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doi: 10.3934/dcdsb.2021318
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Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise

1. 

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

2. 

College of Science, National University of Defense Technology, Changsha, 410073, China

*Corresponding author: Ling Xu

Received  September 2021 Early access January 2022

In this paper, we mainly consider the existence of random attractor and random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. First step, the well-posedness and the existence of a random attractor for the cocycle associated with the considered system is established. Second step, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero. Third step, we prove the regularity of random attractor in a higher regular space by the "iteration" method. Finally, we give the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor.

Citation: Ling Xu, Jianhua Huang, Qiaozhen Ma. Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021318
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, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[4]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-314.  doi: 10.1007/BF02219225.

[5]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab, Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[6]

X. M. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76.  doi: 10.2140/pjm.2004.216.63.

[7]

X. M. Fan, Attractors for a damped stochastic wave equation with sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767–793. doi: 10.1080/07362990600751860.

[8]

X. M. Fan and Y. G. Wang, Fractal dimensional of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25 (2007), 381-396.  doi: 10.1080/07362990601139602.

[9]

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.

[10]

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.

[11]

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.

[12]

Q. Z. MaS. P. Wang and X. B. Chen, Uniform attractors for the coupled suspension bridge equations, Appl. Math. Comput., 217 (2011), 6604-6615.  doi: 10.1016/j.amc.2011.01.045.

[13]

Q. Z. 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.

[14]

Q. Z. 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.

[15]

Q. Z. Ma and C. K. 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.

[16]

Q. Z. Ma and C. K. 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.

[17]

W. J. Ma and Q. Z. Ma, Attractors for stochastic strongly damped plate equations with additive noise, Electron. J. Differential Equations, (2013), 1–12.

[18]

P. McKenna and W. Walter, Nonlinear oscillation in a suspension bridges, Arch. Rational Mech. Anal., 98 (1987), 167–177; Results: Nonlinear Anal., 39 (2000) 731–743. doi: 10.1007/BF00251232.

[19]

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.

[20]

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.

[21]

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.

[22]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differntial Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and appliction, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.  doi: 10.1007/s40072-013-0007-1.

[24]

P. Walters, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.

[25]

B. X. Wang, Upper semicontinuty of random attractors for non-compact random fynamical system, Electron. J. Differential Equations, 2009 (2009), 1-18. 

[26]

B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[27]

B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[28]

B. X. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

[29]

Z. J. Wang and S. F. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst, 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210.

[30]

L. XuJ. H. Huang and Q. Z. Ma, Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5959-5979.  doi: 10.3934/dcdsb.2019115.

[31]

L. Xu and Q. Z. Ma, Upper semicontinuity of random attractor for a Kirchhoff type sus pension bridge equation with strong damping and white noise, Taiwanese J. Math., 24 (2020), 911-935.  doi: 10.11650/tjm/190708.

[32]

M. H. YangJ. Q. Duan and P. E. Kloeden, Asymptotic behavior of solutions for random wave equation with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.  doi: 10.1016/j.nonrwa.2010.06.032.

[33]

M. H. Yang and P. E. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equation, Nonlinear Anal. Real World Appl., 12 (2011), 2811-2821.  doi: 10.1016/j.nonrwa.2011.04.007.

[34]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934.  doi: 10.3934/cpaa.2004.3.921.

[35]

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.

[36]

S. F. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative with noise, Discrete and Continuous Dynamical Systems, 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.

show all references

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, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[4]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-314.  doi: 10.1007/BF02219225.

[5]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab, Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[6]

X. M. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76.  doi: 10.2140/pjm.2004.216.63.

[7]

X. M. Fan, Attractors for a damped stochastic wave equation with sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767–793. doi: 10.1080/07362990600751860.

[8]

X. M. Fan and Y. G. Wang, Fractal dimensional of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25 (2007), 381-396.  doi: 10.1080/07362990601139602.

[9]

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.

[10]

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.

[11]

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.

[12]

Q. Z. MaS. P. Wang and X. B. Chen, Uniform attractors for the coupled suspension bridge equations, Appl. Math. Comput., 217 (2011), 6604-6615.  doi: 10.1016/j.amc.2011.01.045.

[13]

Q. Z. 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.

[14]

Q. Z. 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.

[15]

Q. Z. Ma and C. K. 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.

[16]

Q. Z. Ma and C. K. 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.

[17]

W. J. Ma and Q. Z. Ma, Attractors for stochastic strongly damped plate equations with additive noise, Electron. J. Differential Equations, (2013), 1–12.

[18]

P. McKenna and W. Walter, Nonlinear oscillation in a suspension bridges, Arch. Rational Mech. Anal., 98 (1987), 167–177; Results: Nonlinear Anal., 39 (2000) 731–743. doi: 10.1007/BF00251232.

[19]

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.

[20]

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.

[21]

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.

[22]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differntial Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[23]

A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and appliction, Stoch. Partial Differ. Equ. Anal. Comput., 1 (2013), 241-281.  doi: 10.1007/s40072-013-0007-1.

[24]

P. Walters, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.

[25]

B. X. Wang, Upper semicontinuty of random attractors for non-compact random fynamical system, Electron. J. Differential Equations, 2009 (2009), 1-18. 

[26]

B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[27]

B. X. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[28]

B. X. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

[29]

Z. J. Wang and S. F. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst, 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210.

[30]

L. XuJ. H. Huang and Q. Z. Ma, Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 5959-5979.  doi: 10.3934/dcdsb.2019115.

[31]

L. Xu and Q. Z. Ma, Upper semicontinuity of random attractor for a Kirchhoff type sus pension bridge equation with strong damping and white noise, Taiwanese J. Math., 24 (2020), 911-935.  doi: 10.11650/tjm/190708.

[32]

M. H. YangJ. Q. Duan and P. E. Kloeden, Asymptotic behavior of solutions for random wave equation with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.  doi: 10.1016/j.nonrwa.2010.06.032.

[33]

M. H. Yang and P. E. Kloeden, Random attractors for stochastic semi-linear degenerate parabolic equation, Nonlinear Anal. Real World Appl., 12 (2011), 2811-2821.  doi: 10.1016/j.nonrwa.2011.04.007.

[34]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934.  doi: 10.3934/cpaa.2004.3.921.

[35]

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.

[36]

S. F. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative with noise, Discrete and Continuous Dynamical Systems, 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.

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