{
  "id": "2106.02879",
  "version": "v1",
  "published": "2021-06-05T12:10:29.000Z",
  "updated": "2021-06-05T12:10:29.000Z",
  "title": "Global well-posedness to stochastic reaction-diffusion equations on the real line $\\mathbb{R}$ with superlinear drifts driven by multiplicative space-time white noise",
  "authors": [
    "Shijie Shang",
    "Tusheng Zhang"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the stochastic reaction-diffusion equation with logarithmic nonlinearity driven by space-time white noise: \\begin{align}\\label{1.a} \\left\\{ \\begin{aligned} & \\mathrm{d}u(t,x) = \\frac{1}{2}\\Delta u(t,x)\\,\\mathrm{d}t+ b(u(t,x)) \\,\\mathrm{d}t \\nonumber\\\\ & ~~~~~~~~~~~~~~~~ + \\sigma(u(t,x)) \\,W(\\mathrm{d}t,\\mathrm{d}x), \\ t>0, x\\in I , \\\\ & u(0,x)=u_0(x), \\quad x\\in I .\\nonumber \\end{aligned} \\right. \\end{align} When $I$ is a compact interval, say $I=[0,1]$, the well-posedness of the above equation was established in [DKZ] (Ann. Prob. 47:1,2019). The case where $I=\\mathbb{R}$ was left open. The essential obstacle is caused by the explosion of the supremum norm of the solution, $\\sup_{x\\in\\mathbb{R}}|u(t,x)|=\\infty$, making the usual truncation procedure invalid. In this paper, we prove that there exists a unique global solution to the stochastic reaction-diffusion equation on the whole real line $\\mathbb{R}$ with logarithmic nonlinearity. Because of the nature of the nonlinearity, to get the uniqueness, we are forced to work with the first order moment of the solutions on the space $C_{tem}(\\mathbb{R})$ with a specially designed norm $$\\sup_{t\\leq T, x\\in\\mathbb{R}}\\left(|u(t,x)|e^{-\\lambda |x|e^{\\beta t}}\\right),$$ where, unlike the usual norm in $C_{tem}(\\mathbb{R})$, the exponent also depends on time $t$ in a particular way. Our approach depends heavily on the new, precise lower order moment estimates of the stochastic convolution and a new type of Gronwall's inequalities we obtained, which are of interest on their own right.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-05T12:10:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35R60"
    ],
    "keywords": [
      "stochastic reaction-diffusion equation",
      "multiplicative space-time white noise",
      "superlinear drifts driven",
      "real line",
      "global well-posedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}