{
  "id": "2303.10153",
  "version": "v1",
  "published": "2023-03-15T15:07:00.000Z",
  "updated": "2023-03-15T15:07:00.000Z",
  "title": "On the finite time blow-ups for solutions of nonlinear differential equations",
  "authors": [
    "Luan Hoang"
  ],
  "comment": "29 pages, submitted for publication",
  "categories": [
    "math.AP",
    "math.DS"
  ],
  "abstract": "We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\\alpha$ for some $\\alpha>0$. We prove that the asymptotic behavior of a solution $y(t)$ near a finite blow-up time $T_*$ is $(T_*-t)^{-1/\\alpha}\\xi_*$ for some nonzero vector $\\xi_*$. Specific error estimates for $|(T_*-t)^{1/\\alpha}y(t)-\\xi_*|$ are provided. In some typical cases, they can be a positive power of $(T_*-t)$ or $1/|\\ln(T_*-t)|$. This depends on whether the decaying rate of the lower order term, relative to the size of the dominant term, is of a power or logarithmic form. Similar results are obtained for a class of nonlinear differential inequalities with finite time blow-up solutions. Our results cover larger classes of nonlinear equations, differential inequalities and error estimates than those in the previous work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-15T15:07:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34D05",
      "41A60"
    ],
    "keywords": [
      "nonlinear differential equations",
      "nonlinear ordinary differential equations",
      "finite time blow-up solutions",
      "dominant term",
      "differential inequalities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}