{
  "id": "1807.03937",
  "version": "v1",
  "published": "2018-07-11T02:59:37.000Z",
  "updated": "2018-07-11T02:59:37.000Z",
  "title": "Test function method for blow-up phenomena of semilinear wave equations and their weakly coupled systems",
  "authors": [
    "Masahiro Ikeda",
    "Motohiro Sobajima",
    "Kyouhei Wakasa"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider the wave equations with power type nonlinearities including time-derivatives of unknown functions and their weakly coupled systems. We propose a framework of test function method and give a simple proof of the derivation of sharp upper bound of lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical case, we use a family of self-similar solution to the standard wave equation including Gauss's hypergeometric functions which are originally introduced by Zhou (1992). However, our framework is much simpler than that. As a consequence, we found new $(p,q)$-curve for the system $\\partial_t^2u-\\Delta u=|v|^q$, $\\partial_t^2v-\\Delta v=|\\partial_tu|^p$ and lifespan estimate for small solutions for new region.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-11T02:59:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L05",
      "35L51",
      "35B44"
    ],
    "keywords": [
      "test function method",
      "weakly coupled systems",
      "semilinear wave equations",
      "blow-up phenomena",
      "sharp upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}