{
  "id": "2209.10065",
  "version": "v1",
  "published": "2022-09-21T01:48:50.000Z",
  "updated": "2022-09-21T01:48:50.000Z",
  "title": "Infinite time bubble towers in the fractional heat equation with critical exponent",
  "authors": [
    "Li Cai",
    "Jun Wang",
    "Jun-Cheng Wei",
    "Wen Yang"
  ],
  "comment": "44 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the fractional heat equation with critical exponent in $\\mathbb{R}^n$ for $n>6s,s\\in(0,1),$ \\begin{equation*} u_t=-(-\\Delta)^su+|u|^{\\frac{4s}{n-2s}}u,\\quad (x,t)\\in \\mathbb{R}^n\\times\\mathbb{R}. \\end{equation*} We construct a bubble tower type solution both for the forward and backward problem by establishing the existence of the sign-changing solution with multiple blow-up at a single point with the form \\begin{equation*} u(x,t)=(1+o(1))\\sum_{j=1}^{k}(-1)^{j-1}\\mu_j(t)^{-\\frac{n-2s}{2}}U\\left(\\frac{x}{\\mu_j(t)}\\right) \\quad\\mbox{as}\\quad t\\to+\\infty, \\end{equation*} and the positive solution with multiple blow-up at a single point with the form \\begin{equation*} u(x,t)=(1+o(1))\\sum_{j=1}^{k}\\mu_j(t)^{-\\frac{n-2s}{2}}U\\left(\\frac{x}{\\mu_j(t)}\\right) \\quad\\mbox{as}\\quad t\\to-\\infty, \\end{equation*} respectively. Here $k\\ge2$ is a positive integer, $$U(y)=\\alpha_{n,s}\\left(\\frac{1}{1+|y|^2}\\right)^{\\frac{n-2s}{2}},$$ and \\begin{equation*} \\mu_j(t)=\\beta_j |t|^{-\\alpha_j}(1+o(1))~\\mathrm{as}~t\\to\\pm\\infty, \\quad \\alpha_j=\\frac{1}{2s}\\left(\\frac{n-2s}{n-6s}\\right)^{j-1}-\\frac{1}{2s}, \\end{equation*} for some certain positive numbers $\\beta_j,j=1,\\cdots,k.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-21T01:48:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional heat equation",
      "infinite time bubble towers",
      "critical exponent",
      "single point",
      "multiple blow-up"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}