Local and global dynamic bifurcations of nonlinear evolution equations

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.