English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we establish an invariant set bifurcation theory for the nonautonomous dynamical system $(\va_\lam,\0)_{X,\cH}$ generated by the evolution equation \be\label{e0}u_t+Au=\lam u+p(t,u),\hs p\in \cH=\cH[f(\.,u)]\ee on a Hilbert space $X$, where $A$ is a sectorial operator, $\lam$ is the bifurcation parameter, $f(\.,u):\R\ra X$ is translation compact, $f(t,0)\equiv0$ and $\cH[f]$ is the hull of $f(\.,u)$. Denote by $\va_\lam:=\va_\lam(t,p)u$ the cocycle semiflow generated by the equation. Under some other assumptions on $f$, we show that as the parameter $\lam$ crosses an eigenvalue $\lam_0\in\R$ of $A$, the system bifurcates from $0$ to a nonautonomous invariant set $B_\lam(\.)$ on one-sided neighborhood of $\lam_0$. Moreover, $$\lim_{\lam\ra\lam_0}H_{X^\a}$B_\lam(p),0$=0,\hs p\in P,$$ where $H_{X^\a}(\.,\.)$ denotes the Hausdorff semidistance in $X^\a$ (here $X^\alpha$ ($\a\geq0$) 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 $\cM^\lam_{loc}(\.)$.

An invariant set bifurcation theory for nonautonomous nonlinear evolution equations