{
  "id": "2405.07150",
  "version": "v1",
  "published": "2024-05-12T03:44:00.000Z",
  "updated": "2024-05-12T03:44:00.000Z",
  "title": "Asymptotic behavior for the fast diffusion equation with absorption and singularity",
  "authors": [
    "Changping Xie",
    "Shaomei Fang",
    "Ming Mei",
    "Yuming Qin"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with the weak solution for the fast diffusion equation with absorption and singularity in the form of $u_t=\\triangle u^m -u^p$. We first prove the existence and decay estimate of weak solution when the fast diffusion index satisfies $0<m<1$ and the absorption index is $p>1$. Then we show the asymptotic convergence of weak solution to the corresponding Barenblatt solution for $\\frac{n-1}{n}<m<1$ and $p>m+\\frac{2}{n}$ via the entropy dissipation method combining the generalized Shannon's inequality and Csisz$\\mathrm{\\acute{a}}$r-Kullback inequality. The singularity of spatial diffusion causes us the technical challenges for the asymptotic behavior of weak solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-12T03:44:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fast diffusion equation",
      "asymptotic behavior",
      "weak solution",
      "singularity",
      "fast diffusion index satisfies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}