The time-dependent attractor for beam equation with rotational inertia and structural damping

In this paper, we discussed the asymptotic behavior of the solutions to the beam equation with rotational inertia and structural damping:$ \varepsilon(t)(1+(-\Delta) ^{\alpha})\partial^{2}_tu+\Delta^2 u+\gamma(-\Delta)^{\theta}\partial_tu+f(u) = g(x), $where $ \varepsilon(t) $ was a decreasing bounded function. We found a more optimized subcritical exponent $ p^{*} = \frac{N+2\theta}{N-4} $ depending on $ \theta. $ Additionally, we showed that when the growth exponent $ p $ of nonlinear terms $ f(u) $ was within the range $ 1\leqslant p < p^{*}, $ the well-posedness and regularity were established. Finally, within the theory of process on time-dependent spaces, we investigated the existence of the time-dependent attractor by using the contraction function method and more detailed estimates in the time-dependent space $ \mathcal{H}_{t}^{\alpha}. $ The results refined and extended the model and the work in literature [Longtime behavior for an extensible beam equation with rotational inertia and structural nonlinear damping, J. Math. Anal. Appl., 496 (2021), 124785.] from general energy space to time-dependent space in some sense.