{
  "id": "1703.00252",
  "version": "v1",
  "published": "2017-03-01T11:56:17.000Z",
  "updated": "2017-03-01T11:56:17.000Z",
  "title": "Parabolic equations with natural growth approximated by nonlocal equations",
  "authors": [
    "Tommaso Leonori",
    "Alexis Molino",
    "Sergio Segura de León"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study several aspects related with solutions of nonlocal problems whose prototype is $$ u_t =\\displaystyle \\int_{\\mathbb{R}^N} J(x-y) \\big( u(y,t) -u(x,t) \\big) \\mathcal G\\big( u(y,t) -u(x,t) \\big) dy \\qquad \\mbox{ in } \\, \\Omega \\times (0,T)\\,, $$ being $ u (x,t)=0 \\mbox{ in } (\\mathbb{R}^N\\setminus \\Omega )\\times (0,T)\\,$ and $ u(x,0)=u_0 (x) \\mbox{ in } \\Omega$. We take, as the most important instance, $\\mathcal G (s) \\sim 1+ \\frac{\\mu}{2} \\frac{s}{1+\\mu^2 s^2 }$ with $\\mu\\in \\mathbb{R}$ as well as $u_0 \\in L^1 (\\Omega)$, $J$ is a smooth symmetric function with compact support and $\\Omega$ is either a bounded smooth subset of $\\mathbb{R}^N$, with nonlocal Dirichlet boundary condition, or $\\mathbb{R}^N$ itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover we prove that if the kernel rescales in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar-Parisi-Zhang equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-01T11:56:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "natural growth",
      "nonlocal equations",
      "parabolic equations",
      "nonlocal dirichlet boundary condition",
      "smooth symmetric function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}