April  2019, 24(4): 1889-1917. doi: 10.3934/dcdsb.2018247

Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains

School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author

Received  January 2018 Revised  May 2018 Published  August 2018

Fund Project: Ma is supported by NSF grant(11561064, 11361053), and partly supported by NWNU-LKQN- 14-6

In this paper we study asymptotic behavior of a class of stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term. First we introduce a continuous random dynamical system 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, Qiaozhen Ma, Tingting Liu. Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1889-1917. doi: 10.3934/dcdsb.2018247
References:
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L. Arnold and Ludwig, Random Dynamical Systems, Berlin: Spinger-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

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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

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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

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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

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

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B. X. 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.  Google Scholar

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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.  Google Scholar

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B. X. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic J. Differential Equations, 139 (2009), 18pp.  Google Scholar

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Z. J. Wang and S. F. Zhou, Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains, J. Math. Anal. Appl., 384 (2011), 160-172.  doi: 10.1016/j.jmaa.2011.02.082.  Google Scholar

[27]

Z. J. WangS. F. Zhou and A. H. Gu, Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal., 12 (2011), 3468-3482.  doi: 10.1016/j.nonrwa.2011.06.008.  Google Scholar

[28]

Z. J. Wang and S. F Zhou, Random attractor for non-autonomous stochastic strongly damped wave equation on unbounded domains, J. Appl. Anal. Comput., 5 (2015), 363-387.   Google Scholar

[29]

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

[30]

H. B. 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

[31]

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

[32]

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

[33]

Z. J. Yang, A global attractor for the elastic waveguide model in $\mathbb{R}^n$, Nonlinear Anal., 74 (2011), 6640-6661.  doi: 10.1016/j.na.2011.06.045.  Google Scholar

[34]

Z. J. YangN. Feng and T. F. Ma, Global attractor for the generalized double dispersion equation, Nonlinear Anal., 115 (2015), 103-116.  doi: 10.1016/j.na.2014.12.006.  Google Scholar

[35]

B. X. Yao and Q. Z. Ma, Global attractors for a Kirchhoff type plate equation with memory, Kodai Math. J., 40 (2017), 63-78.  doi: 10.2996/kmj/1490083224.  Google Scholar

[36]

B. X. Yao and Q. Z. Ma, Global attractors of the extensible plate equations with nonlinear damping and memory, J. Funct. Spaces, 2017 (2017), Art. ID 4896161, 10 pp. doi: 10.1155/2017/4896161.  Google Scholar

[37]

G. C. Yue and C. K. 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

[38]

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

show all references

References:
[1]

L. Arnold and Ludwig, Random Dynamical Systems, Berlin: Spinger-Verlag, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

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

[3]

P. W. BatesK. N. Lu and B. X. 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.  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]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[6]

H. Crauel, Random Probability Measure on Polish Spaces, Taylor and Francis, London, 2002.  Google Scholar

[7]

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

[8]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stochastics-an International Journal of Probability & Stochastic Processes, 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[9]

P. G. Geredeli and I. Lasiecka, Asymptotic analysis and upper semicontinuity with respectto rotational inertia of attractors to von Karman plates with geometrically localized dissipation and critical nonlinearity, Nonlinear Anal., 91 (2013), 72-92.  doi: 10.1016/j.na.2013.06.008.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

Q. Z. Ma, Y. Y and X. L. Zhang, Existence of exponential attractors for the plate equations with strong damping, Electron. J. Differential Equations, 114 (2013), 10pp.  Google Scholar

[16]

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

[17]

Z. W. ShenS. F. Zhou and W. X. 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.  Google Scholar

[18]

X. Y. Shen and Q. Z. 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

[19]

M. A. J. Silva and T. F. Ma, Long-time dynamics for a class of Kirchhoff models with memory, J. Math. Phys., 54 (2013), 021505, 15pp. doi: 10.1063/1.4792606.  Google Scholar

[20]

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

[21]

B. X. Wang and X. L. Gao, Random attractors for wave equations on unbounded domains, Discrete & Continuous Dynamical Systems, 2009 (2009), 800-809.  doi: 10.1016/j.nonrwa.2011.06.008.  Google Scholar

[22]

B. X. Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal., 71 (2009), 2811-2828.  doi: 10.1016/j.na.2009.01.131.  Google Scholar

[23]

B. X. 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.  Google Scholar

[24]

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.  Google Scholar

[25]

B. X. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic J. Differential Equations, 139 (2009), 18pp.  Google Scholar

[26]

Z. J. Wang and S. F. Zhou, Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains, J. Math. Anal. Appl., 384 (2011), 160-172.  doi: 10.1016/j.jmaa.2011.02.082.  Google Scholar

[27]

Z. J. WangS. F. Zhou and A. H. Gu, Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal., 12 (2011), 3468-3482.  doi: 10.1016/j.nonrwa.2011.06.008.  Google Scholar

[28]

Z. J. Wang and S. F Zhou, Random attractor for non-autonomous stochastic strongly damped wave equation on unbounded domains, J. Appl. Anal. Comput., 5 (2015), 363-387.   Google Scholar

[29]

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

[30]

H. B. 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

[31]

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

[32]

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

[33]

Z. J. Yang, A global attractor for the elastic waveguide model in $\mathbb{R}^n$, Nonlinear Anal., 74 (2011), 6640-6661.  doi: 10.1016/j.na.2011.06.045.  Google Scholar

[34]

Z. J. YangN. Feng and T. F. Ma, Global attractor for the generalized double dispersion equation, Nonlinear Anal., 115 (2015), 103-116.  doi: 10.1016/j.na.2014.12.006.  Google Scholar

[35]

B. X. Yao and Q. Z. Ma, Global attractors for a Kirchhoff type plate equation with memory, Kodai Math. J., 40 (2017), 63-78.  doi: 10.2996/kmj/1490083224.  Google Scholar

[36]

B. X. Yao and Q. Z. Ma, Global attractors of the extensible plate equations with nonlinear damping and memory, J. Funct. Spaces, 2017 (2017), Art. ID 4896161, 10 pp. doi: 10.1155/2017/4896161.  Google Scholar

[37]

G. C. Yue and C. K. 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

[38]

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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