{
  "id": "1404.3429",
  "version": "v2",
  "published": "2014-04-13T21:24:00.000Z",
  "updated": "2015-04-30T14:35:30.000Z",
  "title": "Topological properties of invariant sets for strongly damped wave equation at resonance",
  "authors": [
    "Piotr Kokocki"
  ],
  "comment": "29 pages. arXiv admin note: substantial text overlap with arXiv:1310.6794",
  "categories": [
    "math.DS"
  ],
  "abstract": "We are interested in the following differential equation $\\ddot u(t) = -A u(t) - c A \\dot u(t) + \\lambda u(t) + F(u(t))$ where $c > 0$ is a damping factor, $A$ is a sectorial operator and $F$ is a continuous map. We consider the situation where the equation is at resonance at infinity, which means that $\\lambda$ is an eigenvalue of $A$ and $F$ is a bounded map. We provide geometrical conditions for the nonlinearity $F$ and use the Conley index methods to prove that the set, which is the union of the bounded orbits of this equation, is compact and non-empty.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-13T21:24:00.000Z",
      "title": "Conley index of invariant sets for strongly damped hyperbolic equations at resonance",
      "abstract": "We prove the existence of compact invariant sets for the strongly damped hyperbolic differential equation $\\ddot u(t) = -A u(t) - c A \\dot u(t) + \\lambda u(t) + F(t,u(t))$ being at resonance at infinity, that is, $A: X\\supset D(A)\\to X$ is a sectorial operator on a Banach space $X$ and $F:[0,+\\infty)\\times X^\\alpha\\to X$ is a continuous bounded map defined on the fractional space $X^\\alpha$, $c > 0$ is a damping factor and $\\lambda$ is an eigenvalue of $A$. More precisely, we provide geometrical assumptions for the nonlinearity $F$, that allow to obtain Conley index formulas stating that the Conley index for the associated semiflow, with respect to large ball, is equal to suspension of the sphere with dimension depending on what of the geometrical assumptions imposed on the nonlinearity is satisfied. It is also proved that the geometrical assumptions generalize well-known Landesman-Lazer conditions, and moreover, cover some other cases where the nonlinearity $F$ exhibits a lower order resonance at infinity.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-30T14:35:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B30",
      "35L10",
      "35P05"
    ],
    "keywords": [
      "strongly damped hyperbolic equations",
      "conley index",
      "invariant sets",
      "assumptions generalize well-known landesman-lazer",
      "damped hyperbolic differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.3429K"
    }
  }
}