Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations

Abstract

The paper investigates the upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with structural damping: \begin{document}$ u_{tt}-M(\|\nabla u\|^2)\Delta u+(-\Delta)^\alpha u_t+f(u) = g(x,t) $\end{document} , where \begin{document}$ \alpha\in(1/2, 1) $\end{document} is said to be a dissipative index. It shows that when the nonlinearity \begin{document}$ f(u) $\end{document} is of supercritical growth \begin{document}$ p: 1 \leq p , the related evolution process has a pullback attractor for each \begin{document}$ \alpha\in(1/2, 1) $\end{document} , and the family of pullback attractors is upper semicontinuous with respect to \begin{document}$ \alpha $\end{document} . These results extend those in [ 27 ] for autonomous Kirchhoff wave models.