Strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation

In this paper, we investigate the global well-posedness and the existence of strong global and exponential attractors for a nonlinear strongly damped hyperbolic equation in \begin{document}$ \Omega\subset{\mathbb R}^N $\end{document}:

\begin{document}$ u_{tt}+\Delta ^2u+\Delta ^2u_t+\Delta \phi (\Delta u) = g(x), $\end{document}

with the hinged boundary condition. We show that (i) when the nonlinearity \begin{document}$ \phi $\end{document} is quasi-monotone and is of at most the critical growth: \begin{document}$ 1\leq p\leq p^{*}: = \frac{N+2}{(N-2)^{+}} \ (N\geq 2) $\end{document} and \begin{document}$ g = 0 $\end{document}, the model has in phase space \begin{document}$ {\mathcal H} = V_3\times L^2 $\end{document} a trivial global and exponential attractor, respectively. (ii) In particular when \begin{document}$ N = 1 $\end{document}, without any polynomial growth restriction for \begin{document}$ \phi $\end{document}, the model has a strong global and a strong exponential attractor, respectively. These results deepen and extend the related researches on this topic in recent literature [16,22]. The method developed here allows us to establish the existence of the strong global and exponential attractor for this nonlinear model.