{
  "id": "1904.10747",
  "version": "v1",
  "published": "2019-04-24T11:13:10.000Z",
  "updated": "2019-04-24T11:13:10.000Z",
  "title": "On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions",
  "authors": [
    "Monica Marras",
    "Nicola Pintus",
    "Giuseppe Viglialoro"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we analyze the porous medium equation \\begin{equation}\\label{ProblemAbstract} \\tag{$\\Diamond$} %\\begin{cases} u_t=\\Delta u^m + a\\io u^p-b u^q -c\\lvert\\nabla\\sqrt{u}\\rvert^2 \\quad \\textrm{in}\\quad \\Omega \\times I,%\\\\ %u_\\nu-g(u)=0 & \\textrm{on}\\; \\partial \\Omega, t>0,\\\\ %u({\\bf x},0)=u_0({\\bf x})&{\\bf x} \\in \\Omega,\\\\ %\\end{cases} \\end{equation} where $\\Omega$ is a bounded and smooth domain of $\\R^N$, with $N\\geq 1$, and $I= [0,t^*)$ is the maximal interval of existence for $u$. The constants $a,b,c$ are positive, $m,p,q$ proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of $u$. Under some hypothesis on the data, including intrinsic relations between $m,p$ and $q$, and assuming that for some positive and sufficiently regular function $u_0(\\nx)$ the Initial Boundary Value Problem (IBVP) associated to \\eqref{ProblemAbstract} possesses a positive classical solution $u=u(\\nx,t)$ on $\\Omega \\times I$: \\begin{itemize} \\item [$\\triangleright$] when $p>q$ and in 2- and 3-dimensional domains, we determine a \\textit{lower bound of} $t^*$ for those $u$ becoming unbounded in $L^{m(p-1)}(\\Omega)$ at such $t^*$; \\item [$\\triangleright$] when $p<q$ and in $N$-dimensional settings, we establish a \\textit{global existence criterion} for $u$. \\end{itemize}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-24T11:13:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear boundary conditions",
      "non-local porous medium problem",
      "classical solution",
      "proper real numbers larger",
      "initial boundary value problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}