{
  "id": "1702.07591",
  "version": "v1",
  "published": "2017-02-24T14:08:59.000Z",
  "updated": "2017-02-24T14:08:59.000Z",
  "title": "On the maximum principle for a time-fractional diffusion equation",
  "authors": [
    "Yuri Luchko",
    "Masahiro Yamamoto"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we discuss the maximum principle for a time-fractional diffusion equation $$ \\partial_t^{\\alpha} u(x,t) = \\sum_{i,j=1}^n \\partial_i(a_{ij}(x)\\partial_j u(x,t)) + c(x)u(x,t) + F(x,t),\\ t>0,\\ x \\in \\Omega \\subset {\\mathbb R}^n $$ with the Caputo time-derivative of the order $\\alpha \\in (0,1)$ in the case of the homogeneous Dirichlet boundary condition. Compared to the already published results, our findings have two important special features. First, we derive a maximum principle for a suitably defined weak solution in the fractional Sobolev spaces, not for the strong solution. Second, for the non-negative source functions $F = F(x,t)$ we prove the non-negativity of the weak solution to the problem under consideration without any restrictions on the sign of the coefficient $c=c(x)$ by the derivative of order zero in the spatial differential operator. Moreover, we prove the monotonicity of the solution with respect to the coefficient $c=c(x)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-24T14:08:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A33",
      "35A05",
      "35B30",
      "35B50",
      "35C05",
      "35E05",
      "35L05",
      "45K05",
      "60E99"
    ],
    "keywords": [
      "time-fractional diffusion equation",
      "maximum principle",
      "spatial differential operator",
      "important special features",
      "homogeneous dirichlet boundary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}