{
  "id": "2307.16197",
  "version": "v1",
  "published": "2023-07-30T10:33:11.000Z",
  "updated": "2023-07-30T10:33:11.000Z",
  "title": "The Cauchy problem associated to the logarithmic Laplacian with an application to the fundamental solution",
  "authors": [
    "Huyuan Chen",
    "Laurent Véron"
  ],
  "comment": "54 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\lnlap$ be the logarithmic Laplacian operator with Fourier symbol $2\\ln |\\zeta|$, we study the expression of the diffusion kernel which is associated to the equation $$\\partial_tu+ \\lnlap u=0 \\ \\ {\\rm in}\\ \\, (0,\\tfrac N2) \\times \\R^N,\\quad\\quad u(0,\\cdot)=0\\ \\ {\\rm in}\\ \\, \\R^N\\setminus \\{0\\}.$$ We apply our results to give a classification of the solutions of $$\\left\\{ \\arraycolsep=1pt \\begin{array}{lll} \\displaystyle \\partial_tu+ \\lnlap u=0\\quad \\ &{\\rm in}\\ \\ (0,T)\\times \\R^N\\\\[2.5mm] \\phantom{ \\lnlap \\ \\, } \\displaystyle u(0,\\cdot)=f\\quad \\ &{\\rm{in}}\\ \\ \\R^N \\end{array} \\right. $$ and obtain an expression of the fundamental solution of the associated stationary equation in $\\R^N$, and of the fundamental solution in a bounded domain, i.e. $$\\lnlap u=k\\delta_0\\quad {\\rm in}\\ \\ \\cD'(\\Omega)\\quad \\text{such that }\\,u=0\\quad {\\rm in}\\ \\ \\R^N\\setminus\\Omega. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-30T10:33:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K05",
      "35A08"
    ],
    "keywords": [
      "fundamental solution",
      "cauchy problem",
      "application",
      "logarithmic laplacian operator",
      "fourier symbol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}