{
  "id": "1902.01680",
  "version": "v1",
  "published": "2019-02-05T14:01:26.000Z",
  "updated": "2019-02-05T14:01:26.000Z",
  "title": "Polynomial bounds on the Sobolev norms of the solutions of the nonlinear wave equation with time dependent potential",
  "authors": [
    "Vesselin Petkov",
    "Nikolay Tzvetkov"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the Cauchy problem for the nonlinear wave equation $u_{tt} - \\Delta_x u +q(t, x) u + u^3 = 0$ with smooth and periodic in time potential $q(t, x) \\geq 0$ having compact support with respect to $x$. The linear equation without the nonlinear term $u^3$ may have solutions with exponentially increasing as $ t \\to \\infty$ norm $H^1({\\mathbb R}^3_x)$. In [2] it was established that adding the nonlinear term $u^3$ the $H^1({\\mathbb R}^3_x)$ of the solution is polynomially bounded for every choice of $q$. In this paper we show that $H^k({\\mathbb R}^3_x)$ norm of this global solution is also polynomially bounded. To prove this we apply a different argument based on the analysis of a sequence $\\{Y_k(n\\tau_k)\\}_{n = 0}^{\\infty}$ with suitably defined energy norm $Y_k(t)$ and $0 < \\tau_k <1.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-05T14:01:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L71",
      "35L15"
    ],
    "keywords": [
      "nonlinear wave equation",
      "time dependent potential",
      "polynomial bounds",
      "sobolev norms",
      "nonlinear term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}