{
  "id": "2407.20133",
  "version": "v1",
  "published": "2024-07-29T16:01:43.000Z",
  "updated": "2024-07-29T16:01:43.000Z",
  "title": "Global gradient estimates for solutions of parabolic equations with nonstandard growth",
  "authors": [
    "Rakesh Arora",
    "Sergey Shmarev"
  ],
  "comment": "30 pages. Comments are welcome",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study how the smoothness of the initial datum and the free term affect the global regularity properties of solutions to the Dirichlet problem for the class of parabolic equations of $p(x,t)$-Laplace type %with nonlinear sources depending on the solution and its gradient: \\[ u_t-\\Delta_{p(\\cdot)}u=f(z)+F(z,u,\\nabla u),\\quad z=(x,t)\\in Q_T=\\Omega\\times (0,T), \\] with the nonlinear source $F(z,u,\\nabla u)=a(z)|u|^{q(z)-2}u+|\\nabla u|^{s(z)-2}(\\vec c,\\nabla u)$. It is proven the existence of a solution such that if $|\\nabla u(x,0)|\\in L^r(\\Omega)$ for some $r\\geq \\max\\{2,\\max p(z)\\}$, then the gradient preserves the initial order of integrability in time, gains global higher integrability, and the solution acquires the second-order regularity in the following sense: \\[ \\text{$|\\nabla u(x,t)|\\in L^r(\\Omega)$ for a.e. $t \\in (0,T)$}, \\qquad \\text{$|\\nabla u|^{p(z)+\\rho+r-2} \\in L^1(Q_T)$ for any $\\rho \\in \\left(0, \\frac{4}{N+2}\\right)$}, \\] and \\[ |\\nabla u|^{\\frac{p(z)+r}{2}-2}\\nabla u\\in L^2(0,T;W^{1,2}(\\Omega))^N. \\] The exponent $r$ is arbitrary and independent of $p(z)$ if $f\\in L^{N+2}(Q_T)$, while for $f\\in L^\\sigma(Q_T)$ with $\\sigma \\in (2,N+2)$ the exponent $r$ belongs to a bounded interval whose endpoints are defined by $\\max p(z)$, $\\min p(z)$, $N$, and $\\sigma$. An integration by parts formula is also proven, which is of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-29T16:01:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K65",
      "35K67",
      "35B65",
      "35K55",
      "35K99"
    ],
    "keywords": [
      "global gradient estimates",
      "parabolic equations",
      "nonstandard growth",
      "gains global higher integrability",
      "nonlinear source"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}