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

Abstract

<p style='text-indent:20px;'>The paper investigates the existence and the continuity of uniform attractors for the non-autonomous Kirchhoff wave equations with strong damping: <inline-formula><tex-math id="M1">\begin{document}$ u_{tt}-(1+\epsilon\|\nabla u\|^{2})\Delta u-\Delta u_{t}+f(u) = g(x,t) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \epsilon\in [0,1] $\end{document}</tex-math></inline-formula> is an extensibility parameter. It shows that when the nonlinearity <inline-formula><tex-math id="M3">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> is of optimal supercritical growth <inline-formula><tex-math id="M4">\begin{document}$ p: \frac{N+2}{N-2} = p^*&lt;p&lt;p^{**} = \frac{N+4}{(N-4)^+} $\end{document}</tex-math></inline-formula>: (ⅰ) the related evolution process has in natural energy space <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{H} = (H^1_0\cap L^{p+1})\times L^2 $\end{document}</tex-math></inline-formula> a compact uniform attractor <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{A}^{\epsilon}_{\Sigma} $\end{document}</tex-math></inline-formula> for each <inline-formula><tex-math id="M7">\begin{document}$ \epsilon\in [0,1] $\end{document}</tex-math></inline-formula>; (ⅱ) the family of compact uniform attractor <inline-formula><tex-math id="M8">\begin{document}$ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $\end{document}</tex-math></inline-formula> is continuous on <inline-formula><tex-math id="M9">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> in a residual set <inline-formula><tex-math id="M10">\begin{document}$ I^*\subset [0,1] $\end{document}</tex-math></inline-formula> in the sense of <inline-formula><tex-math id="M11">\begin{document}$ \mathcal{H}_{ps} ( = (H^1_0\cap L^{p+1,w})\times L^2) $\end{document}</tex-math></inline-formula>-topology; (ⅲ) <inline-formula><tex-math id="M12">\begin{document}$ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $\end{document}</tex-math></inline-formula> is upper semicontinuous on <inline-formula><tex-math id="M13">\begin{document}$ \epsilon\in [0,1] $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M14">\begin{document}$ \mathcal{H}_{ps} $\end{document}</tex-math></inline-formula>-topology.</p>