{
  "id": "1611.06346",
  "version": "v1",
  "published": "2016-11-19T11:46:56.000Z",
  "updated": "2016-11-19T11:46:56.000Z",
  "title": "Quasi-Poincaré compactifications and blow-up solutions of ordinary differential equations",
  "authors": [
    "Kaname Matsue"
  ],
  "comment": "31 pages, 6 figures",
  "categories": [
    "math.DS"
  ],
  "abstract": "We construct a generalized compactification of Euclidean spaces as well as dynamical systems on them for studying blow-up solutions of ordinary differential equations. The basic idea is based on the quasi-homogeneous desingularization (blowing-up) of singularities. We apply this desingularization to infinity so that divergent solutions including grow-up and blow-up solutions for differential equations whose asymptotic form is not necessarily homogeneous can be treated on compact spaces. As a prototype, we define the quasi-homogeneous version of Poincar\\'e compactifications and discuss fundamental properties which play important roles to study blow-up solutions of inhomogeneous vector fields. As a demonstration of applicability, we consider singular shock waves consisting of several pieces of blow-up solutions and bounded solutions near infinity. We see that the quasi-Poincar\\'e compactification well describes singular profiles of singular shocks observed in preceding studies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-19T11:46:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34A26",
      "34C08",
      "35B44",
      "35L67",
      "58K55"
    ],
    "keywords": [
      "ordinary differential equations",
      "play important roles",
      "study blow-up solutions",
      "singular shock waves",
      "desingularization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}