{
  "id": "1611.08877",
  "version": "v1",
  "published": "2016-11-27T17:30:38.000Z",
  "updated": "2016-11-27T17:30:38.000Z",
  "title": "On the stability of type II blowup for the 1-corotational harmonic heat flow in supercritical dimensions",
  "authors": [
    "Tej-Eddine Ghoul",
    "Slim Ibrahim",
    "Van Tien Nguyen"
  ],
  "comment": "67 pages. arXiv admin note: text overlap with arXiv:1407.1415 by other authors",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We consider the energy-supercritical harmonic heat flow from $\\mathbb{R}^d$ into the $d$-sphere $\\mathbb{S}^d$ with $d \\geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear heat equation $$\\partial_t u = \\partial^2_r u + \\frac{(d-1)}{r}\\partial_r u - \\frac{(d-1)}{2r^2}\\sin(2u).$$ We construct for this equation a family of $\\mathcal{C}^{\\infty}$ solutions which blow up in finite time via concentration of the universal profile $$u(r,t) \\sim Q\\left(\\frac{r}{\\lambda(t)}\\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the quantized rates $$\\lambda(t) \\sim c_u(T-t)^\\frac{\\ell}{\\gamma}, \\quad \\ell \\in \\mathbb{N}^*, \\;\\; 2\\ell > \\gamma = \\gamma(d) \\in (1,2].$$ The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Rapha\\\"el and Rodnianski [Camb. Jour. Math, 3(4):439-617, 2015] for the energy supercritical nonlinear Schr\\\"odinger equation and by Rapha\\\"el and Schweyer [Anal. PDE, 7(8):1713-1805, 2014] for the energy critical harmonic heat flow, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem. Moreover, our constructed solutions are in fact $(\\ell - 1)$ codimension stable under perturbations of the initial data. As a consequence, the case $\\ell = 1$ corresponds to a stable type II blowup regime.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-27T17:30:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "supercritical dimensions",
      "robust universal energy method",
      "dimensional semilinear heat equation",
      "energy critical harmonic heat flow",
      "brouwer fixed point theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 67,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}