{
  "id": "1808.02706",
  "version": "v1",
  "published": "2018-08-08T10:11:01.000Z",
  "updated": "2018-08-08T10:11:01.000Z",
  "title": "An application of $L^1$ estimates for oscillating integrals to parabolic like semi-linear structurally damped $σ$-evolution models",
  "authors": [
    "Tuan Anh Dao",
    "Michael Reissig"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the following Cauchy problems for semi-linear structurally damped $\\sigma$-evolution models: \\begin{equation*} u_{tt}+ (-\\Delta)^\\sigma u+ \\mu (-\\Delta)^\\delta u_t = f(u,u_t),\\, u(0,x)= u_0(x),\\, u_t(0,x)=u_1(x) \\end{equation*} with $\\sigma \\ge 1$, $\\mu>0$ and $\\delta \\in (0,\\frac{\\sigma}{2})$. Here the function $f(u,u_t)$ stands for the power nonlinearities $|u|^{p}$ and $|u_t|^{p}$ with a given number $p>1$. We are interested in investigating $L^{1}$ estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand sides by applying the theory of modified Bessel functions and Fa\\`{a} di Bruno's formula. By assuming additional $L^{m}$ regularity on the initial data, we use $(L^{m}\\cap L^{q})- L^{q}$ and $L^{q}- L^{q}$ estimates with $q\\in (1,\\infty)$ and $m\\in [1,q)$, to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on $L^q$ spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-08T10:11:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "evolution models",
      "oscillating integrals",
      "semi-linear",
      "small data sobolev solutions",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}