{
  "id": "2312.15419",
  "version": "v1",
  "published": "2023-12-24T06:43:47.000Z",
  "updated": "2023-12-24T06:43:47.000Z",
  "title": "Gradient estimates on graphs with the $CDψ(n,-K)$condition",
  "authors": [
    "Yi Li",
    "Qianwei Zhang"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "This paper investigates gradient estimates on graphs satisfying the $CD\\psi(n,-K)$ condition with positive constants $n,K$, and concave $C^{1}$ functions $\\psi:(0,+\\infty)\\rightarrow\\mathbb{R}$. Our study focuses on gradient estimates for positive solutions of the heat equation $\\partial_{t}u=\\Delta u$. Additionally, the estimate is extended to a heat-type equation $\\partial_{t}u=\\Delta u+cu^{\\sigma}$, where $\\sigma$ is a constant and $c$ is a continuous function defined on $[0,+\\infty)$. Furthermore, we utilize these estimates to derive heat kernel bounds and Harnack inequalities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-24T06:43:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gradient estimates",
      "derive heat kernel bounds",
      "harnack inequalities",
      "study focuses",
      "heat equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}