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Continuity of Global Attractors for a Suspension Bridge Equation

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Abstract

Our purpose is to study the continuity of global attractors under small perturbations \(\delta ,\ \epsilon \in [0, 1]\) for a suspension bridge equation on a bounded smooth domain of \(\mathbb{R}^{n}\)

$$ u_{tt} + \Delta ^{2}\,u +\kappa \, u^{+}-(a+\delta \|\nabla \,u\|^{2}) \Delta \,u+ \textrm{g}_{\epsilon }(\|u_{t}\|)u_{t} + f (u) =0, $$

with hinged (clamped) boundary condition. Based on nonlinear semigroups and the theory of monotone operators, we prove the well-posedness of the considered problem. Then we show the existence of an absorbing set. Exponential stability and global boundedness of solutions are immediate consequences. The existence of global attractors with finite fractal dimension is achieved by the useful property of quasi-stability in the theory of infinite-dimensional dynamical systems. Finally we analysis the continuity of global attractors with respect to a pair of perturbation parameters \((\delta ,\epsilon )\) in a residual dense set and their upper semicontinuity in a complete metric space. In particular, we demonstrate an explicit control over semidistances between trajectories in the weak energy phase space in terms of the perturbation parameters.

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Acknowledgements

The author would like to thank the referees for their critical review and valuable comments that allowed to improve the paper.

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  1. UR Systèmes dynamiques et leurs applications, UR 17ES21, Ecole Nationale d’Ingénieurs de Bizerte, Université de Carthage, 7035, BP 66, Bizerte, Tunisia

    Moncef Aouadi

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Aouadi, M. Continuity of Global Attractors for a Suspension Bridge Equation. Acta Appl Math 176, 16 (2021). https://doi.org/10.1007/s10440-021-00462-x

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