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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 |
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.
References:
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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. |
[2] |
H. Crauel, Random Probability Measure on Polish Spaces, Taylor and Francis, London, 2002. |
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H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
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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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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.
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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. |
[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. |
show all references
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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