Time-dependent global attractor for the extensible beam equation with rotational inertia and nonlinear damping

In this paper, we investigate the well-posedness and long-time dynamics of the non-autonomous extensible beam equation with rotational inertia and nonlinear damping$ \varepsilon(t)(1+(-\Delta) ^{\alpha})u_{tt}+\Delta^2 u-M(\|\nabla u\|^{2})\Delta u+g(u_t)+f(u) = h(x), $where $ \varepsilon(t) $ is a bounded decreasing function. As the growth exponent $ p $ of the nonlinearity $ f(u) $ satisfies $ 1\leqslant p< p^{*} = \frac{N+2\alpha}{N-4} $ with $ N>4 $ and the growth index $ \gamma $ of the nonlinear damping term $ g(u_{t}) $ satisfies $ \frac{N+2\alpha}{N-2\alpha}\leqslant\gamma\leqslant\frac{N+4}{N-4}, $ we obtain an invariant time-dependent global attractor by the method of contraction function in the natural energy space $ \mathcal{H}_{t}^{\alpha} $(the definition of function space $ \mathcal{H}^{\alpha}_{\tau}\; $ can be found on page 5, line -15).