{
  "id": "1504.01455",
  "version": "v1",
  "published": "2015-04-07T01:49:17.000Z",
  "updated": "2015-04-07T01:49:17.000Z",
  "title": "Regularity and geometric character of solution of a degenerate parabolic equation",
  "authors": [
    "Jiaqing Pan"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation $u_{t}=\\Delta{}u^{m}$. Our main objective is to improve the H$\\ddot{o}$lder estimate obtained by pioneers and then, to show the geometric characteristic of free boundary of degenerate parabolic equation. To be exact, the present work will show that: (1) the weak solution $u(x,t)\\in{}C^{\\alpha,\\frac{\\alpha}{2}}(\\mathbb{R}^{n}\\times\\mathbb{R}^{+})$, where $\\alpha\\in(0,1)$ when $m\\geq2$ and $\\alpha=1$ when $m\\in(1,2)$; (2) the surface $\\phi=(u(x,t))^{\\beta}$ is a complete Riemannian manifold, which is tangent to $\\mathbb{R}^{n}$ at the boundary of the positivity set of $u(x,t)$. (3) the function $(u(x,t))^{\\beta}$ is a classical solution to another degenerate parabolic equation if $ \\beta$ is large sufficiently; Moreover, some explicit expressions about the speed of propagation of $u(x,t)$ and the continuous dependence on the nonlinearity of the equation are obtained. Recalling the older H$\\ddot{o}$lder estimate ($u(x,t)\\in{}C^{\\alpha,\\frac{\\alpha}{2}}(\\mathbb{R}^{n}\\times\\mathbb{R}^{+})$ with $0<\\alpha<1$ for all $m>1$), we see our result (1) improves the older result and, based on this conclusion, we can obtain (2), which shows the geometric characteristic of free boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-07T01:49:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K15",
      "35K55",
      "35K65",
      "53C25"
    ],
    "keywords": [
      "degenerate parabolic equation",
      "regularity",
      "free boundary",
      "geometric characteristic",
      "lder estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150401455P"
    }
  }
}