{
  "id": "1810.10232",
  "version": "v1",
  "published": "2018-10-24T07:57:58.000Z",
  "updated": "2018-10-24T07:57:58.000Z",
  "title": "Global existence and lifespan for semilinear wave equations with mixed nonlinear terms",
  "authors": [
    "Wei Dai",
    "Daoyuan Fang",
    "Chengbo Wang"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "Firstly, we study the equation $\\square u = |u|^{q_c}+ |\\partial u|^p$ with small data, where $q_c$ is the critical power of \\emph{Strauss} conjecture and $p\\geq q_c.$ We obtain the optimal lifespan $\\ln({T_\\varepsilon})\\approx\\varepsilon^{-q_c(q_c-1)}$ in $n=3$, and improve the lower-bound of $T_\\varepsilon$ from $\\exp({c\\varepsilon^{-(q_c-1)}})$ to $\\exp({c\\varepsilon^{-(q_c-1)^2/2}})$ in $n=2$. Then, we study the Cauchy problem with small initial data for a system of semilinear wave equations $\\square u = |v|^q,$ $ \\square v = |\\partial_t u|^p$ in 3-dimensional space with $q<2$. We obtain that this system admits a global solution above a $p-q$ curve for spherically symmetric data. On the contrary, we get a new region where the solution will blow up.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-24T07:57:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L05",
      "35L15",
      "35L70"
    ],
    "keywords": [
      "semilinear wave equations",
      "mixed nonlinear terms",
      "global existence",
      "small initial data",
      "small data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}