{
  "id": "1601.04906",
  "version": "v1",
  "published": "2016-01-19T13:07:49.000Z",
  "updated": "2016-01-19T13:07:49.000Z",
  "title": "Structure of $ω$-limit Sets for Almost-periodic Parabolic Equations on $S^1$ with Reflection Symmetry",
  "authors": [
    "Wenxian Shen",
    "Yi Wang",
    "Dun Zhou"
  ],
  "comment": "30 pages. arXiv admin note: text overlap with arXiv:1507.01709",
  "categories": [
    "math.DS"
  ],
  "abstract": "The structure of the $\\omega$-limit sets is thoroughly investigated for the skew-product semiflow which is generated by a scalar reaction-diffusion equation \\begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\\,\\,t>0,\\,x\\in S^{1}=\\mathbb{R}/2\\pi \\mathbb{Z}, \\end{equation*} where $f$ is uniformly almost periodic in $t$ and satisfies $f(t,u,u_x)=f(t,u,-u_x)$. We show that any $\\omega$-limit set $\\Omega$ contains at most two minimal sets. Moreover, any hyperbolic $\\omega$-limit set $\\Omega$ is a spatially-homogeneous $1$-cover of hull $H(f)$. When $\\dim V^c(\\Omega)=1$ ($V^c(\\Omega)$ is the center space associated with $\\Omega$), it is proved that either $\\Omega$ is a spatially-homogeneous, or $\\Omega$ is a spatially-inhomogeneous $1$-cover of $H(f)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-19T13:07:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "limit set",
      "almost-periodic parabolic equations",
      "reflection symmetry",
      "scalar reaction-diffusion equation",
      "skew-product semiflow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}