# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2021089
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## Stability of a suspension bridge with a localized structural damping

 1 Department of Mathematics, University of Gabès, Gabès, Tunisia 2 The Preparatory Year Program and The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran, KSA 3 Department of Mathematics, College of Sciences, University of Sharjah, P.O.Box 27272, Sharjah, UAE

Received  February 2021 Revised  June 2021 Early access August 2021

Fund Project: The second and third authors are supported by KFUPM-Project #SB201003

Strong vibrations can cause lots of damage to structures and break materials apart. The main reason for the Tacoma Narrows Bridge collapse was the sudden transition from longitudinal to torsional oscillations caused by a resonance phenomenon. There exist evidences that several other bridges collapsed for the same reason. To overcome unwanted vibrations and prevent structures from resonating during earthquakes, winds, ..., features and modifications such as dampers are used to stabilize these bridges. In this work, we use a minimum amount of dissipation to establish exponential decay- rate estimates to the following nonlocal evolution equation
 $u_{tt}(x,y,t)+\Delta^2 u(x,y,t) - \phi(u) u_{xx}- \left(\alpha(x, y) u_{xt}(x,y,t)\right)_x = 0,$
which models the deformation of the deck of either a footbridge or a suspension bridge.
Citation: Zayd Hajjej, Mohammad Al-Gharabli, Salim Messaoudi. Stability of a suspension bridge with a localized structural damping. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021089
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Function $\psi$
Smooth function $\eta.$