{
  "id": "1202.2674",
  "version": "v2",
  "published": "2012-02-13T09:46:03.000Z",
  "updated": "2013-03-22T09:55:37.000Z",
  "title": "$L^{\\infty}$ estimates and uniqueness results for nonlinear parabolic equations with gradient absorption terms",
  "authors": [
    "Marie-Françoise Bidaut-Véron",
    "Nguyen Anh Dao"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \\[ \\left\\{\\begin{array} [c]{c}% u_{t}-\\nu\\Delta u+|\\nabla u|^{q}=0, u(0)=u_{0}, \\end{array} \\right. \\] in $Q_{\\Omega,T}=\\Omega\\times\\left(0,T\\right) ,$ where $q>1,\\nu\\geqq 0,T\\in\\left(0,\\infty\\right] ,$ and $\\Omega=\\mathbb{R}^{N}$ or $\\Omega$ is a smooth bounded domain, and $u_{0}\\in L^{r}(\\Omega),r\\geqq1,$ or $u_{0}% \\in\\mathcal{M}_{b}(\\Omega).$ We show $L^{\\infty}$ decay estimates, valid for \\textit{any weak solution}, \\textit{without any conditions a}s $\\left\\| x\\right\\| \\rightarrow\\infty,$ and \\textit{without uniqueness assumptions}. As a consequence we obtain new uniqueness results, when $u_{0}\\in \\mathcal{M}_{b}(\\Omega)$ and $q<(N+2)/(N+1),$ or $u_{0}\\in L^{r}(\\Omega)$ and $q<(N+2r)/(N+r).$ We also extend some decay properties to quasilinear equations of the model type \\[ u_{t}-\\Delta_{p}u+\\left\\| u\\right\\| ^{\\lambda-1}u|\\nabla u|^{q}=0 \\] where $p>1,\\lambda\\geqq0,$ and $u$ is a signed solution.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-03-22T09:55:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear parabolic equations",
      "gradient absorption terms",
      "uniqueness results",
      "model type",
      "smooth bounded domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.2674B"
    }
  }
}