{
  "id": "2203.11616",
  "version": "v1",
  "published": "2022-03-22T11:11:12.000Z",
  "updated": "2022-03-22T11:11:12.000Z",
  "title": "Deterministic KPZ equations with nonlocal \"gradient terms\"",
  "authors": [
    "Boumediene Abdellaoui",
    "Antonio J. Fernández",
    "Tommaso Leonori",
    "Abdelbadie Younes"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The main goal of this paper is to prove existence and non-existence results for deterministic Kardar-Parisi-Zhang type equations involving non-local \"gradient terms\". More precisely, let $\\Omega \\subset \\mathbb{R}^N$, $N \\geq 2$, be a bounded domain with boundary $\\partial \\Omega$ of class $C^2$. For $s \\in (0,1)$, we consider problems of the form \\[ \\tag{KPZ} \\left\\{ \\begin{aligned} (-\\Delta)^s u & = \\mu(x) |\\mathbb{D}(u)|^q + \\lambda f(x), \\quad && \\mbox{ in } \\Omega,\\\\ u & = 0, && \\mbox{ in } \\mathbb{R}^N \\setminus \\Omega, \\end{aligned} \\right. \\] where $q > 1$ and $\\lambda > 0$ are real parameters, $f$ belongs to a suitable Lebesgue space, $\\mu$ belongs to $L^{\\infty}(\\Omega)$ and $\\mathbb{D}$ represents a nonlocal \"gradient term\". Depending on the size of $\\lambda > 0$, we derive existence and non-existence results. In particular, we solve several open problems posed in [4, Section 6] and [2, Section 7].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-22T11:11:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gradient term",
      "deterministic kpz equations",
      "non-existence results",
      "deterministic kardar-parisi-zhang type equations",
      "main goal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}