{
  "id": "1511.01885",
  "version": "v1",
  "published": "2015-11-05T20:37:25.000Z",
  "updated": "2015-11-05T20:37:25.000Z",
  "title": "Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity",
  "authors": [
    "Johannes Lankeit"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove convergence of positive solutions to \\[ u_t = u\\Delta u + u\\int_{\\Omega} |\\nabla u|^2, \\qquad u\\rvert_{\\partial\\Omega} =0, \\qquad u(\\cdot,0)=u_0 \\] in a bounded domain $\\Omega\\subset \\mathbb{R}^n$, $n\\ge 1$, with smooth boundary in the case of $\\int_\\Omega u_0=1$ and identify the $W_0^{1,2}(\\Omega)$-limit of $u(t)$ as $t\\to \\infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\\int_\\Omega u_0<1$ and $\\int_\\Omega u_0>1$ which are known to result in convergence to zero or blow-up in finite time, respectively. The proof is based on a monotonicity property of $\\int_{\\Omega} |\\nabla u|^2$ along trajectories and the analysis of an associated constrained minimization problem. Keywords: degenerate diffusion, nonlocal nonlinearity, long-term behaviour",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-05T20:37:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B40",
      "35K55",
      "35K65"
    ],
    "keywords": [
      "degenerate parabolic equation",
      "unit mass solutions",
      "nonlocal gradient nonlinearity",
      "equilibration",
      "degenerate diffusion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}