{
  "id": "1009.4092",
  "version": "v1",
  "published": "2010-09-21T13:41:25.000Z",
  "updated": "2010-09-21T13:41:25.000Z",
  "title": "On the energy-minimizing steady states of a thin film equation",
  "authors": [
    "Almut Burchard",
    "Marina Chugunova",
    "Benjamin K. Stephens"
  ],
  "comment": "14 pages, 5 figures",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Steady states of the thin film equation $u_t+[u^3 (u_xxx + \\alpha^2 u_x -\\sin(x) )]_x=0$ are considered on the periodic domain $\\Omega = (-\\pi,\\pi)$. The equation defines a generalized gradient flow for an energy functional that controls the $H^1$-norm. The main result establishes that there exists for each given mass a unique nonnegative function of minimal energy. This minimizer is symmetric decreasing about $x=0$. For $\\alpha<1$ there is a critical value for the mass at which the minimizer has a touchdown zero. If the mass exceeds this value, the minimizer is strictly positive. Otherwise, it is supported on a proper subinterval of the domain and meets the dry region at zero contact angle. A second result explores the relation between strict positivity and exponential convergence for steady states. It is shown that positive minimizers are locally exponentially attractive, while the distance from a steady state with a dry region cannot decay faster than a power law.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-21T13:41:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K25",
      "35K35",
      "35Q35",
      "37L05",
      "76A20"
    ],
    "keywords": [
      "thin film equation",
      "energy-minimizing steady states",
      "dry region",
      "zero contact angle",
      "main result establishes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.4092B"
    }
  }
}