# American Institute of Mathematical Sciences

## 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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar [2] H. Crauel, Random Probability Measure on Polish Spaces, Taylor and Francis, London, 2002.  Google Scholar [3] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar [4] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar [5] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [15] W. Ma and Q. Ma, Attractors for the stochastic strongly damped plate equations with additive noise, J. Differential Equations, 111 (2013), 12 pp.  Google Scholar [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.   Google Scholar [17] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst., 2009 (2009), 800-809.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [2] H. Crauel, Random Probability Measure on Polish Spaces, Taylor and Francis, London, 2002.  Google Scholar [3] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar [4] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar [5] C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [15] W. Ma and Q. Ma, Attractors for the stochastic strongly damped plate equations with additive noise, J. Differential Equations, 111 (2013), 12 pp.  Google Scholar [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.   Google Scholar [17] V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discrete Contin. Dyn. Syst., 2009 (2009), 800-809.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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