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doi: 10.3934/era.2021028

Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise

1. 

School of Mathematics and Compute Science, Liupanshui Normal University, Liupanshui, Guizhou 553004, China

2. 

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

3. 

College of Sciences, Beijing Information Science and Technology University, Beijing 100192, China

Received  November 2020 Revised  March 2021 Published  April 2021

The global well-posedness and long-time mean random dynamics are studied for a high-dimensional non-autonomous stochastic nonlinear lattice pseudo-parabolic equation with locally Lipschitz drift and diffusion terms. The existence and uniqueness of three different types of weak pullback mean random attractors as well as their relations are established for the mean random dynamical systems generated by the solution operators. This is the first paper to study the well-posedness and dynamics of the stochastic lattice pseudo-parabolic equation even when the nonlinear noise reduces to the linear one.

Citation: Lianbing She, Nan Liu, Xin Li, Renhai Wang. Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise. Electronic Research Archive, doi: 10.3934/era.2021028
References:
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P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.  Google Scholar

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X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

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S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.  Google Scholar

[21]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.  Google Scholar

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X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

[23]

R. WangL. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^N$, Nonlinearity, 32 (2019), 4524-4556.  doi: 10.1088/1361-6544/ab32d7.  Google Scholar

[24]

R. Wang and B. Wang, Random dynamics of $p$-Laplacian Lattice systems driven by infinite-dimensional nonlinear noise, Stochastic Process. Appl., 130 (2020), 7431-7462.  doi: 10.1016/j.spa.2020.08.002.  Google Scholar

[25]

X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141.  Google Scholar

[26]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

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R. Xu and Y. Niu, Addendum to "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations" [J. Func. Anal. 264 (2013), 2732–2763], J. Funct. Anal., 270 (2016), 4039-4041.  doi: 10.1016/j.jfa.2016.02.026.  Google Scholar

[28]

C. Zhang and L. Zhao, The attractors for $2$nd-order stochastic delay lattice systems, Discrete Contin. Dyn. Syst., 37 (2017), 575-590.  doi: 10.3934/dcds.2017023.  Google Scholar

[29]

W. Zhao and S. Song, Dynamics of stochastic nonclassical diffusion equations on unbounded domains, Electronic J. Differential Equations, 282 (2015), 22 pp.  Google Scholar

[30]

C. Zhao and S. Zhou, Limiting behavior of a global attractor for lattice nonclassical parabolic equations, Appl. Math. Lett., 20 (2007), 829-834.  doi: 10.1016/j.aml.2006.06.019.  Google Scholar

[31]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.  Google Scholar

[32]

X. ZhuF. Li and Y. Li, Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term, Appl. Math. Comput., 329 (2018), 38-51.  doi: 10.1016/j.amc.2018.02.003.  Google Scholar

show all references

References:
[1]

E. C. Aifantis, On the problem of diffusion in solids, Acta Mech., 37 (1980), 265-296.  doi: 10.1007/BF01202949.  Google Scholar

[2]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.  Google Scholar

[3]

T. Caraballo, B. Guo, N. H. Tuan and R. Wang, Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, (2020), 1–31. doi: 10.1017/prm.2020.77.  Google Scholar

[4]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.  Google Scholar

[5]

S. Cooper and A. Savostianov, Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations, Adv. Nonlinear Anal., 9 (2020), 745-787.  doi: 10.1515/anona-2020-0024.  Google Scholar

[6]

M. J. Dos SantosB. FengD. S. Almeida Jénior and M. L. Santos, Global and exponential attractors for a nonlinear porous elastic system with delay term, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2805-2828.   Google Scholar

[7]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction-diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I.  Google Scholar

[8]

M. J. Garrido-AtienzaB. Maslowski and J. Šnupárková, Semilinear stochastic equations with bilinear fractional noise, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3075-3094.  doi: 10.3934/dcdsb.2016088.  Google Scholar

[9]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.  Google Scholar

[10]

P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.  doi: 10.1016/j.jde.2012.05.016.  Google Scholar

[11]

K. Kuttler and E. Aifantis, Quasilinear evolution equations in nonclassical diffusion, SIAM J. Math. Anal., 19 (1988), 110-120.  doi: 10.1137/0519008.  Google Scholar

[12]

D. LiK. LuB. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009.  Google Scholar

[13]

M. Sui and Y. Wang, Upper semicontinuity of pullback attractors for lattice nonclassical diffusion delay equations under singular perturbations, Appl. Math. Comput., 242 (2014), 315-327.  doi: 10.1016/j.amc.2014.05.045.  Google Scholar

[14]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.  Google Scholar

[15]

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.  Google Scholar

[16]

B. Wang, Weak pullback attractors for mean random dynamical systems in Bochner spaces, J. Dynam. Differential Equations, 31 (2019), 2177-2204.  doi: 10.1007/s10884-018-9696-5.  Google Scholar

[17]

B. Wang, Dynamics of fractional stochastic reaction-diffusion equations on unbounded domains driven by nonlinear noise, J. Differential Equations, 268 (2019), 1-59.  doi: 10.1016/j.jde.2019.08.007.  Google Scholar

[18]

R. Wang, Long-time dynamics of stochastic lattice plate equations with nonlinear noise and damping, J. Dynam. Differential Equations, (2020). doi: 10.1007/s10884-020-09830-x.  Google Scholar

[19]

R. Wang, B. Guo and B. Wang, Well-posedness and dynamics of fractional FitzHugh-Nagumo systems on $\mathbb{R}^N$ driven by nonlinear noise, Sci. China Math., (2020). doi: 10.1007/s11425-019-1714-2.  Google Scholar

[20]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.  Google Scholar

[21]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.  Google Scholar

[22]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.  Google Scholar

[23]

R. WangL. Shi and B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on $\mathbb{R}^N$, Nonlinearity, 32 (2019), 4524-4556.  doi: 10.1088/1361-6544/ab32d7.  Google Scholar

[24]

R. Wang and B. Wang, Random dynamics of $p$-Laplacian Lattice systems driven by infinite-dimensional nonlinear noise, Stochastic Process. Appl., 130 (2020), 7431-7462.  doi: 10.1016/j.spa.2020.08.002.  Google Scholar

[25]

X. Wang and R. Xu, Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.  doi: 10.1515/anona-2020-0141.  Google Scholar

[26]

R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[27]

R. Xu and Y. Niu, Addendum to "Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations" [J. Func. Anal. 264 (2013), 2732–2763], J. Funct. Anal., 270 (2016), 4039-4041.  doi: 10.1016/j.jfa.2016.02.026.  Google Scholar

[28]

C. Zhang and L. Zhao, The attractors for $2$nd-order stochastic delay lattice systems, Discrete Contin. Dyn. Syst., 37 (2017), 575-590.  doi: 10.3934/dcds.2017023.  Google Scholar

[29]

W. Zhao and S. Song, Dynamics of stochastic nonclassical diffusion equations on unbounded domains, Electronic J. Differential Equations, 282 (2015), 22 pp.  Google Scholar

[30]

C. Zhao and S. Zhou, Limiting behavior of a global attractor for lattice nonclassical parabolic equations, Appl. Math. Lett., 20 (2007), 829-834.  doi: 10.1016/j.aml.2006.06.019.  Google Scholar

[31]

S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.  Google Scholar

[32]

X. ZhuF. Li and Y. Li, Global solutions and blow up solutions to a class of pseudo-parabolic equations with nonlocal term, Appl. Math. Comput., 329 (2018), 38-51.  doi: 10.1016/j.amc.2018.02.003.  Google Scholar

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