LAUSR

This work is concerned with the following nonautonomous evolutionary system on a Banach space \begin{document}$X$\end{document} , \begin{document}${x_t} + Ax = f\left( {x, h\left( t \right)} \right), \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {0.1} \right)$ \end{document} where \begin{document}$A$\end{document} is a hyperbolic sectorial operator on \begin{document}$X$\end{document} , the nonlinearity \begin{document}$f \in C({X^\alpha } \times X,X)$\end{document} is Lipschitz in the first variable, the nonautonomous forcing \begin{document}$h \in C(\mathbb{R},X)$\end{document} is \begin{document}$\mu $\end{document} -subexponentially growing for some \begin{document}$\mu >0$\end{document} (see (3.4) below for definition). Under some reasonable assumptions, we first establish an existence result for a unique nonautonomous hyperbolic equilibrium for the system in the framework of cocycle semiflows. We then demonstrate that the system exhibits a global synchronising behavior with the nonautonomous forcing \begin{document}$h$\end{document} as time varies. Finally, we apply the abstract results to stochastic partial differential equations with additive white noise and obtain stochastic hyperbolic equilibria for the corresponding systems.