{
  "id": "1612.08128",
  "version": "v1",
  "published": "2016-12-24T03:31:41.000Z",
  "updated": "2016-12-24T03:31:41.000Z",
  "title": "Local and Global Dynamic Bifurcations of Nonlinear Evolution Equations",
  "authors": [
    "Desheng Li",
    "Zhi-Qiang Wang"
  ],
  "comment": "42 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We present new local and global dynamic bifurcation results for nonlinear evolution equations of the form $u_t+A u=f_\\lambda(u)$ on a Banach space $X$, where $A$ is a sectorial operator, and $\\lambda\\in R$ is the bifurcation parameter. Suppose the equation has a trivial solution branch $\\{(0,\\lambda):\\,\\,\\lambda\\in R\\}$. Denote $\\Phi_\\lambda$ the local semiflow generated by the initial value problem of the equation. It is shown that if the crossing number $n$ at a bifurcation value $\\lambda=\\lambda_0$ is nonzero and moreover, $S_0=\\{0\\}$ is an isolated invariant set of $\\Phi_{\\lambda_0}$, then either there is a one-sided neighborhood $I_1$ of $\\lambda_0$ such that $\\Phi_\\lambda$ bifurcates a topological sphere $\\mathbb{S}^{n-1}$ for each $\\lambda\\in I_1\\setminus\\{\\lambda_0\\}$, or there is a two-sided neighborhood $I_2$ of $\\lambda_0$ such that the system $\\Phi_\\lambda$ bifurcates from the trivial solution an isolated nonempty compact invariant set $K_\\lambda$ with $0\\not\\in K_\\lambda$ for each $\\lambda\\in I_2\\setminus\\{\\lambda_0\\}$. We also prove that the bifurcating invariant set has nontrivial Conley index. Building upon this fact we establish a global dynamical bifurcation theorem. Roughly speaking, we prove that for any given neighborhood $\\Omega$ of the bifurcation point $(0,\\lambda_0)$, the connected bifurcation branch $\\Gamma$ from $(0,\\lambda_0)$ either meets the boundary $\\partial\\Omega$ of $\\Omega$, or meets another bifurcation point $(0,\\lambda_1)$. This result extends the well-known Rabinowitz's Global Bifurcation Theorem to the setting of dynamic bifurcations of evolution equations without requiring the crossing number to be odd. As an illustration example, we consider the well-known Cahn-Hilliard equation. Some global features on dynamical bifurcations of the equation are discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-24T03:31:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C23",
      "34K18",
      "35B32",
      "37G99"
    ],
    "keywords": [
      "nonlinear evolution equations",
      "global dynamic bifurcation",
      "nonempty compact invariant set",
      "well-known rabinowitzs global bifurcation theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}