# American Institute of Mathematical Sciences

January  2022, 27(1): 443-468. doi: 10.3934/dcdsb.2021050

## Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains

 School of Mathematics and Statistics, Qinghai Nationalities University, Xi'ning, Qinghai 810007, China

* Corresponding author

Received  October 2020 Revised  December 2020 Published  January 2022 Early access  February 2021

Fund Project: Yao is supported by NSFC grant (11561064, 11361053), Key projects of university level planning in Qinghai Nationalities University grant(2021XJGH01), Scientific Research Innovation Team in Qinghai Nationalities University

In this paper we study asymptotic behavior of a class of stochastic plate equations with memory and additive noise. First we introduce a continuous cocycle for the equation and establish the pullback asymptotic compactness of solutions. Second we consider the existence and upper semicontinuity of random attractors for the equation.

Citation: Xiaobin Yao. Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 443-468. doi: 10.3934/dcdsb.2021050
##### References:
 [1] A. R. A. Barbosaa 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. [2] H. Crauel, Random Probability Measure on Polish Spaces, Taylor and Francis, London, 2002. [3] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225. [4] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [5] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [6] F. Flandoli and B. Schmalfuss, Random attractors for the 3$D$ stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083. [7] M. A. Jorge Silva and T. F. Ma, Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys., 54 (2013), 021505, 15 pp. doi: 10.1063/1.4792606. [8] N. Ju, The $H^1$-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains, Nonlinearity, 13 (2000), 1227-1238.  doi: 10.1088/0951-7715/13/4/313. [9] A. Kh. Khanmamedov, A global attractor for the plate equation with displacement-dependent damping, Nonlinear Anal., 74 (2011), 1607-1615.  doi: 10.1016/j.na.2010.10.031. [10] A. Kh. Khanmamedov, Existence of global attractor for the plate equation with the critical exponent in an unbounded domain, Appl. Math. Lett., 18 (2005), 827-832.  doi: 10.1016/j.aml.2004.08.013. [11] A. Kh. Khanmamedov, Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain, J. Differential Equations, 225 (2006), 528-548.  doi: 10.1016/j.jde.2005.12.001. [12] T. Liu and Q. Ma, Time-dependent asymptotic behavior of the solution for plate equations with linear memory, Discrete Contin. Dyn. Syst. Ser., 23 (2018), 4595-4616.  doi: 10.3934/dcdsb.2018178. [13] T. Liu and Q. Ma, Time-dependent attractor for plate equations on $\mathbb{R}^n$, J. Math. Anal. Appl., 479 (2019), 315-332.  doi: 10.1016/j.jmaa.2019.06.028. [14] T. T. Liu and Q. Z. Ma, Existence of time-dependent strong pullback attractors for non-autonomous plate equations, Chinese Journal of Contemporary Mathematics, 38 (2017), 101-118. [15] W. Ma and Q. Ma, Attractors for the stochastic strongly damped plate equations with additive noise, J. Differential Equations, 111 (2013), 12 pp. [16] W. J. Ma and Q. Z. Ma, Asymptotic behavior of solutions for stochastic plate equations with strongly damped and white noise, J. Northwest Norm. Univ. Nat. Sci., 50 (2014), 6-17. [17] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. [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] X. Y. Shen and Q. Z. Ma, The existence of random attractors for plate equations with memory and additive white noise, Korean J. Math., 24 (2016), 447-467.  doi: 10.11568/kjm.2016.24.3.447. [20] X. Shen and Q. Ma, Existence of random attractors for weakly dissipative plate equations with memory and additive white noise, Comput. Math. Appl., 73 (2017), 2258-2271.  doi: 10.1016/j.camwa.2017.03.009. [21] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8. [22] 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. [23] B. 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. [24] B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst., 2009 (2009), 800-809. [25] 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. [26] H. Wu, Long-time behavior for a nonlinear plate equation with thermal memory, J. Math. Anal. Appl., 348 (2008), 650-670.  doi: 10.1016/j.jmaa.2008.08.001. [27] H. Xiao, Asymptotic dynamics of plate equations with a critical exponent on unbounded domain, Nonlinear Anal., 70 (2009), 1288-1301.  doi: 10.1016/j.na.2008.02.012. [28] L. Yang, Uniform attractor for non-autonomous plate equations with a localized damping and a critical nonlinearity, J. Math. Anal. Appl., 338 (2008), 1243-1254.  doi: 10.1016/j.jmaa.2007.06.011. [29] L. Yang and C.-K. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Anal., 69 (2008), 3802-3810.  doi: 10.1016/j.na.2007.10.016. [30] X. Yao, Existence of a random attractor for non-autonomous stochastic plate equations with additive noise and nonlinear damping on $\mathbb{R}^n$, Boundary Value Problems, 49 (2020), Paper No. 49, 27 pp. doi: 10.1186/s13661-020-01346-z. [31] X. Yao, Random attractors for non-autonomous stochastic plate equations with multiplicative noise and nonlinear damping, AIMS Math., 5 (2020), 2577-2607.  doi: 10.3934/math.2020169. [32] X. Yao and X. Liu, Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains, Open Math., 17 (2019), 1281-1302.  doi: 10.1515/math-2019-0092. [33] X. Yao, Q. Ma and T. Liu, Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1889-1917.  doi: 10.3934/dcdsb.2018247. [34] G. Yue and C. Zhong, Global attractors for plate equations with critical exponent in locally uniform spaces, Nonlinear Anal., 71 (2009), 4105-4114.  doi: 10.1016/j.na.2009.02.089. [35] J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput., 265 (2015), 807-818.  doi: 10.1016/j.amc.2015.05.098.

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##### References:
 [1] A. R. A. Barbosaa 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. [2] H. Crauel, Random Probability Measure on Polish Spaces, Taylor and Francis, London, 2002. [3] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225. [4] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705. [5] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609. [6] F. Flandoli and B. Schmalfuss, Random attractors for the 3$D$ stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083. [7] M. A. Jorge Silva and T. F. Ma, Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys., 54 (2013), 021505, 15 pp. doi: 10.1063/1.4792606. [8] N. Ju, The $H^1$-compact global attractor for the solutions to the Navier-Stokes equations in two-dimensional unbounded domains, Nonlinearity, 13 (2000), 1227-1238.  doi: 10.1088/0951-7715/13/4/313. [9] A. Kh. Khanmamedov, A global attractor for the plate equation with displacement-dependent damping, Nonlinear Anal., 74 (2011), 1607-1615.  doi: 10.1016/j.na.2010.10.031. [10] A. Kh. Khanmamedov, Existence of global attractor for the plate equation with the critical exponent in an unbounded domain, Appl. Math. Lett., 18 (2005), 827-832.  doi: 10.1016/j.aml.2004.08.013. [11] A. Kh. Khanmamedov, Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain, J. Differential Equations, 225 (2006), 528-548.  doi: 10.1016/j.jde.2005.12.001. [12] T. Liu and Q. Ma, Time-dependent asymptotic behavior of the solution for plate equations with linear memory, Discrete Contin. Dyn. Syst. Ser., 23 (2018), 4595-4616.  doi: 10.3934/dcdsb.2018178. [13] T. Liu and Q. Ma, Time-dependent attractor for plate equations on $\mathbb{R}^n$, J. Math. Anal. Appl., 479 (2019), 315-332.  doi: 10.1016/j.jmaa.2019.06.028. [14] T. T. Liu and Q. Z. Ma, Existence of time-dependent strong pullback attractors for non-autonomous plate equations, Chinese Journal of Contemporary Mathematics, 38 (2017), 101-118. [15] W. Ma and Q. Ma, Attractors for the stochastic strongly damped plate equations with additive noise, J. Differential Equations, 111 (2013), 12 pp. [16] W. J. Ma and Q. Z. Ma, Asymptotic behavior of solutions for stochastic plate equations with strongly damped and white noise, J. Northwest Norm. Univ. Nat. Sci., 50 (2014), 6-17. [17] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529. [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] X. Y. Shen and Q. Z. Ma, The existence of random attractors for plate equations with memory and additive white noise, Korean J. Math., 24 (2016), 447-467.  doi: 10.11568/kjm.2016.24.3.447. [20] X. Shen and Q. Ma, Existence of random attractors for weakly dissipative plate equations with memory and additive white noise, Comput. Math. Appl., 73 (2017), 2258-2271.  doi: 10.1016/j.camwa.2017.03.009. [21] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8. [22] 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. [23] B. 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. [24] B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst., 2009 (2009), 800-809. [25] 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. [26] H. Wu, Long-time behavior for a nonlinear plate equation with thermal memory, J. Math. Anal. Appl., 348 (2008), 650-670.  doi: 10.1016/j.jmaa.2008.08.001. [27] H. Xiao, Asymptotic dynamics of plate equations with a critical exponent on unbounded domain, Nonlinear Anal., 70 (2009), 1288-1301.  doi: 10.1016/j.na.2008.02.012. [28] L. Yang, Uniform attractor for non-autonomous plate equations with a localized damping and a critical nonlinearity, J. Math. Anal. Appl., 338 (2008), 1243-1254.  doi: 10.1016/j.jmaa.2007.06.011. [29] L. Yang and C.-K. Zhong, Global attractor for plate equation with nonlinear damping, Nonlinear Anal., 69 (2008), 3802-3810.  doi: 10.1016/j.na.2007.10.016. [30] X. Yao, Existence of a random attractor for non-autonomous stochastic plate equations with additive noise and nonlinear damping on $\mathbb{R}^n$, Boundary Value Problems, 49 (2020), Paper No. 49, 27 pp. doi: 10.1186/s13661-020-01346-z. [31] X. Yao, Random attractors for non-autonomous stochastic plate equations with multiplicative noise and nonlinear damping, AIMS Math., 5 (2020), 2577-2607.  doi: 10.3934/math.2020169. [32] X. Yao and X. Liu, Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains, Open Math., 17 (2019), 1281-1302.  doi: 10.1515/math-2019-0092. [33] X. Yao, Q. Ma and T. Liu, Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1889-1917.  doi: 10.3934/dcdsb.2018247. [34] G. Yue and C. Zhong, Global attractors for plate equations with critical exponent in locally uniform spaces, Nonlinear Anal., 71 (2009), 4105-4114.  doi: 10.1016/j.na.2009.02.089. [35] J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput., 265 (2015), 807-818.  doi: 10.1016/j.amc.2015.05.098.
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