{
  "id": "2204.08231",
  "version": "v1",
  "published": "2022-04-18T09:34:54.000Z",
  "updated": "2022-04-18T09:34:54.000Z",
  "title": "Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order",
  "authors": [
    "Jonas Jansen",
    "Christina Lienstromberg",
    "Katerina Nik"
  ],
  "comment": "25 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the $H^1(\\Omega)$-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate $1/t^{1/\\beta}$ for some $\\beta > 0$. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in $H^1(\\Omega)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-18T09:34:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76A05",
      "76A20",
      "35B40",
      "35Q35",
      "35K35",
      "35K65"
    ],
    "keywords": [
      "quasilinear doubly degenerate parabolic equations",
      "long-time behaviour",
      "steady state",
      "higher order",
      "quasilinear doubly degenerate parabolic problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}