{
  "id": "2108.11473",
  "version": "v1",
  "published": "2021-08-25T20:59:19.000Z",
  "updated": "2021-08-25T20:59:19.000Z",
  "title": "Interpolating the Stochastic Heat and Wave Equations with Time-independent Noise: Solvability and Exact Asymptotics",
  "authors": [
    "Le Chen",
    "Nicholas Eisenberg"
  ],
  "comment": "41 pages, 6 figures",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global $L^p(\\Omega)$-solution exits for all $p\\ge 2$. In this case, we derive exact moment asymptotics following the same strategy in a recent work by Balan et al [1]. In the case when there exits only a local solution, we determine the precise deterministic time, $T_2$, before which a unique $L^2(\\Omega)$-solution exits, but after which the series corresponding to the $L^2(\\Omega)$ moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-25T20:59:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H07",
      "37H15"
    ],
    "keywords": [
      "wave equations",
      "stochastic heat",
      "time-independent noise",
      "exact asymptotics",
      "solvability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}