{
  "id": "2108.11216",
  "version": "v1",
  "published": "2021-08-25T13:08:05.000Z",
  "updated": "2021-08-25T13:08:05.000Z",
  "title": "Hamilton-Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown",
  "authors": [
    "Hitoshi Ishii",
    "Kaizhi Wang",
    "Lin Wang",
    "Jun Yan"
  ],
  "comment": "42 pages",
  "categories": [
    "math.AP",
    "math.DS"
  ],
  "abstract": "We study the Hamilton-Jacobi equations $H(x,Du,u)=0$ in $M$ and $\\partial u/\\partial t +H(x,D_xu,u)=0$ in $M\\times(0,\\infty)$, where the Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on the variable $u$. In the framework of the semicontinuous viscosity solutions due to Barron-Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-25T13:08:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35F21",
      "35D40",
      "35B51",
      "35B40",
      "49H25"
    ],
    "keywords": [
      "hamilton-jacobi equations",
      "hamiltonian",
      "cauchy problem",
      "semicontinuous viscosity solutions",
      "long-time behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}