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November  2019, 24(11): 5959-5979. doi: 10.3934/dcdsb.2019115

## Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory

 1 College of Science, National University of Defense Technology, Changsha 410073, China 2 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

* Corresponding author: Ling Xu

Received  October 2018 Revised  December 2018 Published  November 2019 Early access  June 2019

Fund Project: The authors are supported by the NSF of China (11361053,11771449), the NSF of Gansu Province (17JR5RA069), the University Project of Gansu Province(2017B-90) and the Project of Northwest Normal University(NWNU-LKQN-16-16; NWNU-LKQN-18-14).

This paper is devoted to the well-posedness and long-time behavior of a stochastic suspension bridge equation with memory effect. The existence of the random attractor for the stochastic suspension bridge equation with memory is established. Moreover, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero.

Citation: Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115
##### References:
 [1] N. Ahmed and H. Harbi, Mathematical analysis of dynamical models of suspension bridges, SIAM J. Appl. Math., 58 (1998), 853-874.  doi: 10.1137/S0036139996308698. [2] L. Arnold, Random Dynamical Systems, Spring-verlag, New York, 1998. doi: 10.1007/978-3-662-12878-7. [3] S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277. [4] I. Chueshov, Monotone Random Systems Theory and Applications, in: Lecture Notes in Mathematics, , vol. 1779, Springer, Berlin, 2002. doi: 10.1007/b83277. [5] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab, Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [6] C. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [7] L. Humphreys, Numercial mountain pass solutions of a suspension bridge equation, Nonlinear Anal., 28 (1997), 1811-1826.  doi: 10.1016/S0362-546X(96)00020-X. [8] J. 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. [9] A. Lazer and P. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connection with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120. [10] Q. Ma, S. Wang and X. Chen, Uniform attractors for the coupled suspension bridge equations, Appl. Math. Comput., 217 (2011), 6604-6615.  doi: 10.1016/j.amc.2011.01.045. [11] Q. Ma and L. Xu, Random attractors for the extensible suspension bridge equation with white noise, Comput. Math. Appl., 70 (2015), 2895-2903.  doi: 10.1016/j.camwa.2015.09.029. [12] Q. Ma and L. Xu, Random attractors for the coupled suspension bridge equations with white noises, Appl. Math. Comput., 306 (2017), 38-48.  doi: 10.1016/j.amc.2017.02.019. [13] Q. Ma and C. Zhong, Existence of global attractors for the coupled suspension bridge equations, J. Math. Anal. Appl., 308 (2005), 365-379.  doi: 10.1016/j.jmaa.2005.01.036. [14] Q. Ma and C. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differential Equations, 246 (2009), 3755-3775.  doi: 10.1016/j.jde.2009.02.022. [15] P. McKenna and W. Walter, Nonlinear oscillation in a suspension bridges, Results: Nonlinear Anal., 39 (2000), 731-743.  doi: 10.1007/BF00251232. [16] J. Park and J. Kang, Global attractors for the suspension bridge equations with nonlinear damping, Quart. Appl. Math., 69 (2011), 465-475.  doi: 10.1090/S0033-569X-2011-01259-1. [17] J. Park and J. Kang, Pullback $\mathcal{D}$-attractors for non-autonomous suspension bridge equations, Nonlinear Anal., 71 (2009), 4618-4623.  doi: 10.1016/j.na.2009.03.025. [18] J. Park and J. Kang, Uniform attractor for non-autonomous suspension bridge equations with localized damping, Math. Methods Appl. Sci., 34 (2011), 487-496.  doi: 10.1002/mma.1376. [19] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. [20] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differntial Equations, Appl. Math. Sci. Berlin, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [21] B. Wang, Upper semicontinuty of random attractors for non-compact random dynamical system, Electron. J. Differential Equations, 2009 (2009), 1-18. [22] C. Zhong, Q. Ma and C. 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. [23] S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal., 120 (2015), 202-226.  doi: 10.1016/j.na.2015.03.009.

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##### References:
 [1] N. Ahmed and H. Harbi, Mathematical analysis of dynamical models of suspension bridges, SIAM J. Appl. Math., 58 (1998), 853-874.  doi: 10.1137/S0036139996308698. [2] L. Arnold, Random Dynamical Systems, Spring-verlag, New York, 1998. doi: 10.1007/978-3-662-12878-7. [3] S. Borini and V. Pata, Uniform attractors for a strongly damped wave equation with linear memory, Asymptot. Anal., 20 (1999), 263-277. [4] I. Chueshov, Monotone Random Systems Theory and Applications, in: Lecture Notes in Mathematics, , vol. 1779, Springer, Berlin, 2002. doi: 10.1007/b83277. [5] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab, Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [6] C. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [7] L. Humphreys, Numercial mountain pass solutions of a suspension bridge equation, Nonlinear Anal., 28 (1997), 1811-1826.  doi: 10.1016/S0362-546X(96)00020-X. [8] J. 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. [9] A. Lazer and P. McKenna, Large-amplitude periodic oscillations in suspension bridges: Some new connection with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.  doi: 10.1137/1032120. [10] Q. Ma, S. Wang and X. Chen, Uniform attractors for the coupled suspension bridge equations, Appl. Math. Comput., 217 (2011), 6604-6615.  doi: 10.1016/j.amc.2011.01.045. [11] Q. Ma and L. Xu, Random attractors for the extensible suspension bridge equation with white noise, Comput. Math. Appl., 70 (2015), 2895-2903.  doi: 10.1016/j.camwa.2015.09.029. [12] Q. Ma and L. Xu, Random attractors for the coupled suspension bridge equations with white noises, Appl. Math. Comput., 306 (2017), 38-48.  doi: 10.1016/j.amc.2017.02.019. [13] Q. Ma and C. Zhong, Existence of global attractors for the coupled suspension bridge equations, J. Math. Anal. Appl., 308 (2005), 365-379.  doi: 10.1016/j.jmaa.2005.01.036. [14] Q. Ma and C. Zhong, Existence of strong solutions and global attractors for the coupled suspension bridge equations, J. Differential Equations, 246 (2009), 3755-3775.  doi: 10.1016/j.jde.2009.02.022. [15] P. McKenna and W. Walter, Nonlinear oscillation in a suspension bridges, Results: Nonlinear Anal., 39 (2000), 731-743.  doi: 10.1007/BF00251232. [16] J. Park and J. Kang, Global attractors for the suspension bridge equations with nonlinear damping, Quart. Appl. Math., 69 (2011), 465-475.  doi: 10.1090/S0033-569X-2011-01259-1. [17] J. Park and J. Kang, Pullback $\mathcal{D}$-attractors for non-autonomous suspension bridge equations, Nonlinear Anal., 71 (2009), 4618-4623.  doi: 10.1016/j.na.2009.03.025. [18] J. Park and J. Kang, Uniform attractor for non-autonomous suspension bridge equations with localized damping, Math. Methods Appl. Sci., 34 (2011), 487-496.  doi: 10.1002/mma.1376. [19] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. [20] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differntial Equations, Appl. Math. Sci. Berlin, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [21] B. Wang, Upper semicontinuty of random attractors for non-compact random dynamical system, Electron. J. Differential Equations, 2009 (2009), 1-18. [22] C. Zhong, Q. Ma and C. 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. [23] S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal., 120 (2015), 202-226.  doi: 10.1016/j.na.2015.03.009.
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