{
  "id": "2304.11879",
  "version": "v1",
  "published": "2023-04-24T07:46:33.000Z",
  "updated": "2023-04-24T07:46:33.000Z",
  "title": "$L_p$-regularity theory for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity",
  "authors": [
    "Beom-Seok Han",
    "Jaeyun Yi"
  ],
  "comment": "21 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the existence, uniqueness, and regularity of the solution to the stochastic reaction-diffusion equation (SRDE) with colored noise $\\dot{F}$: $$ \\partial_t u = a^{ij}u_{x^ix^j} + b^i u_{x^i} + cu - \\bar{b} u^{1+\\beta} + \\xi u^{1+\\gamma}\\dot F,\\quad (t,x)\\in \\mathbb{R}_+\\times\\mathbb{R}^d; \\quad u(0,\\cdot) = u_0, $$ where $a^{ij},b^i,c, \\bar{b}$ and $\\xi$ are $C^2$ or $L_\\infty$ bounded random coefficients. Here $\\beta>0$ denotes the degree of the strong dissipativity and $\\gamma>0$ represents the degree of stochastic force. Under the reinforced Dalang's condition on $\\dot{F}$, we show the well-posedness of the SRDE provided $\\gamma < \\frac{\\kappa(\\beta +1)}{d+2}$ where $\\kappa>0$ is the constant related to $\\dot F$. Our result assures that strong dissipativity prevents the solution from blowing up. Moreover, we provide the maximal H\\\"older regularity of the solution in time and space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-24T07:46:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35R60"
    ],
    "keywords": [
      "stochastic reaction-diffusion equation",
      "super-linear multiplicative noise",
      "regularity theory",
      "strong dissipativity prevents",
      "result assures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}