{
  "abstract": "In this paper, we study the family of inhomogeneous discounted Hamilton-Jacobi equations \\begin{equation}\\label{hjs1} λ(x)u+h(x,d_x u)=c \\quad \\tag{$\\ast$} \\end{equation} on a closed manifold $M$ with a non-identically vanishing discount factor $λ(x)$. There is a critical value $c_0\\in[-\\infty,\\infty)$ such that \\eqref{hjs1} admits a viscosity solution if $c>c_0$ and no solution if $c<c_0$. Inspired by the recent development \\cite{RWY} on the stability theory of viscosity solutions, we show that the equation admits an asymptotically stable solution if and only if $c>c_0$. In this case, we determine the basin of the stable solution and investigate the long-time behavior of the solution semigroup associated with \\eqref{hjs1}. In particular, we obtain a formula relating the lowest convergence rate of the solution semigroup near a stable solution to a minimizing problem concerning the integral of $λ$ over Mather measures supporting on the $1$-graph of the stable solution. This formula has two direct consequences: 1) it yields a description of their asymptotic behavior as $c$ tends to infinity; 2) it helps to classify the ergodic Mather measures and locate their distribution in the phase space. The second consequence leads to a dynamical necessary condition for $c=c_0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.05670",
  "author_total": 3,
  "authors": [
    {
      "name": "Liang Jin"
    },
    {
      "name": "Jun Yan"
    },
    {
      "name": "Kai Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7gp7xgk843hwe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.05670",
  "fulltext_url": null,
  "id": "7gp7xgk843hwe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.05670",
  "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.DS"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7160601378",
  "pdf_url": "https://arxiv.org/pdf/2605.05670",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7gp7xgk843hwe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2605.05670",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.05670"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "On the inhomogeneous discounted Hamilton-Jacobi equations",
  "title_text": "On the inhomogeneous discounted Hamilton-Jacobi equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/7gp7xgk843hwe",
  "venue": "arXiv",
  "versions": []
}