Local and Global Dynamic Bifurcations of Nonlinear Evolution Equations

TL;DR

This paper develops new local and global bifurcation results for nonlinear evolution equations, extending classical theorems to include dynamic bifurcations without the odd crossing number restriction, with applications to the Cahn-Hilliard equation.

Contribution

It introduces a comprehensive framework for dynamic bifurcations in evolution equations, generalizing Rabinowitz's theorem to cases with arbitrary crossing numbers.

Findings

Bifurcation from trivial solutions can produce topological spheres or isolated invariant sets.

The bifurcating invariant sets have nontrivial Conley index.

Global bifurcation branches either reach the boundary or connect to other bifurcation points.

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.