{
  "id": "1805.01911",
  "version": "v1",
  "published": "2018-05-04T18:34:37.000Z",
  "updated": "2018-05-04T18:34:37.000Z",
  "title": "Infinite time blow-up for the fractional heat equation with critical exponent",
  "authors": [
    "Y. Sire",
    "J. Wei",
    "Z. Zheng",
    "Y. Zhou"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider positive solutions for the fractional heat equation with critical exponent \\begin{equation*} \\begin{cases} u_t = -(-\\Delta)^{s}u + u^{\\frac{n+2s}{n-2s}}\\text{ in } \\Omega\\times (0, \\infty), u = 0\\text{ on } (\\mathbb{R}^n\\setminus \\Omega)\\times (0, \\infty), u(\\cdot, 0) = u_0\\text{ in }\\mathbb{R}^n, \\end{cases} \\end{equation*} where $\\Omega$ is a smooth bounded domain in $\\mathbb{R}^n$, $n > 4s$, $s\\in (0, 1)$, $u:\\mathbb{R}^n\\times [0, \\infty)\\to \\mathbb{R}$ and $u_0$ is a positive smooth initial datum with $u_0|_{\\mathbb{R}^n\\setminus \\Omega} = 0$. We prove the existence of $u_0$ such that the solution blows up precisely at prescribed distinct points $q_1,\\cdots, q_k$ in $\\Omega$ as $t\\to +\\infty$. The main ingredient of the proofs is a new inner-outer gluing scheme for the fractional parabolic problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-04T18:34:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional heat equation",
      "infinite time blow-up",
      "critical exponent",
      "positive smooth initial datum",
      "fractional parabolic problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}