{
  "id": "1803.06250",
  "version": "v1",
  "published": "2018-03-16T14:38:29.000Z",
  "updated": "2018-03-16T14:38:29.000Z",
  "title": "On the nonlinear wave equation with time periodic potential",
  "authors": [
    "Vesselin Petkov Nikolay Tzvetkov"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "It is known that for some time periodic potentials $q(t, x) \\geq 0$ having compact support with respect to $x$ some solutions of the Cauchy problem for the wave equation $\\partial_t^2 u - \\Delta_x u + q(t,x)u = 0$ have exponentially increasing energy as $t \\to \\infty$. We show that if one adds a nonlinear defocusing interaction $|u|^ru, 2\\leq r < 4,$ then the solution of the nonlinear wave equation exists for all $t \\in {\\mathbb R}$ and its energy is polynomially bounded as $t \\to \\infty$ for every choice of $q$. Moreover, we prove that the zero solution of the nonlinear wave equation is instable if the corresponding linear equation has the property mentioned above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-16T14:38:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L71",
      "35L15"
    ],
    "keywords": [
      "nonlinear wave equation",
      "time periodic potential",
      "compact support",
      "corresponding linear equation",
      "zero solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}