{
  "id": "1910.09805",
  "version": "v1",
  "published": "2019-10-22T07:51:00.000Z",
  "updated": "2019-10-22T07:51:00.000Z",
  "title": "Inward/outward Energy Theory of Non-radial Solutions to 3D Semi-linear Wave Equation",
  "authors": [
    "Ruipeng Shen"
  ],
  "comment": "34 pages, 8 figures",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\\partial_t^2 u - \\Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $3\\leq p<5$. We generalize inward/outward energy theory and weighted Morawetz estimates for radial solutions to the non-radial case. As an application we show that if $3<p<5$ and $\\kappa>\\frac{5-p}{2}$, then the solution scatters as long as the initial data $(u_0,u_1)$ satisfy \\[ \\int_{{\\mathbb R}^3} (|x|^\\kappa+1)\\left(\\frac{1}{2}|\\nabla u_0|^2 + \\frac{1}{2}|u_1|^2+\\frac{1}{p+1}|u_0|^{p+1}\\right) dx < +\\infty. \\] If $p=3$, we can also prove the scattering result if initial data $(u_0,u_1)$ are contained in the critical Sobolev space and satisfy the inequality \\[ \\int_{{\\mathbb R}^3} |x|\\left(\\frac{1}{2}|\\nabla u_0|^2 + \\frac{1}{2}|u_1|^2+\\frac{1}{4}|u_0|^{p+1}\\right) dx < +\\infty. \\] These assumptions on the decay rate of initial data as $|x| \\rightarrow \\infty$ are weaker than previously known scattering results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-22T07:51:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L71",
      "35L05"
    ],
    "keywords": [
      "3d semi-linear wave equation",
      "non-radial solutions",
      "initial data",
      "scattering result",
      "generalize inward/outward energy theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}