{
  "id": "1104.0306",
  "version": "v1",
  "published": "2011-04-02T09:16:17.000Z",
  "updated": "2011-04-02T09:16:17.000Z",
  "title": "A general fractional porous medium equation",
  "authors": [
    "Arturo de Pablo",
    "Fernando Quirós",
    "Ana Rodríguez",
    "Juan Luis Vázquez"
  ],
  "comment": "43 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \\{ll} \\dfrac{\\partial u}{\\partial t} + (-\\Delta)^{\\sigma/2} (|u|^{m-1}u)=0, & \\qquad x\\in\\mathbb{R}^N,\\; t>0, [8pt] u(x,0) = f(x), & \\qquad x\\in\\mathbb{R}^N.%. $$ We consider data $f\\in L^1(\\mathbb{R}^N)$ and all exponents $0<\\sigma<2$ and $m>0$. Existence and uniqueness of a weak solution is established for $m> m_*=(N-\\sigma)_+ /N$, giving rise to an $L^1$-contraction semigroup. In addition, we obtain the main qualitative properties of these solutions. In the lower range $0<m\\le m_*$ existence and uniqueness of solutions with good properties happen under some restrictions, and the properties are different from the case above $m_*$. We also study the dependence of solutions on $f,m$ and $\\sigma$. Moreover, we consider the above questions for the problem posed in a bounded domain.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-02T09:16:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A33",
      "35A05",
      "35K55",
      "76S05"
    ],
    "keywords": [
      "general fractional porous medium equation",
      "uniqueness",
      "properties happen",
      "lower range",
      "main qualitative properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.0306D"
    }
  }
}