{
  "id": "2405.12210",
  "version": "v1",
  "published": "2024-05-20T17:55:53.000Z",
  "updated": "2024-05-20T17:55:53.000Z",
  "title": "Blow-up solutions of the \"bad\" Boussinesq equation",
  "authors": [
    "Christophe Charlier"
  ],
  "comment": "31 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study blow-up solutions of the ``bad\" Boussinesq equation, and prove that a wide range of asymptotic scenarios can happen. For example, for each $T>0$, $x_{0}\\in \\mathbb{R}$ and $\\delta \\in (0,1)$, we prove that there exist Schwartz class solutions $u(x,t)$ on $\\mathbb{R} \\times [0,T)$ such that $|u(x,t)| \\leq C \\frac{1+x^{2}}{(x-x_{0})^{2}}$ and $u(x_{0},t)\\asymp (T-t)^{-\\delta}$ as $t\\to T$. We also prove that for any $q\\in \\mathbb{N}$, $T>0$, $x_{0}\\in \\mathbb{R}$, $\\delta \\in (0,\\frac{1}{2})$, there exist Schwartz class solutions $u(x,t)$ on $\\mathbb{R} \\times [0,T)$ such that (i) $|\\partial_{x}^{q_{1}}\\partial_{t}^{q_{2}}u(x,t)|\\leq C$ for each $q_{1},q_{2}\\in \\mathbb{N}$ such that $q_{1}+2q_{2}\\leq q$, (ii) $|\\partial_{x}^{q_{1}}\\partial_{t}^{q_{2}}u(x,t)| \\leq C \\frac{1+|x|}{|x-x_{0}|}$ for each $q_{1},q_{2}\\in \\mathbb{N}$ such that $q_{1}+2q_{2}= q+1$, (iii) $|\\partial_{x}^{q_{1}}\\partial_{t}^{q_{2}}u(x_{0},t)| \\asymp (T-t)^{-\\delta}$ as $t\\to T$ for each $q_{1},q_{2}\\in \\mathbb{N}$ such that $q_{1}+2q_{2}= q+1$. In particular, when $q=0$, this result establishes the existence of wave-breaking solutions, i.e. solutions that remain bounded but whose $x$-derivative blows up in finite time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-20T17:55:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "boussinesq equation",
      "schwartz class solutions",
      "study blow-up solutions",
      "wide range",
      "finite time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}