{
  "id": "2203.10911",
  "version": "v1",
  "published": "2022-03-21T12:05:03.000Z",
  "updated": "2022-03-21T12:05:03.000Z",
  "title": "Higher order evolution inequalities with nonlinear convolution terms",
  "authors": [
    "Roberta Filippucci",
    "Marius Ghergu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We are concerned with the study of existence and nonexistence of weak solutions to $$ \\begin{cases} &\\displaystyle \\frac{\\partial^k u}{\\partial t^k}+(-\\Delta)^m u\\geq (K\\ast |u|^p)|u|^q \\quad\\mbox{ in } \\mathbb R^N \\times \\mathbb R_+,\\\\[0.1in] &\\displaystyle \\frac{\\partial^i u}{\\partial t^i}(x,0) = u_i(x) \\,\\, \\text{ in } \\mathbb R^N,\\, 0\\leq i\\leq k-1,\\\\ \\end{cases} $$ where $N,k,m\\geq 1$ are positive integers, $p,q>0$ and $u_i\\in L^1_{\\rm loc}(\\mathbb{R}^N)$ for $0\\leq i\\leq k-1$. We assume that $K$ is a radial positive and continuous function which decreases in a neighbourhood of infinity. In the above problem, $K\\ast |u|^p$ denotes the standard convolution operation between $K(|x|)$ and $|u|^p$. We obtain necessary conditions on $N,m,k,p$ and $q$ such that the above problem has solutions. Our analysis emphasizes the role played by the sign of $\\displaystyle \\frac{\\partial^{k-1} u}{\\partial t^{k-1}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-21T12:05:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35G20",
      "35K30",
      "35L30",
      "35B45"
    ],
    "keywords": [
      "higher order evolution inequalities",
      "nonlinear convolution terms",
      "standard convolution operation",
      "weak solutions",
      "necessary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}