Robustness of Global Attractors for Extensible Coupled Suspension Bridge Equations with Fractional Damping

Abstract

In this paper we study the long-time dynamics of the perturbed system of suspension bridge equations

$$\begin{aligned} m_c u_{tt}-\beta u_{xx}+\kappa (u-w)^+ +f_1(u,w)+ (-\partial _{xx})^{\gamma }u_{t}= & {} 0,\\ m_b w_{tt}+\mu w_{xxxx}+(p-\epsilon \Vert w_x\Vert ^2)w_{xx}-\kappa (u-w)^++f_2(u,w)+ (-\partial _{xx})^{\gamma }w_{t}= & {} 0, \end{aligned}$$

where $\epsilon \in (0, 1]$ is a perturbed parameter and $\gamma \in (0, 1)$ is said to be a fractional exponent. Under quite general assumptions on source terms and based on semigroup theory, we establish the global well-posedness and the existence of global attractors with finite fractal dimension. We analysis the upper semicontinuity of global attractors on the perturbed parameter $\epsilon$ in some sense. Moreover, we demonstrate an explicit control over semidistances between trajectories in the weak energy phase space in terms of $\epsilon$. Finally, we prove that the family of global attractors is upper semicontinuous with respect to the fractional exponent $\gamma \in (0,1/2)$.