{
  "id": "1403.4526",
  "version": "v3",
  "published": "2014-03-18T16:39:26.000Z",
  "updated": "2014-08-20T20:20:57.000Z",
  "title": "On convergence of solutions to equilibria for fully nonlinear parabolic problems with nonlinear boundary conditions",
  "authors": [
    "Helmut Abels",
    "Nasrin Arab",
    "Harald Garcke"
  ],
  "comment": "49 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Convergence to stationary solutions in fully nonlinear parabolic systems with general nonlinear boundary conditions is shown in situations where the set of stationary solutions creates a $C^2$-manifold of finite dimension which is normally stable. We apply the parabolic H\\\"older setting which allows to deal with nonlocal terms including highest order point evaluation. In this direction some theorems concerning the linearized systems is also extended. As an application of our main result we prove that the lens-shaped networks generated by circular arcs are stable under the surface diffusion flow.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-29T17:27:25.000Z",
      "abstract": "We show convergence of solutions to equilibria for fully nonlinear parabolic evolution systems with nonlinear boundary conditions in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^2$-manifold which is normally stable. We apply the parabolic H\\\"older setting for such problems which allows to deal with nonlocal terms. As an illustration of the scope of our result we show that the lens-shaped networks generated by circular arcs are stable under the surface diffusion flow.",
      "comment": "48 pages. arXiv admin note: text overlap with arXiv:0807.1539 by other authors",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-08-20T20:20:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35B35",
      "37L15",
      "53C44",
      "35K50",
      "37L10",
      "35B65"
    ],
    "keywords": [
      "nonlinear boundary conditions",
      "fully nonlinear parabolic problems",
      "equilibria",
      "convergence",
      "fully nonlinear parabolic evolution systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.4526A"
    }
  }
}