{
  "id": "math/0312395",
  "version": "v2",
  "published": "2003-12-20T14:40:34.000Z",
  "updated": "2005-04-07T15:56:30.000Z",
  "title": "A blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain",
  "authors": [
    "Konstantin Khanin",
    "Dmitry Khmelev",
    "Andrei Sobolevskii"
  ],
  "comment": "19 pages, no figures; based on a talk given at the workshop \"Idempotent Mathematics and Mathematical Physics\" at the E. Schroedinger Institute for Mathematical Physics in Vienna in February 2003. A dedication indicating the untimely death of Dmitry Khmelev is added",
  "journal": "\"Idempotent Mathematics and Mathematical Physics\", G. L. Litvinov, V. P. Maslov (eds.), AMS, Providence, 2005, ISBN 0-8218-3538-6, p. 161-179",
  "categories": [
    "math.OC",
    "math.DS"
  ],
  "abstract": "We construct an example of blow-up in a flow of min-plus linear operators arising as solution operators for a Hamilton-Jacobi equation with a Hamiltonian of the form |p|^alpha+U(x,t), where alpha>1 and the potential U(x,t) is uniformly bounded together with its gradient. The construction is based on the fact that, for a suitable potential defined on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O((log T)^(2-2/alpha)). We also show that this growth estimate cannot be surpassed. Implications of this example for existence of global generalized solutions to randomly forced Hamilton-Jacobi or Burgers equations are discussed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-04-07T15:56:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L67",
      "49L99"
    ],
    "keywords": [
      "hamilton-jacobi equation",
      "blow-up phenomenon",
      "unbounded domain",
      "global generalized solutions",
      "growth estimate"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....12395K"
    }
  }
}