{
  "id": "2108.05667",
  "version": "v1",
  "published": "2021-08-12T11:21:03.000Z",
  "updated": "2021-08-12T11:21:03.000Z",
  "title": "On the critical exponent and sharp lifespan estimates for semilinear damped wave equations with data from Sobolev spaces of negative order",
  "authors": [
    "Wenhui Chen",
    "Michael Reissig"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study semilinear damped wave equations with power nonlinearity $|u|^p$ and initial data belonging to Sobolev spaces of negative order $\\dot{H}^{-\\gamma}$. In the present paper, we obtain a new critical exponent $p=p_{\\mathrm{crit}}(n,\\gamma):=1+\\frac{4}{n+2\\gamma}$ for some $\\gamma\\in(0,\\frac{n}{2})$ and low dimensions in the framework of Soblev spaces of negative order. Precisely, global (in time) existence of small data Sobolev solutions of lower regularity is proved for $p>p_{\\mathrm{crit}}(n,\\gamma)$, and blow-up of weak solutions in finite time even for small data if $1<p<p_{\\mathrm{crit}}(n,\\gamma)$. Furthermore, in order to more accurately describe the blow-up time, we investigate sharp upper bound and lower bound estimates for the lifespan in the subcritical case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-12T11:21:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sharp lifespan estimates",
      "negative order",
      "sobolev spaces",
      "critical exponent",
      "study semilinear damped wave equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}