{
  "id": "1603.06104",
  "version": "v1",
  "published": "2016-03-19T15:26:27.000Z",
  "updated": "2016-03-19T15:26:27.000Z",
  "title": "Continuity of attractors for a family of $C^1$ perturbations of the square",
  "authors": [
    "Pricila S. Barbosa",
    "Antônio L. Pereira",
    "Marcone C. Pereira"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider here the family of semilinear parabolic problems \\begin{equation*} \\begin{array}{rcl} \\left\\{ \\begin{array}{rcl} u_t(x,t)&=&\\Delta u(x,t) -au(x,t) + f(u(x,t)) ,\\,\\,\\ x \\in \\Omega_\\epsilon \\,\\,\\,\\mbox{and}\\,\\,\\,\\,\\,\\,t>0\\,, \\\\ \\displaystyle\\frac{\\partial u}{\\partial N}(x,t)&=&g(u(x,t)), \\,\\, x \\in \\partial\\Omega_\\epsilon \\,\\,\\,\\mbox{and}\\,\\,\\,\\,\\,\\,t>0\\,, \\end{array} \\right. \\end{array} \\end{equation*} where $ {\\Omega} $ is the unit square, $\\Omega_{\\epsilon}=h_{\\epsilon}(\\Omega)$ and $h_{\\epsilon}$ is a family of diffeomorphisms converging to the identity in the $C^1$-norm. We show that the problem is well posed for $\\epsilon>0$ sufficiently small in a suitable phase space, the associated semigroup has a global attractor $\\mathcal{A}_{\\epsilon}$ and the family $\\{\\mathcal{A}_{\\epsilon}\\}$ is continuous at $\\epsilon = 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-19T15:26:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B41",
      "35K20",
      "58D25"
    ],
    "keywords": [
      "continuity",
      "perturbations",
      "semilinear parabolic problems",
      "unit square",
      "suitable phase space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}