{
  "id": "0912.3506",
  "version": "v1",
  "published": "2009-12-17T20:10:29.000Z",
  "updated": "2009-12-17T20:10:29.000Z",
  "title": "On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations",
  "authors": [
    "Guy Barles",
    "Alessio Porretta",
    "Thierry Wilfried Tabet Tchamba"
  ],
  "journal": "Journal de Math\\'ematiques Pures et Appliqu\\'ees (9) 94, 5 (2010) 497-519",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article, we are interested in the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi Equations. In the superquadratic case, the third author has proved that these solutions can have only two different behaviors: either the solution of the evolution equation converges to the solution of the associated stationary generalized Dirichlet problem (provided that it exists) or it behaves like $-ct+\\varphi (x)$ where $c\\geq0$ is a constant, often called the \"ergodic constant\" and $\\varphi$ is a solution of the so-called \"ergodic problem\". In the present subquadratic case, we show that the situation is slightly more complicated: if the gradient-growth in the equation is like $|Du|^m$ with $m>3/2,$ then analogous results hold as in the superquadratic case, at least if $c>0.$ But, on the contrary, if $m\\leq 3/2$ or $c=0,$ then another different behavior appears since $u(x,t) + ct$ can be unbounded from below where $u$ is the solution of the subquadratic viscous Hamilton-Jacobi Equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-17T20:10:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35B40",
      "49L25"
    ],
    "keywords": [
      "subquadratic viscous hamilton-jacobi equations",
      "large time behavior",
      "superquadratic case",
      "associated stationary generalized dirichlet problem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.3506B"
    }
  }
}