{
  "abstract": "We study observability and stabilization for the whole-space Grushin heat equation $$ \\partial_t u-Δ_γu=0,\\qquad Δ_γ:=Δ_x+|x|^{2γ}Δ_y,\\qquad (t,x,y)\\in(0,T)\\times\\mathbb{R}^n\\times\\mathbb{R}^m,\\quad γ>0. $$ For product sets $E=E_1\\times E_2$, we identify three observability regimes in which thickness, distance from the degeneracy set $\\{x=0\\}$, and local density near that set play different roles. (i) In the subcritical regime $0<γ<1$, the observability inequality holds for every $T>0$ if and only if both $E_1$ and $E_2$ are thick; hence $T_{\\min}(E)=0$. (ii) At the critical exponent $γ=1$, if $E_2=\\mathbb{R}^m$ and $E_1\\subset\\mathbb{R}^n$ is thick, we determine the exact minimal observable time: $$ T_{\\min}(E)=\\frac{a_{\\mathrm{ess}}(E_1)^2}{2n},\\qquad a_{\\mathrm{ess}}(E_1):=\\operatorname*{ess\\,inf}_{x\\in E_1}|x|. $$ Thus, the threshold depends on the essential distance of $E_1$ from the degeneracy set. (iii) In the supercritical regime $γ>1$, we consider $n=1$, $E_2=\\mathbb{R}^m$ and thick sets $E_1\\subset\\mathbb{R}$. If $a_{\\mathrm{ess}}(E_1)>0$, observability fails for every $T>0$, so that $T_{\\min}(E)=+\\infty$. If $a_{\\mathrm{ess}}(E_1)=0$, observability depends on the density of $E_1$ near $0$, measured by $$ D_{E_1}(r):=\\log\\!\\left(\\frac{2r}{|E_1\\cap(-r,r)|}\\right),\\qquad L_{E_1}:=\\limsup_{r\\downarrow0}r^2D_{E_1}(r). $$ If $L_{E_1}=0$, then $T_{\\min}(E)=0$; if $0<L_{E_1}<\\infty$, a finite sufficient observation time exists. Moreover, observability at time $T$ requires $$ \\limsup_{r\\downarrow0}r^{2(1+γ)/(γ-1)}D_{E_1}(r)\\le C_{T,γ}. $$ In contrast, for every $γ>0$, exponential and complete stabilizability hold if and only if both $E_1$ and $E_2$ are thick. The stabilization criterion is therefore independent of $γ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05821",
  "author_total": 3,
  "authors": [
    {
      "name": "Shanlin Huang"
    },
    {
      "name": "Gengsheng Wang"
    },
    {
      "name": "Jiaqi Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9r68zsgjrdpay.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05821",
  "fulltext_url": null,
  "id": "9r68zsgjrdpay",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05821",
  "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"
    ],
    [
      "arxiv",
      "math.OC"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05821",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9r68zsgjrdpay.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05821",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05821"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Observability and stabilizability for Grushin-Type parabolic equations in the whole space",
  "title_text": "Observability and stabilizability for Grushin-Type parabolic equations in the whole space",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:29Z",
  "url": "https://research.draconitic.com/paper/9r68zsgjrdpay",
  "venue": "arXiv",
  "versions": []
}