{
  "id": "1102.1259",
  "version": "v1",
  "published": "2011-02-07T10:12:21.000Z",
  "updated": "2011-02-07T10:12:21.000Z",
  "title": "Buckling and longterm dynamics of a nonlinear model for the extensible beam",
  "authors": [
    "Ivana Bochicchio",
    "Elena Vuk"
  ],
  "comment": "14 pages, 2 figures",
  "journal": "Mathematical and Computer Modelling, Vol. 51 (2010), pp. 833-846",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "This work is focused on the longtime behavior of a non linear evolution problem describing the vibrations of an extensible elastic homogeneous beam resting on a viscoelastic foundation with stiffness k>0 and positive damping constant. Buckling of solutions occurs as the axial load exceeds the first critical value, \\beta_c, which turns out to increase piecewise-linearly with k. Under hinged boundary conditions and for a general axial load P, the existence of a global attractor, along with its characterization, is proved by exploiting a previous result on the extensible viscoelastic beam. As P<\\beta_c, the stability of the straight position is shown for all values of k. But, unlike the case with null stiffness, the exponential decay of the related energy is proved if P<\\bar\\beta(k), where \\bar\\beta(k) < \\beta_c(k) and the equality holds only for small values of k.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-07T10:12:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear model",
      "longterm dynamics",
      "extensible beam",
      "elastic homogeneous beam resting",
      "general axial load"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.1259B"
    }
  }
}