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 {document}eqnarray* uₜₜ+Δ² u-m(\|∇ u\|²)Δ u +\| uₜ\|ᵖuₜ+f(u) = h, in\; Ω×R⁺, p≥0 eqnarray*{document} on a bounded smooth domain Ωⁿ with hinged (clamped) boundary condition. Under some suitable conditions on the Kirchhoff coefficient m(\|∇ u\|²) and the nonlinear term $ f(u) $, 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.