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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 & 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. Google Scholar

[2]

R. O. AraújoT. 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.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Amsterdam, North-Holland, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. LiuH. 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.  Google Scholar

[16]

Q. Z. Ma and C. K. Zhong, Existence of global attractors for the suspension bridge equations, J. Sichuan Univ., 43 (2006), 271-276.   Google Scholar

[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.  Google Scholar

[18]

P. J. McKenna and W. Walter, Nonlinear oscillation in a suspension bridge, Nonlinear Anal., 39 (2000), 731-743.   Google Scholar

[19]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differ. Equ., 41 (2011), 1-20.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. Q. XuS. 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.  Google Scholar

[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.  Google Scholar

[26]

C. K. ZhongQ. 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.  Google Scholar

show all references

References:
[1]

Y. An, On the suspension bridge equations and the relevant problems, Ph.D thesis, 2001. Google Scholar

[2]

R. O. AraújoT. 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.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Amsterdam, North-Holland, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. LiuH. 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.  Google Scholar

[16]

Q. Z. Ma and C. K. Zhong, Existence of global attractors for the suspension bridge equations, J. Sichuan Univ., 43 (2006), 271-276.   Google Scholar

[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.  Google Scholar

[18]

P. J. McKenna and W. Walter, Nonlinear oscillation in a suspension bridge, Nonlinear Anal., 39 (2000), 731-743.   Google Scholar

[19]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differ. Equ., 41 (2011), 1-20.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

G. Q. XuS. 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.  Google Scholar

[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.  Google Scholar

[26]

C. K. ZhongQ. 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.  Google Scholar

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