{
  "abstract": "We prove conjectured-optimal large-time upper decay estimates for solutions of weakly coupled systems of $m\\ge2$ time-fractional diffusion equations. Our result establishes the upper-bound assertion in the conjecture of Li, Liu, and Wada~\\cite{LLW26}. The sharpness of these upper exponents for $m\\ge3$ remains open in general; see Section~\\ref{sec:discussion}. Assume that the diffusion matrices are uniformly elliptic, the coupling coefficients are bounded and may depend on time, and the matrix of $L^\\infty$-norms of the coupling coefficients satisfies a spectral-radius stability condition. Let $α^*$ denote the smallest fractional order among the components with nonzero initial data. Then the $L^2(Ω)^m$-norm of the solution is $O(t^{-α^*})$ as $t\\to\\infty$ if $α^*<1$, whereas it is $O(t^{-(1+α_m)})$ if $α^*=1$, where $α_m$ is the smallest fractional order occurring in the system. For $m=2$, the result yields the same upper decay exponents as \\cite[Theorem~1.1]{LLW26}; in the absence of self-coupling, the matrix stability condition is weaker than the corresponding small-coupling condition in that work. The case $m\\ge3$ is new. Our proof uses a large-time weighted Volterra framework. If $α^*<1$, the weight is a regularly varying power weight; if $α^*=1$, it is constructed by convolution with the resolvent kernel associated with the smallest fractional order. These weights yield the required large-time bounds for all resolvent kernels in the system. In particular, the argument avoids the Laplace-transform and residue-calculus methods used in \\cite{LLW26}, which become difficult to implement for systems with three or more components.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41993",
  "author_total": 3,
  "authors": [
    {
      "name": "Huong Nguyen"
    },
    {
      "name": "Thang Nguyen"
    },
    {
      "name": "Ingo Witt"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7w2zvpgycbdcf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41993",
  "fulltext_url": null,
  "id": "7w2zvpgycbdcf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41993",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41993",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7w2zvpgycbdcf.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41993",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41993"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Large-Time Upper Decay Estimates for Weakly Coupled Time-Fractional Diffusion Systems",
  "title_text": "Large-Time Upper Decay Estimates for Weakly Coupled Time-Fractional Diffusion Systems",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/7w2zvpgycbdcf",
  "venue": "arXiv",
  "versions": []
}