An invariant set bifurcation theory for nonautonomous nonlinear evolution equations

In this paper we establish an invariant set bifurcation theory for the nonautonomous dynamical system $(\varphi_\lambda,\theta)_{X,\mathcal H}$ generated by the evolution equation \begin{equation}\label{e0}u_t+Au=\lambda u+p(t,u),\hspace{0.4cm} p\in \mathcal H=\mathcal H[f(\cdot,u)]\end{equation} on a Hilbert space $X$, where $A$ is a sectorial operator, $\lambda$ is the bifurcation parameter, $f(\cdot,u):\mathbb{R}\rightarrow X$ is translation compact, $f(t,0)\equiv0$ and $\mathcal H[f]$ is the hull of $f(\cdot,u)$. Denote by $\varphi_\lambda:=\varphi_\lambda(t,p)u$ the cocycle semiflow generated by the equation. Under some other assumptions on $f$, we show that as the parameter $\lambda$ crosses an eigenvalue $\lambda_0\in\mathbb{R}$ of $A$, the system bifurcates from $0$ to a nonautonomous invariant set $B_\lambda(\cdot)$ on one-sided neighborhood of $\lambda_0$. Moreover, $$\lim_{\lambda\rightarrow\lambda_0}H_{X^\alpha}\left(B_\lambda(p),0\right)=0,\hspace{0.4cm} p\in P,$$ where $H_{X^\alpha}(\cdot,\cdot)$ denotes the Hausdorff semidistance in $X^\alpha$ (here $X^\alpha$ ($\alpha\geq0$) defined below is the fractional power spaces associated with $A$). Our result is based on the pullback attractor bifurcation on the local central invariant manifolds $\mathcal {M}^\lambda_{\operatorname{loc}}(\cdot)$.