Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models

The paper investigates the existence and the continuity of uniform attractors for the non-autonomous Kirchhoff wave equations with strong damping: \begin{document}$ u_{tt}-(1+\epsilon\|\nabla u\|^{2})\Delta u-\Delta u_{t}+f(u) = g(x,t) $\end{document} , where \begin{document}$ \epsilon\in [0,1] $\end{document} is an extensibility parameter. It shows that when the nonlinearity \begin{document}$ f(u) $\end{document} is of optimal supercritical growth \begin{document}$ p: \frac{N+2}{N-2} = p^*<p<p^{**} = \frac{N+4}{(N-4)^+} $\end{document} : (ⅰ) the related evolution process has in natural energy space \begin{document}$ \mathcal{H} = (H^1_0\cap L^{p+1})\times L^2 $\end{document} a compact uniform attractor \begin{document}$ \mathcal{A}^{\epsilon}_{\Sigma} $\end{document} for each \begin{document}$ \epsilon\in [0,1] $\end{document} ; (ⅱ) the family of compact uniform attractor \begin{document}$ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $\end{document} is continuous on \begin{document}$ \epsilon $\end{document} in a residual set \begin{document}$ I^*\subset [0,1] $\end{document} in the sense of \begin{document}$ \mathcal{H}_{ps} ( = (H^1_0\cap L^{p+1,w})\times L^2) $\end{document} -topology; (ⅲ)\begin{document}$ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $\end{document} is upper semicontinuous on \begin{document}$ \epsilon\in [0,1] $\end{document} in \begin{document}$ \mathcal{H}_{ps} $\end{document} -topology.