{
  "id": "1602.01279",
  "version": "v1",
  "published": "2016-02-03T12:26:46.000Z",
  "updated": "2016-02-03T12:26:46.000Z",
  "title": "Attractors for Damped Semilinear Wave Equations with Singularly Perturbed Acoustic Boundary Conditions",
  "authors": [
    "Joseph L. Shomberg"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1503.01821",
  "categories": [
    "math.AP"
  ],
  "abstract": "Under consideration is the damped semilinear wave equation \\[ u_{tt}+u_t-\\Delta u+u+f(u)=0 \\] in a bounded domain $\\Omega$ in $\\mathbb{R}^3$ subject to an acoustic boundary condition with a singular perturbation, which we term \"massless acoustic perturbation,\" \\[ \\ep\\delta_{tt}+\\delta_t+\\delta = -u_t\\quad\\text{for}\\quad \\ep\\in[0,1]. \\] By adapting earlier work by S. Frigeri, we prove the existence of a family of global attractors for each $\\ep\\in[0,1]$. We also establish the optimal regularity for the global attractors, as well as the existence of an exponential attractor, for each $\\ep\\in[0,1].$ The later result insures the global attractors possess finite (fractal) dimension, however, we cannot yet guarantee that this dimension is independent of the perturbation parameter $\\ep.$ The family of global attractors are upper-semicontinuous with respect to the perturbation parameter $\\ep$, a result which follows by an application of a new abstract result also contained in this article. Finally, we show that it is possible to obtain the global attractors using weaker assumptions on the nonlinear term $f$, however, in that case, the optimal regularity, the finite dimensionality, and the upper-semicontinuity of the global attractors does not necessarily hold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-03T12:26:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B25",
      "35B41",
      "35L20",
      "35L71",
      "35Q40",
      "35Q70"
    ],
    "keywords": [
      "damped semilinear wave equation",
      "singularly perturbed acoustic boundary conditions",
      "global attractors",
      "perturbation parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160201279S"
    }
  }
}