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May  2017, 37(5): 2787-2812. doi: 10.3934/dcds.2017120

Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise

Zhaojuan Wang 1,, and  Shengfan Zhou 2,

1. 

School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, China
2. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

* Corresponding author: Zhaojuan Wang

Received  July 2016 Revised  January 2017 Published  February 2017

Fund Project: The authors are supported by NSFC grant No. 11326114, 11401244, 11471290. Natural Science Research Project of Ordinary Universities in Jiangsu Province grant No. 14KJB110003. Zhejiang Natural Science Foundation grant No. LY14A010012 and Zhejiang Normal University Foundation grant No. ZC304014012.

In this paper we study the asymptotic behavior of solutions of the non-autonomous stochastic strongly damped wave equation driven by multiplicative noise defined on unbounded domains. We first introduce a continuous cocycle for the equation. Then we consider the existence of a tempered pullback random attractor for the cocycle. Finally we establish the upper semicontinuity of random attractors as the coefficient of the white noise term tends to zero.

Keywords: Stochastic strongly damped wave equation, unbounded domains, random attractor, upper semicontinuity.
Mathematics Subject Classification: Primary: 37L55; Secondary: 35B41, 35B40.
Citation: Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120
References:
[1]

L. Arnold, Random Dynamical Systems Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[3]

P. Chow, Stochastic wave equation with polynomial nonlinearity, Ann. Appl. Probab, 12 (2002), 361-381.  doi: 10.1214/aoap/1015961168.

[4]

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

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H. Crauel, Random Probability Measure on Polish Spaces Taylor & Francis, London, 2002. doi: 10.4324/9780203219119.

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

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

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H. Crauel, P. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stochastics and Dynamics, 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.

[9]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.

[10]

X. 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.

[11]

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

[12]

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

[13]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741.

[14]

R. Jones and B. Wang, Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. Real World Appl., 14 (2013), 1308-1322.  doi: 10.1016/j.nonrwa.2012.09.019.

[15]

H. Li, Y. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.

[16]

K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024.

[17]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.

[18]

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

[19]

Z. Shen, S. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[21]

B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discr. Contin. Dyn. Syst. Supplement, (2009), 800–809. doi: 10.3934/proc.2009.2009.800.

[22]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[23]

B. 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.

[24]

B. 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.

[25]

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

[26]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrodinger equations and its applications, Physica D, 88 (1995), 167-175.  doi: 10.1016/0167-2789(95)00196-B.

[27]

Z. Wang, S. Zhou and A. Gu, Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal. Real World Appl., 12 (2011), 3468-3482.  doi: 10.1016/j.nonrwa.2011.06.008.

[28]

Z. Wang, S. Zhou and A. Gu, Random attractor of the stochastic strongly damped wave equation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1649-1658.  doi: 10.1016/j.cnsns.2011.09.001.

[29]

Z. Wang and S. Zhou, Random attractors for non-autonomous stochastic strongly damped wave equation on unbounded domains, J. Appl. Anal. Comput., 5 (2015), 363-387.  doi: 10.11948/2015031.

[30]

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

[31]

S. Zhou, F. Yin and Z. Ouyang, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903.  doi: 10.1137/050623097.

[32]

S. Zhou and M. Zhao, Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonlinear. Anal., 133 (2016), 292-318.  doi: 10.1016/j.na.2015.12.013.

[33]

S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic damped wave equation with multiplicative noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[3]

P. Chow, Stochastic wave equation with polynomial nonlinearity, Ann. Appl. Probab, 12 (2002), 361-381.  doi: 10.1214/aoap/1015961168.

[4]

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

[5]

H. Crauel, Random Probability Measure on Polish Spaces Taylor & Francis, London, 2002. doi: 10.4324/9780203219119.

[6]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[7]

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

[8]

H. Crauel, P. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stochastics and Dynamics, 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.

[9]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.  doi: 10.1214/aop/1068646380.

[10]

X. 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.

[11]

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

[12]

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

[13]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741.

[14]

R. Jones and B. Wang, Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. Real World Appl., 14 (2013), 1308-1322.  doi: 10.1016/j.nonrwa.2012.09.019.

[15]

H. Li, Y. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.

[16]

K. Lu and B. Schmalfuß, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492.  doi: 10.1016/j.jde.2006.09.024.

[17]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.  doi: 10.1016/j.jde.2007.10.009.

[18]

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

[19]

Z. Shen, S. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.

[20]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[21]

B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discr. Contin. Dyn. Syst. Supplement, (2009), 800–809. doi: 10.3934/proc.2009.2009.800.

[22]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.

[23]

B. 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.

[24]

B. 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.

[25]

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

[26]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrodinger equations and its applications, Physica D, 88 (1995), 167-175.  doi: 10.1016/0167-2789(95)00196-B.

[27]

Z. Wang, S. Zhou and A. Gu, Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal. Real World Appl., 12 (2011), 3468-3482.  doi: 10.1016/j.nonrwa.2011.06.008.

[28]

Z. Wang, S. Zhou and A. Gu, Random attractor of the stochastic strongly damped wave equation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1649-1658.  doi: 10.1016/j.cnsns.2011.09.001.

[29]

Z. Wang and S. Zhou, Random attractors for non-autonomous stochastic strongly damped wave equation on unbounded domains, J. Appl. Anal. Comput., 5 (2015), 363-387.  doi: 10.11948/2015031.

[30]

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

[31]

S. Zhou, F. Yin and Z. Ouyang, Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903.  doi: 10.1137/050623097.

[32]

S. Zhou and M. Zhao, Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonlinear. Anal., 133 (2016), 292-318.  doi: 10.1016/j.na.2015.12.013.

[33]

S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic damped wave equation with multiplicative noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.

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