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 $u_t+A u=f_λ(u)$ on a Banach space $X$, where $A$ is a sectorial operator, and $λ\in R$ is the bifurcation parameter. Suppose the equation has a trivial solution branch $\{(0,λ):\,\,λ\in R\}$. Denote $Φ_λ$ 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 $λ=λ_0$ is nonzero and moreover, $S_0=\{0\}$ is an isolated invariant set of $Φ_{λ_0}$, then either there is a one-sided neighborhood $I_1$ of $λ_0$ such that $Φ_λ$ bifurcates a topological sphere $\mathbb{S}^{n-1}$ for each $λ\in I_1\setminus\{λ_0\}$, or there is a two-sided neighborhood $I_2$ of $λ_0$ such that the system $Φ_λ$ bifurcates from the trivial solution an isolated nonempty compact invariant set $K_λ$ with $0\not\in K_λ$ for each $λ\in I_2\setminus\{λ_0\}$. 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 $Ω$ of the bifurcation point $(0,λ_0)$, the connected bifurcation branch $Γ$ from $(0,λ_0)$ either meets the boundary $\partialΩ$ of $Ω$, or meets another bifurcation point $(0,λ_1)$. 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.