{
  "id": "1402.6365",
  "version": "v2",
  "published": "2014-02-25T22:59:26.000Z",
  "updated": "2014-10-11T09:05:15.000Z",
  "title": "Impacts of Noise on a Class of Partial Differential Equations",
  "authors": [
    "Guangying Lv",
    "Jinqiao Duan"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with effects of noise on the solutions of partial differential equations. We first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic parabolic equations. Then, we prove that noise could induce singularities (finite time blow up of solutions). Finally, we show that a stochastic Allen-Cahn equation does not have finite time singularities and the unique solution exists globally.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-25T22:59:26.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-11T09:05:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K20",
      "60H15",
      "60H40"
    ],
    "keywords": [
      "partial differential equations",
      "stochastic allen-cahn equation",
      "stochastic parabolic equations",
      "finite time blow",
      "finite time singularities"
    ],
    "publication": {
      "doi": "10.1016/j.jde.2014.12.002",
      "journal": "Journal of Differential Equations",
      "year": 2015,
      "month": "Mar",
      "volume": 258,
      "number": 6,
      "pages": 2196
    },
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JDE...258.2196L"
    }
  }
}