{
  "id": "1809.01690",
  "version": "v1",
  "published": "2018-09-05T18:49:00.000Z",
  "updated": "2018-09-05T18:49:00.000Z",
  "title": "Continuity of attractors for $\\mathcal{C}^1$ perturbations of a smooth domain",
  "authors": [
    "Antônio L. Pereira"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider a family of semilinear parabolic problems with nonlinear boundary conditions \\[ \\left\\{ \\begin{aligned} 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{aligned} \\right. \\] where $\\Omega_0 \\subset \\R^n$ is a smooth (at least $\\mathcal{C}^2$) domain , $\\Omega_{\\epsilon} = h_{\\epsilon}(\\Omega_0)$ and $h_{\\epsilon}$ is a family of diffeomorphisms converging to the identity in the $\\mathcal{C}^1$-norm. Assuming suitable regularity and dissipative conditions for the nonlinearites, we show that the problem is well posed for $\\epsilon>0$ sufficiently small in a suitable scale of fractional spaces, 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": "2018-09-05T18:49:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B41",
      "35K20",
      "58D25"
    ],
    "keywords": [
      "smooth domain",
      "continuity",
      "perturbations",
      "nonlinear boundary conditions",
      "semilinear parabolic problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}