic second-order operator with variable, $x_n$-independent coefficients and without zero-order term. For $L=Δ$ this problem was studied in dimensions $n=1,2,3$ by methods based on pointwise bounds of the kernels, which are not available for $n\\ge4$. Our approach relies on the Eidelman--Kochubei fundamental solution: although its singularity on the diagonal depends on the dimension, we show that it satisfies integral bounds of the same form for all $n\\ge1$. Using these bounds, we prove existence, uniqueness and stability of regular solutions of the direct problem with a Hölder continuous reaction coefficient. Without assuming additional smoothness of the solution, we show that the inverse problem is equivalent to a closed system of integral equations for $q$ and the derivatives $\\partial_{x_n}u$, $\\partial^2_{x_n}u$. For the inverse problem we establish a conditional Lipschitz stability estimate on the whole time interval and global uniqueness in every dimension, and prove local existence of a solution. In the case $a_{in}\\equiv0$, $b_n\\equiv0$ we obtain a stability estimate with explicit constants, which reduces to the known estimates for the Laplacian. An explicit two-dimensional example with a mixed coefficient and a normal drift shows that the hypotheses of the main theorems are compatible and that the corresponding terms in the trace condition cannot be omitted.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29763",
  "author_total": 1,
  "authors": [
    {
      "name": "D. K. Durdiev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vpzvpngwsxak.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29763",
  "fulltext_url": null,
  "id": "5vpzvpngwsxak",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29763",
  "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|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7223228961",
  "pdf_url": "https://arxiv.org/pdf/2610.29763",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vpzvpngwsxak.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29763",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29763"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Recovering the reaction coefficient in a time-fractional diffusion equation with variable coefficients in any spatial dimension",
  "title_text": "Recovering the reaction coefficient in a time-fractional diffusion equation with variable coefficients in any spatial dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/5vpzvpngwsxak",
  "venue": "arXiv",
  "versions": []
}