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, doi: 10.3934/dcdsb.2018247
References:
[1]

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

[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.

[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.

[4]

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

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

[6]

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

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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.

[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.

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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.

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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.

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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.

[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.

[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.

[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.

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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.

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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.

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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.

[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.

[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.

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R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4684-0313-8.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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