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October  2020, 19(10): 4995-5013. doi: 10.3934/cpaa.2020224

## Longtime dynamics for a type of suspension bridge equation with past history and time delay

 1 College of Science, Henan University of Technology, Zhengzhou 450001, China 2 Department of Economic Mathematics, Southwestern University of Finance and Economics, , Chengdu, 611130, China 3 College of Mathematics and Information Science Henan Normal University, Xinxiang 453007, China

* Corresponding author

Received  February 2020 Revised  May 2020 Published  July 2020

Fund Project: Gongwei Liu is partially supported by NSFC (No. 11801145), Key Scientific Research Foundation of the Higher Education Institutions of Henan Province, China (No.19A110004) and the Fund of Young Backbone Teacher in Henan Province (No.2018GGJS068). Baowei Feng is partially supported by NSFC (No. 11701465). Xinguang Yang is partially supported by the Fund of Young Backbone Teacher in Henan Province (No.2018GGJS039)

In this paper, we investigate a suspension bridge equation with past history and time delay effects, defined in a bounded domain $\Omega$ of $\mathbb{R}^N$. Many researchers have considered the well-posedness, energy decay of solution and existence of global attractors for suspension bridge equation without memory or delay. But as far as we know, there are no results on the suspension bridge equation with both memory and time delay. The purpose of this paper is to show the existence of a global attractor which has finite fractal dimension by using the methods developed by Chueshov and Lasiecka. Result on exponential attractors is also proved. We also establish the exponential stability under some conditions. These results are extension and improvement of earlier results.

Citation: Gongwei Liu, Baowei Feng, Xinguang Yang. Longtime dynamics for a type of suspension bridge equation with past history and time delay. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4995-5013. doi: 10.3934/cpaa.2020224
##### References:
 [1] Y. An, On the suspension bridge equations and the relevant problems, Ph.D thesis, 2001. [2] R. O. Araújo, T. F. Ma and Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differ. Equ., 254 (2013), 4066-4087.  doi: 10.1016/j.jde.2013.02.010. [3] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Amsterdam, North-Holland, 1992. [4] A. R. A. Barbosa and T. F. Ma, long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416 (2014), 143-165.  doi: 10.1016/j.jmaa.2014.02.042. [5] I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monogr. Math., Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9. [6] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [7] Q. Dai and Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6. [8] B. Feng and X. Yang, Long-term dynamics for a nonlinear Timoshenko system with delay, Appl. Anal., 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139. [9] B. Feng, On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors, Discrete Contin. Dyn. Syst., 37 (2017), 4729-4751.  doi: 10.3934/dcds.2017203. [10] J. R. Kang, Long-time behavior of a suspension bridge equations with past history, Appl. Math. Comput., 265 (2015), 509-519.  doi: 10.1016/j.amc.2015.04.116. [11] J. R. Kang, Global attractor for suspension bridge equations with memeory, Math. Meth. Appl. Sci., 39 (2016), 762-775.  doi: 10.1002/mma.3520. [12] M. Kirane and B. Said Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0. [13] A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridge: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120. [14] G. Liu and H. Zhang, Well-posedness for a class of wave equation with past history and a delay, Z. Angew. Math. Phys., 67 (2016), 1-14.  doi: 10.1007/s00033-015-0593-z. [15] G. Liu, H. Yue and H. Zhang, Long time behavior for a wave equation with time delay, Taiwan. J. Math., 27 (2017), 2017-129.  doi: 10.11650/tjm.21.2017.7246. [16] Q. Z. Ma and C. K. Zhong, Existence of global attractors for the suspension bridge equations, J. Sichuan Univ., 43 (2006), 271-276. [17] Q. Z. Ma and C. K. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differ. Equ., 246 (2009), 3755-3775.  doi: 10.1016/j.jde.2009.02.022. [18] P. J. McKenna and W. Walter, Nonlinear oscillation in a suspension bridge, Nonlinear Anal., 39 (2000), 731-743. [19] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differ. Equ., 41 (2011), 1-20. [20] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891. [21] J. Y. Park and J. R. Kang, Pullback D-attractors for non-autonomous suspension bridge equations, Nonlinear Anal., 71 (2009), 4618-4623.  doi: 10.1016/j.na.2009.03.025. [22] J. Y. Park and J. R. Kang, Global attractors for the suspension bridge equations with nonlinear damping, Q. Appl. Math., 69 (2011), 465-475.  doi: 10.1090/S0033-569X-2011-01259-1. [23] S. H. Park, Long-time behavior for suspension bridge equations with time delay, Z. Angew. Math. Phys., 69 (2018), Art. 45. doi: 10.1007/s00033-018-0934-9. [24] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021. [25] Z. Yang, Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay, Z. Angew. Math. Phys., 66 (2015), 727-745.  doi: 10.1007/s00033-014-0429-2. [26] C. K. Zhong, Q. Z. Ma and C. Y. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Anal., 67 (2007), 442-454.  doi: 10.1016/j.na.2006.05.018.

show all references

##### References:
 [1] Y. An, On the suspension bridge equations and the relevant problems, Ph.D thesis, 2001. [2] R. O. Araújo, T. F. Ma and Y. Qin, Long-time behavior of a quasilinear viscoelastic equation with past history, J. Differ. Equ., 254 (2013), 4066-4087.  doi: 10.1016/j.jde.2013.02.010. [3] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Amsterdam, North-Holland, 1992. [4] A. R. A. Barbosa and T. F. Ma, long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416 (2014), 143-165.  doi: 10.1016/j.jmaa.2014.02.042. [5] I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer Monogr. Math., Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9. [6] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [7] Q. Dai and Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6. [8] B. Feng and X. Yang, Long-term dynamics for a nonlinear Timoshenko system with delay, Appl. Anal., 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139. [9] B. Feng, On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors, Discrete Contin. Dyn. Syst., 37 (2017), 4729-4751.  doi: 10.3934/dcds.2017203. [10] J. R. Kang, Long-time behavior of a suspension bridge equations with past history, Appl. Math. Comput., 265 (2015), 509-519.  doi: 10.1016/j.amc.2015.04.116. [11] J. R. Kang, Global attractor for suspension bridge equations with memeory, Math. Meth. Appl. Sci., 39 (2016), 762-775.  doi: 10.1002/mma.3520. [12] M. Kirane and B. Said Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0. [13] A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridge: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120. [14] G. Liu and H. Zhang, Well-posedness for a class of wave equation with past history and a delay, Z. Angew. Math. Phys., 67 (2016), 1-14.  doi: 10.1007/s00033-015-0593-z. [15] G. Liu, H. Yue and H. Zhang, Long time behavior for a wave equation with time delay, Taiwan. J. Math., 27 (2017), 2017-129.  doi: 10.11650/tjm.21.2017.7246. [16] Q. Z. Ma and C. K. Zhong, Existence of global attractors for the suspension bridge equations, J. Sichuan Univ., 43 (2006), 271-276. [17] Q. Z. Ma and C. K. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differ. Equ., 246 (2009), 3755-3775.  doi: 10.1016/j.jde.2009.02.022. [18] P. J. McKenna and W. Walter, Nonlinear oscillation in a suspension bridge, Nonlinear Anal., 39 (2000), 731-743. [19] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differ. Equ., 41 (2011), 1-20. [20] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891. [21] J. Y. Park and J. R. Kang, Pullback D-attractors for non-autonomous suspension bridge equations, Nonlinear Anal., 71 (2009), 4618-4623.  doi: 10.1016/j.na.2009.03.025. [22] J. Y. Park and J. R. Kang, Global attractors for the suspension bridge equations with nonlinear damping, Q. Appl. Math., 69 (2011), 465-475.  doi: 10.1090/S0033-569X-2011-01259-1. [23] S. H. Park, Long-time behavior for suspension bridge equations with time delay, Z. Angew. Math. Phys., 69 (2018), Art. 45. doi: 10.1007/s00033-018-0934-9. [24] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021. [25] Z. Yang, Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay, Z. Angew. Math. Phys., 66 (2015), 727-745.  doi: 10.1007/s00033-014-0429-2. [26] C. K. Zhong, Q. Z. Ma and C. Y. Sun, Existence of strong solutions and global attractors for the suspension bridge equations, Nonlinear Anal., 67 (2007), 442-454.  doi: 10.1016/j.na.2006.05.018.
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