{
  "id": "2408.02893",
  "version": "v1",
  "published": "2024-08-06T01:50:54.000Z",
  "updated": "2024-08-06T01:50:54.000Z",
  "title": "A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source",
  "authors": [
    "Wenguo Liang",
    "Zhengce Zhang"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation $u_t-\\Delta u=u^p+M|\\nabla u|^q$ in $\\Omega\\times I\\subset \\R^N\\times \\R$, where $M>0$, and $p,q>1$. We first establish the local pointwise gradient estimates when $q$ is subcritical, critical and supercritical with respect to $p$. With these estimates, we can prove the parabolic Liouville-type theorems for time-decreasing ancient solutions. Next, we use Gidas-Spruck type integral methods to prove the Liouville-type theorem for the entire solutions when $q$ is critical. Finally, as an application of the Liouville-type theorem, we use the doubling lemma to derive universal priori estimates for local solutions of parabolic equations with general nonlinearities. Our approach relies on a parabolic differential inequality containing a suitable auxiliary function rather than Keller-Osserman type inequality, which allows us to generalize and extend the partial results of the elliptic equation (Bidaut-V\\'{e}ron, Garcia-Huidobro and V\\'{e}ron (2020) \\cite{veron-sum}) to the parabolic case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-06T01:50:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville-type theorem",
      "semilinear parabolic equations",
      "nonlinear gradient source",
      "gidas-spruck type integral methods",
      "derive universal priori estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}