Document Type

Article

Journal/Book Title/Conference

Indiana University Mathematics Journal

Volume

67

Issue

2

Publisher

IUMJ

Publication Date

2018

First Page

1

Last Page

39

Abstract

We present new local and global dynamic bifurcation results for nonlinear evolution equations of the form ut + Au = fλ(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 Φλ 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 = λ0 is nonzero and moreover, S0 = {0} is an isolated invariant set of Φλ0 , then either there is a one-sided neighborhood I1 of λ0 such that Φλ bifurcates a topological sphere Sn−1 for each λ ∈ I1 \ {λ0}, or there is a two-sided neighborhood I2 of λ0 such that the system Φλ bifurcates from the trivial solution an isolated nonempty compact invariant set Kλ with 0 6∈ Kλ for each λ ∈ I2 \ {λ0}. 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, λ0), the connected bifurcation branch Γ from (0, λ0) either meets the boundary ∂Ω of Ω, or meets another bifurcation point (0, λ1). 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.

Download

Included in

Mathematics Commons

Share

COinS