Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback

Wenjun Liu; Hefeng Zhuang

doi:10.3934/dcdsb.2020147

Article Contents

2021, Volume 26, Issue 2: 907-942. Doi: 10.3934/dcdsb.2020147

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Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback

Wenjun Liu^, and
Hefeng Zhuang

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

^* Corresponding author: Wenjun Liu
^* Corresponding author: Wenjun Liu

Received: March 31, 2019

Revised: October 31, 2019

Early access: May 2020

Published: February 2021

The first author is supported by the National Natural Science Foundation of China [grant number 11771216], the Key Research and Development Program of Jiangsu Province (Social Development) [grant number BE2019725], the Six Talent Peaks Project in Jiangsu Province [grant number 2015-XCL-020] and the Qing Lan Project of Jiangsu Province

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Abstract

In the present paper, we consider a suspension bridge problem with a nonlinear delay term in the internal feedback. Namely, we investigate the following equation:

$ \begin{equation*} u_{tt}+ \Delta^2 u + \delta_1 g_1 (u_t (x,y,t))+ \delta_2 g_2 (u_t (x,y, t-\tau))+ h(u(x,y,t)) = f(x,y), \end{equation*} $

together with some suitable initial data and boundary conditions. We prove the global existence of solutions by means of the energy method combined with the Faedo-Galerkin procedure under a certain relation between the weight of the delay term in the feedback and the weight of the nonlinear frictional damping term without delay. Moreover, we establish the existence of a global attractor for the above-mentioned system by proving the existence of an absorbing set and the asymptotic smoothness of the semigroup $ S(t) $.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 35L57.

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References

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