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

Xiaobin Yao

doi:10.3934/dcdsb.2021050

Article Contents

2022, Volume 27, Issue 1: 443-468. Doi: 10.3934/dcdsb.2021050

This issue Previous Article Limit cycles and global dynamic of planar cubic semi-quasi-homogeneous systems Next Article Existence of global weak solutions of

$ p $

-Navier-Stokes equations

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

Xiaobin Yao^,

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

^* Corresponding author
^* Corresponding author

Received: September 30, 2020

Revised: November 30, 2020

Early access: February 2021

Published: January 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B25, 37L30; Secondary: 45K05.

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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.