The existence of time-dependent attractor for wave equation with fractional damping and lower regular forcing term

We investigate the well-posedness and longtime dynamics of fractional damping wave equation whose coefficient \begin{document}$ \varepsilon $\end{document} depends explicitly on time. First of all, when \begin{document}$ 1\leq p\leq p^{\ast\ast} = \frac{N+2}{N-2}\; (N\geq3) $\end{document} , we obtain existence of solution for the fractional damping wave equation with time-dependent decay coefficient in \begin{document}$ H_{0}^{1}(\Omega)\times L^{2}(\Omega) $\end{document} . Furthermore, when \begin{document}$ 1\leq p<p^{*} = \frac{N+4\alpha}{N-2} $\end{document} , \begin{document}$ u_{t} $\end{document} is proved to be of higher regularity in \begin{document}$ H^{1-\alpha}\; (t>\tau) $\end{document} and show that the solution is quasi-stable in weaker space \begin{document}$ H^{1-\alpha}\times H^{-\alpha} $\end{document} . Finally, we get the existence and regularity of time-dependent attractor.