{
  "id": "1805.04466",
  "version": "v1",
  "published": "2018-05-11T16:07:09.000Z",
  "updated": "2018-05-11T16:07:09.000Z",
  "title": "Perturbations of self-similar solutions",
  "authors": [
    "Thierry Cazenave",
    "Flávio Dickstein",
    "Ivan Naumkin",
    "Fred B. Weissler"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the nonlinear heat equation $u_t = \\Delta u + |u|^\\alpha u$ with $\\alpha >0$, either on ${\\mathbb R}^N $, $N\\ge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) \\alpha <4$, for every $\\mu \\in {\\mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = \\mu |x-x_0|^{-\\frac {2} {\\alpha }}$ in a neighborhood of some $x_0\\in \\Omega $ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $\\mu |x-x_0|^{-\\frac {2} {\\alpha }}$ on ${\\mathbb R}^N $. Moreover, if $\\mu \\ge \\mu _0$ for a certain $ \\mu _0( N, \\alpha )\\ge 0$, and $u_0 I\\ge 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-11T16:07:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-similar solutions",
      "initial value",
      "initial condition",
      "dirichlet boundary conditions",
      "nonlinear heat equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}