{
  "id": "2106.12286",
  "version": "v1",
  "published": "2021-06-23T10:14:43.000Z",
  "updated": "2021-06-23T10:14:43.000Z",
  "title": "Effect of different additional $L^{m}$ regularity on semi-linear damped $σ$-evolution models",
  "authors": [
    "Khaldi Said",
    "Arioui Fatima Zahra"
  ],
  "comment": "This paper deals with a generalized class of Cauchy data and its influence on the so-called critical exponent. All comments are welcome!",
  "categories": [
    "math.AP"
  ],
  "abstract": "The motivation of the present study is to discuss the global (in time) existence of small data solutions to the following semi-linear structurally damped $\\sigma$-evolution models: \\begin{equation*} \\partial_{tt}u+(-\\Delta)^{\\sigma}u+(-\\Delta)^{\\sigma/2}\\partial_{t}u=\\left|u\\right| ^{p}, \\ \\sigma\\geq 1, \\ \\ p>1, \\end{equation*} where the Cauchy data $(u(0,x), \\partial_{t}u(0,x))$ will be chosen from energy space on the base of $L^{q}$ with different additional $L^{m}$ regularity, namely \\begin{equation*} u(0,x)\\in H^{\\sigma,q}(\\mathbb{R}^{n})\\cap L^{m_{1}}(\\mathbb{R}^{n}) , \\ \\ \\partial_{t}u(0,x)\\in L^{q}(\\mathbb{R}^{n})\\cap L^{m_{2}}(\\mathbb{R}^{n}), \\ \\ q\\in(1,\\infty),\\ \\ m_{1}, m_{2}\\in [1,q). \\end{equation*} Our new results will show that the critical exponent which guarantees the global (in time) existence is really affected by these different additional regularities and will take \\textit{two different values} under some restrictions on $m_{1}, m_{2}$, $q$, $\\sigma$ and the space dimension $n\\geq1$. Moreover, in each case, we have no loss of decay estimates of the unique solution with respect to the corresponding linear models.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-23T10:14:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35L30",
      "35B33",
      "35B45",
      "35B44"
    ],
    "keywords": [
      "evolution models",
      "regularity",
      "semi-linear",
      "small data solutions",
      "unique solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}