Asymptotic behavior of plate equations with memory driven by colored noise on unbounded domains

Xiao Bin Yao; Chan Yue; Xiao Bin Yao; Chan Yue

doi:10.3934/math.20221017

AIMS Mathematics

2022, Volume 7, Issue 10: 18497-18531. doi: 10.3934/math.20221017

Previous Article Next Article

Research article Special Issues

Asymptotic behavior of plate equations with memory driven by colored noise on unbounded domains

Xiao Bin Yao ^,,
Chan Yue

School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China

Received: 14 June 2022 Revised: 09 August 2022 Accepted: 15 August 2022 Published: 18 August 2022
MSC : 35B40, 60H15

The paper investigates mainly the asymptotic behavior of the non-autonomous random dynamical systems generated by the plate equations with memory driven by colored noise defined on $ \mathbb{R}^n $. Firstly, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.
- plate equation,
- colored noise,
- memory,
- asymptotic compactness,
- random attractor,
- unbounded domain
Citation: Xiao Bin Yao, Chan Yue. Asymptotic behavior of plate equations with memory driven by colored noise on unbounded domains[J]. AIMS Mathematics, 2022, 7(10): 18497-18531. doi: 10.3934/math.20221017

Related Papers:

Abstract

The paper investigates mainly the asymptotic behavior of the non-autonomous random dynamical systems generated by the plate equations with memory driven by colored noise defined on $ \mathbb{R}^n $. Firstly, we prove the well-posedness of the equation in the natural energy space. Secondly, we define a continuous cocycle associated with the solution operator. Finally, we establish the existence and uniqueness of random attractors of the equation by the uniform tail-ends estimates methods and the splitting technique.

References

[1]	L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer Verlag, Berlin, 1998. http://dx.doi.org/10.1007/BFb0095238
[2]	A. R. A. Barbosaa, T. F. Ma, Long-time dynamics of an extensible plate equation with thermal memory, J. Math. Anal. Appl., 416 (2014), 143–165. http://dx.doi.org/10.1016/j.jmaa.2014.02.042 doi: 10.1016/j.jmaa.2014.02.042
[3]	P. W. Bates, K. Lu, B. X. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domain, J. Differential Equations, 246 (2009), 845–869. http://dx.doi.org/10.1016/j.jde.2008.05.017 doi: 10.1016/j.jde.2008.05.017
[4]	C. Chen, F. S. Alotaibi, R. E. E. Omer, 3D Mathematical Modelling Technology in Visual Rehearsal System of Sports Dance, Appl. Math. Nonlin. Sci., 7 (2022), 113–122. http://dx.doi.org/10.2478/amns.2021.2.00078 doi: 10.2478/amns.2021.2.00078
[5]	C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297–308. http://dx.doi.org/10.1007/BF00251609 doi: 10.1007/BF00251609
[6]	A. Gu, B. Wang, Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1689-1720. http://dx.doi.org/10.3934/dcdsb.2018072 doi: 10.3934/dcdsb.2018072
[7]	A. Gu, B. Guo, B. Wang, Long term behavior of random Navier-Stokes equations driven by colored noise, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2495–2532. http://dx.doi.org/10.3934/dcdsb.2020020 doi: 10.3934/dcdsb.2020020
[8]	A. Kh. Khanmamedov, A global attractor for the plate equation with displacement-dependent damping, Non. Anal., 74 (2011), 1607–1615. http://dx.doi.org/10.1016/j.na.2010.10.031 doi: 10.1016/j.na.2010.10.031
[9]	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. http://dx.doi.org/10.1016/j.aml.2004.08.013 doi: 10.1016/j.aml.2004.08.013
[10]	A. Kh. Khanmamedov, Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain, J. Differ. Equ., 225 (2006), 528–548. http://dx.doi.org/10.1016/j.jde.2005.12.001 doi: 10.1016/j.jde.2005.12.001
[11]	J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Gauthier-Villars, Paris, 1969.
[12]	T. T. Liu, Q. Z. Ma, Time-dependent asymptotic behavior of the solution for plate equations with linear memory, Discrete Cont. Dyn-S., 23 (2018), 4595–4616. http://dx.doi.org/10.3934/dcdsb.2018178 doi: 10.3934/dcdsb.2018178
[13]	T. T. Liu, Q. Z. Ma, Time-dependent attractor for plate equations on $\mathbb{R}^n$, J. Math. Anal. Appl., 479 (2019), 315–332. http://dx.doi.org/10.1016/j.jmaa.2019.06.028 doi: 10.1016/j.jmaa.2019.06.028
[14]	T. T. Liu, Q. Z. Ma, The existence of time-dependent strong pullback attractors for non-autonomous plate equations, Chinese J. Contemporary Math.(English), 2 (2017), 101–118. http://dx.doi.org/10.16205/j.cnki.cama.2017.0011 doi: 10.16205/j.cnki.cama.2017.0011
[15]	W. J. Ma, Q. Z. Ma, Attractors for the stochastic strongly damped plate equations with additive noise, Electron. J. Differ. Equ., 111 (2013), 1–12.
[16]	Q. Z. Ma, W. J. 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]	Yanran Ma, Nan Chen and Han Lv, Back propagation mathematical model for stock price prediction, Appl. Math. Nonlin. Sci., 7 (2022), 165–174. http://dx.doi.org/10.2478/amns.2021.2.00144 doi: 10.2478/amns.2021.2.00144
[18]	M. Mohan Raja, V. Vijayakumar, A. Shukla, K. S. Nisar, H. M. Baskonus, On the approximate controllability results for fractional integrodifferential systems of order $1<r<2$ with sectorial operators, J. Comput. Appl. Math., 415 (2022), 165–174. http://dx.doi.org/10.1016/j.cam.2022.114492 doi: 10.1016/j.cam.2022.114492
[19]	V. Pata, A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2011), 505–529. http://dx.doi.org/10.1002/(SICI)1099-1476(20000510)23:7<633::AID-MMA135>3.0.CO;2-C doi: 10.1002/(SICI)1099-1476(20000510)23:7<633::AID-MMA135>3.0.CO;2-C
[20]	L. Ridolfi, P. D'Odorico, F. Laio, Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, Cambridge, 2011.
[21]	X. Y. Shen, Q. Z. Ma, The existence of random attractors for plate equations with memory and additive white noise, Korean J. Math., 24 (2016), 447–467. http://dx.doi.org/10.11568/kjm.2016.24.3.447 doi: 10.11568/kjm.2016.24.3.447
[22]	X. Y. Shen, 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. http://dx.doi.org/10.1016/j.camwa.2017.03.009 doi: 10.1016/j.camwa.2017.03.009
[23]	G. Uhlenbeck, L. Ornstein, On the theory of Brownian motion, Phys. Rev., 36 (1930), 823–841. http://dx.doi.org/10.1103/PhysRev.36.823 doi: 10.1103/PhysRev.36.823
[24]	M. Wang, G. Uhlenbeck, On the theory of Brownian motion. II, Rev. Modern Phys., 17 (1945), 323–342. http://dx.doi.org/10.1103/RevModPhys.17.323 doi: 10.1103/RevModPhys.17.323
[25]	B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41–52. http://dx.doi.org/10.1016/S0167-2789(98)00304-2 doi: 10.1016/S0167-2789(98)00304-2
[26]	R. Wang, L. Shi, B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^n$, Nonlinearity, 32 (2019), 4524–4556. http://dx.doi.org/10.1088/1361-6544/ab32d7 doi: 10.1088/1361-6544/ab32d7
[27]	X. Wang, K. Lu, B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2018), 378–424. http://dx.doi.org/10.1016/j.jde.2017.09.006 doi: 10.1016/j.jde.2017.09.006
[28]	B. Wang, Asymptotic behavior of supercritical wave equations driven by colored noise on unbounded domains, Discrete Cont. Dyn-S., 27 (2022), 4185–4229. http://dx.doi.org/10.3934/dcdsb.2021223 doi: 10.3934/dcdsb.2021223
[29]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544–1583. http://dx.doi.org/10.1016/j.jde.2012.05.015 doi: 10.1016/j.jde.2012.05.015
[30]	B. Wang, Long-time behavior for a nonlinear plate equation with thermal memory, J. Math. Anal. Appl., 348 (2008), 650–670. http://dx.doi.org/10.1016/j.jmaa.2008.08.001 doi: 10.1016/j.jmaa.2008.08.001
[31]	H. B. Xiao, Asymptotic dynamics of plate equations with a critical exponent on unbounded domain, Non. Anal., 70 (2009), 1288–1301. http://dx.doi.org/10.1016/j.na.2008.02.012 doi: 10.1016/j.na.2008.02.012
[32]	L. Yang, C. K. Zhong, Uniform attractor for non-autonomous plate equations with a localized damping and a critical nonlinearity, J. Math. Anal. Appl., 338 (2008), 1243–1254. http://dx.doi.org/10.1016/j.jmaa.2007.06.011 doi: 10.1016/j.jmaa.2007.06.011
[33]	L. Yang, C. K. Zhong, Global attractor for plate equation with nonlinear damping, Non. Anal., 69 (2008), 3802–3810. http://dx.doi.org/10.1016/j.na.2007.10.016 doi: 10.1016/j.na.2007.10.016
[34]	B. X. Yao, Q. Z. Ma, Ling Xu, Global attractors for a Kirchhoff type plate equation with memory, Kodai Math. J., 40 (2017), 63–78. http://dx.doi.org/10.2996/kmj/1490083224 doi: 10.2996/kmj/1490083224
[35]	B. X. Yao, Q. Z. Ma, Global attractors of the extensible plate equations with nonlinear damping and memory, J. Funct. Spaces, 2017 (2017), 1–11. http://dx.doi.org/10.1155/2017/4896161 doi: 10.1155/2017/4896161
[36]	X. B. Yao, Q. Z. Ma, T. T. Liu, Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains, Discrete Cont. Dyn-S., 24 (2019), 1889–1917. http://dx.doi.org/10.3934/dcdsb.2018247 doi: 10.3934/dcdsb.2018247
[37]	X. B. Yao, X. Liu, Asymptotic behavior for non-autonomous stochastic plate equation on unbounded domains, Open Math., 17 (2019), 1281–1302. http://dx.doi.org/10.1515/math-2019-0092 doi: 10.1515/math-2019-0092
[38]	X. B. Yao, Existence of a random attractor for non-autonomous stoc- hastic plate equations with additive noise and nonlinear damping on $\mathbb{R}^n$, Bound. Value Probl., 49 (2020), 1–27. http://dx.doi.org/10.1186/s13661-020-01346-z doi: 10.1186/s13661-020-01346-z
[39]	X. B. Yao, Random attractors for non-autonomous stochastic plate equations with multiplicative noise and nonlinear damping, AIMS Math., 5 (2020), 2577–2607. http://dx.doi.org/10.3934/math.2020169 doi: 10.3934/math.2020169
[40]	X. B. Yao, Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains, Discrete Cont. Dyn-S., 27 (2022), 443–468. http://dx.doi.org/10.3934/dcdsb.2021050 doi: 10.3934/dcdsb.2021050
[41]	X. B. Yao, Random attractors for stochastic plate equations with memory in unbounded domains, Open Math., 19 (2021), 1435–1460. http://dx.doi.org/10.1515/math-2021-0097 doi: 10.1515/math-2021-0097
[42]	G. C. Yue, C. K. Zhong, Global attractors for plate equations with critical exponent in locally uniform spaces, Non. Anal., 21 (2009), 4105–4114. http://dx.doi.org/10.1016/j.na.2009.02.089 doi: 10.1016/j.na.2009.02.089
[43]	Y. Zhang, J. Huang, J. Zhang, S. Liu, S. S. Huang, Analysis and prediction of second-hand house price based on random forest, Appl. Math. Nonlin. Sci., 7 (2022), 27–42. http://dx.doi.org/10.2478/amns.2022.1.00052 doi: 10.2478/amns.2022.1.00052
[44]	J. Zhou, Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping, Appl. Math. Comput., 265 (2015), 807–818. http://dx.doi.org/10.1016/j.amc.2015.05.098 doi: 10.1016/j.amc.2015.05.098

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)