{
  "id": "math/0306160",
  "version": "v1",
  "published": "2003-06-10T13:07:27.000Z",
  "updated": "2003-06-10T13:07:27.000Z",
  "title": "Stability of solutions of quasilinear parabolic equations",
  "authors": [
    "Giuseppe Maria Coclite",
    "Helge Holden"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We bound the difference between solutions $u$ and $v$ of $u_t = a\\Delta u+\\Div_x f+h$ and $v_t = b\\Delta v+\\Div_x g+k$ with initial data $\\phi$ and $ \\psi$, respectively, by $\\Vert u(t,\\cdot)-v(t,\\cdot)\\Vert_{L^p(E)}\\le A_E(t)\\Vert \\phi-\\psi\\Vert_{L^\\infty(\\R^n)}^{2\\rho_p}+ B(t)(\\Vert a-b\\Vert_{\\infty}+ \\Vert \\nabla_x\\cdot f-\\nabla_x\\cdot g\\Vert_{\\infty}+ \\Vert f_u-g_u\\Vert_{\\infty} + \\Vert h-k\\Vert_{\\infty})^{\\rho_p} \\abs{E}^{\\eta_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $x\\in\\R^n$, and $t$. The functions $a$ and $h$ may in addition depend on $\\nabla u$. Identical assumptions hold for the functions that determine the solutions $v$. Furthermore, $E\\subset\\R^n$ is assumed to be a bounded set, and $\\rho_p$ and $\\eta_p$ are fractions that depend on $n$ and $p$. The diffusion coefficients $a$ and $b$ are assumed to be strictly positive and the initial data are smooth.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-06-10T13:07:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quasilinear parabolic equations",
      "initial data",
      "diffusion coefficients",
      "addition depend",
      "identical assumptions hold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......6160C"
    }
  }
}