{
  "id": "1710.06538",
  "version": "v1",
  "published": "2017-10-18T00:38:31.000Z",
  "updated": "2017-10-18T00:38:31.000Z",
  "title": "$L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data",
  "authors": [
    "Masahiro Ikeda",
    "Takahisa Inui",
    "Mamoru Okamoto",
    "Yuta Wakasugi"
  ],
  "comment": "39 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the Cauchy problem of the damped wave equation \\begin{align*} \\partial_{t}^2 u - \\Delta u + \\partial_t u = 0 \\end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\\le q \\le p < \\infty\\ (p\\neq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\\cap H_r^{\\beta}) \\times (H^{s-1} \\cap L^r)$ with $r \\in (1,2]$, $s\\ge 0$, and $\\beta = (n-1)|\\frac{1}{2}-\\frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power $1+\\frac{2r}{n}$, while it is known that the critical power $1+\\frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan and blow-up rates by an ODE argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-18T00:38:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L71",
      "35A01",
      "35B40",
      "35B44"
    ],
    "keywords": [
      "damped wave equation",
      "slowly decaying data",
      "nonlinear problem",
      "critical exponent",
      "global solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}