{
  "id": "2307.16665",
  "version": "v1",
  "published": "2023-07-31T13:45:39.000Z",
  "updated": "2023-07-31T13:45:39.000Z",
  "title": "Simultaneous determination of initial value and source term for time-fractional wave-diffusion equations",
  "authors": [
    "Paola Loreti",
    "Daniela Sforza",
    "Masahiro Yamamoto"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider initial boundary value problems for time fractional diffusion-wave equations: $$ d_t^{\\alpha} u = -Au + \\mu(t)f(x) $$ in a bounded domain where $\\mu(t)f(x)$ describes a source and $\\alpha \\in (0,1) \\cup (1,2)$, and $-A$ is a symmetric ellitpic operator with repect to the spatial variable $x$. We assume that $\\mu(t) = 0$ for $t > T$:some time and choose $T_2>T_1>T$. We prove the uniqueness in simultaneously determining $f$ in $\\Omega$, $\\mu$ in $(0,T)$, and initial values of $u$ by data $u\\vert_{\\omega\\times (T_1,T_2)}$, provided that the order $\\alpha$ does not belong to a countably infinite set in $(0,1) \\cup (1,2)$ which is characterized by $\\mu$. The proof is based on the asymptotic behavior of the Mittag-Leffler functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-31T13:45:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30",
      "35R11",
      "35R25"
    ],
    "keywords": [
      "time-fractional wave-diffusion equations",
      "initial value",
      "source term",
      "simultaneous determination",
      "initial boundary value problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}