{
  "id": "1707.02136",
  "version": "v1",
  "published": "2017-07-07T12:05:42.000Z",
  "updated": "2017-07-07T12:05:42.000Z",
  "title": "Final value problems for parabolic differential equations and their well-posedness",
  "authors": [
    "Ann-Eva Christensen",
    "Jon Johnsen"
  ],
  "comment": "37 pages. Preprint version, submitted 7 July 2017",
  "categories": [
    "math.AP"
  ],
  "abstract": "This article bridges a gap in the understanding of parabolic final value problems, since a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given a certain graph norm, and it is defined by a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax--Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichl\\'et data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-07T12:05:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "47D06"
    ],
    "keywords": [
      "parabolic differential equations",
      "well-posedness",
      "compatibility condition",
      "parabolic final value problems",
      "improper bochner integral"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}