{
  "id": "1706.06057",
  "version": "v1",
  "published": "2017-06-19T16:58:13.000Z",
  "updated": "2017-06-19T16:58:13.000Z",
  "title": "Life-span of smooth solutions to a PDE system with cubic nonlinearity",
  "authors": [
    "Xiangsheng Xu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study the life-span of classical solutions to the initial boundary value problem for the system $-\\mbox{div}\\left[(I+{\\bf m}\\otimes {\\bf m})\\nabla p\\right]=S(x),\\ \\ \\partial_t{\\bf m}-D^2\\Delta {\\bf m}-E^2({\\bf m}\\cdot\\nabla p)\\nabla p+|{\\bf m}|^{2(\\gamma-1)}{\\bf m}=0$, where $S(x)$ is a given function and $D, E, \\gamma$ are given numbers. This problem has been proposed as a PDE model for biological transportation networks. Our investigations reveal that local existence of a classical solution can always be achieved and the life-span of such a solution can be extended as far away as one wishes as long as the term $\\|{\\bf m}(x,0)\\|_{\\infty, \\Omega}+\\|S(x)\\|_{\\frac{2N}{3}, \\Omega}$ is made suitably small, where $N$ is the space dimension and $\\|\\cdot\\|_{q,\\Omega}$ denotes the norm in $L^q(\\Omega)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-19T16:58:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cubic nonlinearity",
      "pde system",
      "smooth solutions",
      "initial boundary value problem",
      "classical solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}