{
  "id": "1804.10156",
  "version": "v1",
  "published": "2018-04-26T16:32:02.000Z",
  "updated": "2018-04-26T16:32:02.000Z",
  "title": "A non-autonomous scalar one-dimensional dissipative parabolic problem: The description of the dynamics",
  "authors": [
    "Rita de Cássia D. S. Broche",
    "Alexandre N. Carvalho",
    "José Valero"
  ],
  "comment": "32 pages, 04 figures",
  "categories": [
    "math.DS",
    "math.AP"
  ],
  "abstract": "The purpose of this paper is to give a characterization of the structure of non-autonomous attractors of the problem $u_t= u_{xx} + \\lambda u - \\beta(t)u^3$ when the parameter $\\lambda > 0$ varies. Also, we answer a question proposed in [11], concerning the complete description of the structure of the pullback attractor of the problem when $1<\\lambda <4$ and, more generally, for $\\lambda \\neq N^2$, $2 \\leq N \\in \\mathbb{N}$. We construct global bounded solutions , \"non-autonomous equilibria\", connections between the trivial solution these \"non-autonomous equilibria\" and characterize the $\\alpha$-limit and $\\omega$-limit set of global bounded solutions. As a consequence, we show that the global attractor of the associated skew-product flow has a gradient structure. The structure of the related pullback an uniform attractors are derived from that.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-26T16:32:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K91",
      "35B32",
      "35B41",
      "37L30"
    ],
    "keywords": [
      "scalar one-dimensional dissipative parabolic problem",
      "non-autonomous scalar one-dimensional dissipative parabolic",
      "description",
      "global bounded solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}