Applied Mathematics and Optimization ( IF 1.6 ) Pub Date : 2021-04-20 , DOI: 10.1007/s00245-021-09774-8 Moncef Aouadi
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 \(\gamma \in (0,1/2)\).
中文翻译：
分数阶阻尼可扩展悬索桥方程整体吸引子的鲁棒性。
在本文中，我们研究了悬索桥方程的摄动系统的长期动力学。
$$ \ begin {aligned} m_c u_ {tt}-\ beta u_ {xx} + \ kappa（uw）^ + + f_1（u，w）+（-\ partial _ {xx}）^ {\ gamma} u_ {t} =＆{} 0，\\ m_b w_ {tt} + \ mu w_ {xxxx} +（p- \ epsilon \ Vert w_x \ Vert ^ 2）w_ {xx}-\ kappa（uw）^ ++ f_2（u，w）+（-\ partial _ {xx}）^ {\ gamma} w_ {t} =＆{} 0，\ end {aligned} $$其中\（\ epsilon \ in（0，1] \）是一个扰动参数，而\（\ gamma \ in（0，1）\）被认为是分数指数。半群理论，建立了整体的适定性和有限分形维数的整体吸引子的存在，在一定意义上分析了整体吸引子在扰动参数\（\ epsilon \）上的上半连续性，并证明了显式控制（\ epsilon \）表示在弱能量相空间中的轨迹之间的半距离。最后，我们证明了全局吸引子族关于分数指数\（\ gamma \ in（0,1 / 2）\）。
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