{
  "id": "1102.4062",
  "version": "v1",
  "published": "2011-02-20T10:47:19.000Z",
  "updated": "2011-02-20T10:47:19.000Z",
  "title": "Dimension of attractors and invariant sets in reaction diffusion equations",
  "authors": [
    "Martino Prizzi"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AP",
    "math.DS"
  ],
  "abstract": "Under fairly general assumptions, we prove that every compact invariant set $\\mathcal I$ of the semiflow generated by the semilinear reaction diffusion equation u_t+\\beta(x)u-\\Delta u&=f(x,u),&&(t,x)\\in[0,+\\infty[\\times\\Omega, u&=0,&&(t,x)\\in[0,+\\infty[\\times\\partial\\Omega} {equation*} in $H^1_0(\\Omega)$ has finite Hausdorff dimension. Here $\\Omega$ is an arbitrary, possibly unbounded, domain in $\\R^3$ and $f(x,u)$ is a nonlinearity of subcritical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\\mathcal I$ needs not to be an an attractor. If $\\Omega$ is regular, $f(x,u)$ is dissipative and $\\mathcal I$ is the global attractor, we give an explicit bound on the Hausdorff dimension of $\\mathcal I$ in terms of the structure parameter of the equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-02-20T10:47:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B41",
      "35K57"
    ],
    "keywords": [
      "semilinear reaction diffusion equation",
      "finite hausdorff dimension",
      "compact invariant set",
      "global attractor",
      "structure parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1102.4062P"
    }
  }
}