{
  "id": "1510.03758",
  "version": "v1",
  "published": "2015-10-13T16:09:18.000Z",
  "updated": "2015-10-13T16:09:18.000Z",
  "title": "Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains",
  "authors": [
    "Matteo Bonforte",
    "Alessio Figalli",
    "Xavier Ros-Oton"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-\\Delta)^su^m=0$ in $(0,\\infty)\\times\\Omega$, for $m>1$ and $s\\in (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,\\infty)\\times({\\mathbb R}^N\\setminus\\Omega)$, and nonnegative initial condition $u(0,\\cdot)=u_0\\geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $\\partial\\Omega$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^\\infty$ in $x$ and $C^{1,\\alpha}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-\\mathcal L F(u)=0$ in $\\Omega$, both in bounded or unbounded domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-13T16:09:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K65",
      "35B65",
      "35K55"
    ],
    "keywords": [
      "fractional porous medium equation",
      "general domains",
      "regularity",
      "propagation",
      "nonlocal parabolic equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151003758B"
    }
  }
}