Long-time dynamics for a coupled system modeling the oscillations of suspension bridges

Yang Liu; Xiao Long; Li Zhang; Yang Liu; Xiao Long; Li Zhang

doi:10.3934/cam.2025002

Communications in Analysis and Mechanics

2025, Volume 17, Issue 1: 15-40. doi: 10.3934/cam.2025002

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Research article

Long-time dynamics for a coupled system modeling the oscillations of suspension bridges

College of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, People's Republic of China

Received: 30 April 2024 Revised: 14 November 2024 Accepted: 14 November 2024 Published: 16 January 2025
35G61, 35A01, 35B30, 35B41, 37N15

This paper is concerned with a coupled system modeling the oscillations of suspension bridges, which consists of a beam equation and a viscoelastic string equation. We first transformed the original initial-boundary value problem into an equivalent one in the history space framework. Then we obtained the global well-posedness and regularity of mild solutions by using the semigroup theory. In addition, we employed the perturbed energy method to establish a stabilizability estimate. By verifying the gradient property and quasi-stability of the corresponding dynamical system, we derived the existence of a global attractor with finite fractal dimension.
- coupled beam-string system,
- global well-posedness,
- global attractors,
- gradient dynamical system,
- quasi-stability
Citation: Yang Liu, Xiao Long, Li Zhang. Long-time dynamics for a coupled system modeling the oscillations of suspension bridges[J]. Communications in Analysis and Mechanics, 2025, 17(1): 15-40. doi: 10.3934/cam.2025002

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Abstract

This paper is concerned with a coupled system modeling the oscillations of suspension bridges, which consists of a beam equation and a viscoelastic string equation. We first transformed the original initial-boundary value problem into an equivalent one in the history space framework. Then we obtained the global well-posedness and regularity of mild solutions by using the semigroup theory. In addition, we employed the perturbed energy method to establish a stabilizability estimate. By verifying the gradient property and quasi-stability of the corresponding dynamical system, we derived the existence of a global attractor with finite fractal dimension.

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