{
  "id": "2002.09967",
  "version": "v1",
  "published": "2020-02-23T18:48:59.000Z",
  "updated": "2020-02-23T18:48:59.000Z",
  "title": "Fine properties of solutions to the Cauchy problem for a Fast Diffusion Equation with Caffarelli-Kohn-Nirenberg weights",
  "authors": [
    "Matteo Bonforte",
    "Nikita Simonov"
  ],
  "comment": "41 pages, 3 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the Fast Diffusion Equation with weights (WFDE) $u_t=|x|^\\gamma\\mathrm{div}\\left(|x|^{-\\beta}\\nabla u^m\\right)$ posed on $(0,+\\infty)\\times\\mathbb{R}^d$, with $d\\ge 3$, in the so-called good fast diffusion range $m_c<m<1$, within the natural range of parameters $\\gamma,\\beta$. It is a natural question to ask in which sense such solutions behave like the Barenblatt $\\mathfrak{B}$ (fundamental solution): for instance, asymptotic convergence, i.e. $\\|u(t)-\\mathfrak{B}(t)\\|_{{\\rm L}^p(\\mathbb{R}^d)}\\xrightarrow[]{t\\to\\infty}0$, is well known for all $1\\le p\\le \\infty$, while only few partial results tackle a finer analysis of the tail behaviour. We characterize the maximal set of data $\\mathcal{X}\\subset{\\rm L}^1_+(\\mathbb{R}^d)$ that produces solutions which are pointwise trapped between two Barenblatt (Global Harnack Principle), and uniformly converge in relative error (REC), i.e. ${\\rm d}_\\infty(u(t))=\\|u(t)/\\mathcal{B}(t)-1\\|_{{\\rm L}^\\infty(\\mathbb{R}^d)}\\xrightarrow[]{t\\to\\infty}0$. Such characterization is in terms of a integral condition on $u(t=0)$. To our best knowledge, analogous issues for the linear heat equation $m=1$, do not possess such clear answers, only partial results. Our characterizations are new also for the classical, non-weighted, FDE. We are able to provide minimal rates of convergence to $\\mathcal{B}$ in different norms. Such rates are almost optimal in the non weighted case, and become optimal for radial solutions. To complete the panorama, we show that solutions with data in ${\\rm L}^1_+(\\mathbb{R}^d)\\setminus\\mathcal{X}$, preserve the same ``fat'' spatial tail for all times, hence REC fails and ${\\rm d}_\\infty(u(t))\\!=\\!\\infty$, even if $\\|u(t)-\\mathcal{B}(t)\\|_{{\\rm L}^1(\\mathbb{R}^d)}\\xrightarrow[]{t\\to\\infty}0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-23T18:48:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35B45",
      "35K55",
      "35K67",
      "35K65"
    ],
    "keywords": [
      "fast diffusion equation",
      "cauchy problem",
      "fine properties",
      "caffarelli-kohn-nirenberg weights",
      "linear heat equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}