The global attractor for a class of extensible beams with nonlocal weak damping

The goal of this paper is to study the long-time behavior of a class of extensible beams equation with the nonlocal weak damping \begin{document}$ \begin{eqnarray*} u_{tt}+\Delta^2 u-m(\|\nabla u\|^2)\Delta u +\| u_t\|^{p}u_t+f(u) = h, \rm{in}\; \Omega\times\mathbb{R^{+}}, p\geq0 \end{eqnarray*} $\end{document} on a bounded smooth domain \begin{document}$ \Omega\subset\mathbb{R}^{n} $\end{document} with hinged (clamped) boundary condition. Under some suitable conditions on the Kirchhoff coefficient \begin{document}$ m(\|\nabla u\|^2) $\end{document} and the nonlinear term \begin{document}$ f(u) $\end{document} , the well-posedness is established by means of the monotone operator theory and the existence of a global attractor is obtained in the subcritical case, where the asymptotic smooothness of the semigroup is verified by the energy reconstruction method.