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 + Au = f λ (u) on a Banach space X, where A is a sectorial operator, and λ ∈ R is the bifurcation parameter. Suppose the equation has a trivial solution branch {(0, λ) | λ ∈ 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, if S 0 = {0} is an isolated invariant set of Φ λ0 , then either there is a one-sided neighborhood I 1 of λ 0 such that Φ λ bifurcates a topological sphere S n−1 for each λ ∈ I 1 \ {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 6∈ K λ for each λ ∈ I 2 \ {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 ∂Ω 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.