{
  "id": "math/0210030",
  "version": "v1",
  "published": "2002-10-02T17:40:01.000Z",
  "updated": "2002-10-02T17:40:01.000Z",
  "title": "Solution Representations for a Wave Equation with Weak Dissipation",
  "authors": [
    "Jens Wirth"
  ],
  "comment": "29 pages, 2 figures",
  "journal": "Math. Meth. Appl. Sci. 27/1 (2004) 101-124",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Cauchy problem for the weakly dissipative wave equation $$ \\bx v+\\frac\\mu{1+t}v_t=0, \\qquad x\\in\\R^n,\\quad t\\ge 0, $$ parameterized by $\\mu>0$, and prove a representation theorem for its solution using the theory of special functions. This representation is used to obtain $L_p$--$L_q$ estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the $L_2$ energy estimate we determine the part of the phase sp which is responsible for the decay rate. It will be shown that the situation d strongly on the value of $\\mu$ and that $\\mu=2$ is critical.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-10-02T17:40:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L05",
      "35L15",
      "35B45"
    ],
    "keywords": [
      "weak dissipation",
      "solution representations",
      "cauchy problem",
      "weakly dissipative wave equation",
      "decay rate"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....10030W"
    }
  }
}