{
  "id": "1605.01323",
  "version": "v1",
  "published": "2016-05-04T15:48:18.000Z",
  "updated": "2016-05-04T15:48:18.000Z",
  "title": "Large time behaviour for the non-linear fractional stochastic heat equation",
  "authors": [
    "Mohammud Foondun",
    "Ngartelbaye Guerngar",
    "Erkan Nane"
  ],
  "comment": "17 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "We consider the following fractional stochastic partial differential equation on $D$ an open bounded subset of $\\mathbb{R}^d$ for $d\\geq 1$ \\begin{equation*} \\partial_t u_t(x)= -\\frac{1}{2}(-\\Delta)^{\\frac{\\alpha}{2}} u_t(x)+ \\xi\\sigma (u_t(x)) \\dot W(t,x) \\ \\ \\text{for} \\ \\alpha \\in (0,2] \\end{equation*} where the fractional Laplacian is the infinitesimal generator of a symmetric $\\alpha$-stable process in $\\mathbb{R}^d$ and can be written in the form \\begin{align*} -(-\\Delta)^{\\frac{\\alpha}{2}} u(x)= c \\lim\\limits_{\\varepsilon \\searrow 0}\\int_{\\{y\\in\\mathbb{R}^d: |y-x|>\\varepsilon\\}} (u(y)-u(x))\\frac{dy}{|y-x|^{d+\\alpha}} \\end{align*} for some constant $c= c(\\alpha, d)$. $\\xi$ is a parameter in $\\mathbb{R}$, $\\sigma$ is a Lipschitz continuous function and $\\dot W(t,x)$ is a Gaussian noise white in time and white or coloured in space. We show that under Dirichlet conditions, in the long run, the $p^{\\text{th}}$-moment of the solution grows exponentially fast for large values of $\\xi$. However when $\\xi$ is very small we observe eventually an exponential decay of the $p^{\\text{th}}$-moment of this same solution. It is important to recall here that the case $\\alpha=2$ is provided in \\cite{f}. Therefore we will consider $\\alpha\\in(0,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-04T15:48:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-linear fractional stochastic heat equation",
      "large time behaviour",
      "fractional stochastic partial differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}