doi: 10.3934/dcdsb.2020250

Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author

Received  February 2020 Revised  July 2020 Published  August 2020

Fund Project: This work is supported by National Natural Science Foundation of China grant 11571283

We establish a new robustness theorem of delayed random attractors at zero-memory and the criteria are given by part convergence of cocycles along with regularity, recurrence and eventual compactness of attractors, where we relax the convergence condition of cocycles in all known robustness theorem of attractors, especially by Wang et al.(Siam-jads, 2015). As an application, we consider the stochastic non-autonomous 2D-Ginzburg-Landau delay equation, whose solutions seem not to be convergent for all initial data as the memory time goes to zero, but we can show the convergence of solutions toward zero-memory for part initial data in the lower-regular space. As a further result, we show that, for each memory time, the delay equation has a pullback random attractor such that it is upper semi-continuous at zero-memory.

Citation: Yangrong Li, Fengling Wang, Shuang Yang. Part-convergent cocycles and semi-convergent attractors of stochastic 2D-Ginzburg-Landau delay equations toward zero-memory. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020250
References:
[1]

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

[2]

H. BessaihM. J. Garrido-Atienza and B. Schmalfuss, On 3D Navier-Stokes equations: Regularization and uniqueness by delays, Phys. D, 376/377 (2018), 228-237.  doi: 10.1016/j.physd.2018.03.004.  Google Scholar

[3]

M. Böehm and B. Schmalfuss, Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3115-3138.  doi: 10.3934/dcdsb.2018303.  Google Scholar

[4]

T. CaraballoM.-J. Garrido-AtienzaJ. López-de-la-Cruz and A. Rapaport, Modeling and analysis of random and stochastic input flows in the chemostat model, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3591-3614.  doi: 10.3934/dcdsb.2018280.  Google Scholar

[5]

T. CaraballoX. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.  doi: 10.1137/14099930X.  Google Scholar

[6]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581.  doi: 10.1080/03605309808821394.  Google Scholar

[7]

G. A. ChechkinV. V. Chepyzhov and L. S. Pankratov, Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1133-1154.  doi: 10.3934/dcdsb.2018145.  Google Scholar

[8]

H. CrauelP. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[9]

H. CuiM. M. Freitas and J. A. Langa, Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1297-1324.  doi: 10.3934/dcdsb.2018152.  Google Scholar

[10]

H. CuiM. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3379-3407.  doi: 10.3934/dcdsb.2017142.  Google Scholar

[11]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374/375 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.  Google Scholar

[12]

H. CuiJ. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar

[13]

J. García-Luengo and P. Marín-Rubio, Reaction-diffusion equations with non-autonomous force in $H^{-1}$ and delays under measurability conditions on the driving delay term, J. Math. Anal. Appl., 417 (2014), 80-95.  doi: 10.1016/j.jmaa.2014.03.026.  Google Scholar

[14]

A. Gu, Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations, Discrete Contin. Dyn. Syst. B, 24 (2019), 5737-5767.  doi: 10.3934/dcdsb.2019104.  Google Scholar

[15]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.  Google Scholar

[16]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.  doi: 10.3934/cpaa.2014.13.2543.  Google Scholar

[17]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[18]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[19]

D. LiK. LuB. Wang and X. Wang, Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747.  doi: 10.3934/dcds.2019151.  Google Scholar

[20]

Y. LiL. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058.  Google Scholar

[21]

D. Li and L. Shi, Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay, J. Math. Phys., 59 (2018), 032703, 35 pp. doi: 10.1063/1.4994869.  Google Scholar

[22]

D. Li and L. Shi, Upper semicontinuity of random attractors of stochastic discrete complex Ginzburg-Landau equations with time-varying delays in the delay, J. Difference Equ. Appl., 24 (2018), 872-897.  doi: 10.1080/10236198.2018.1437913.  Google Scholar

[23]

D. Li and X. Wang, Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 449-465.  doi: 10.3934/dcdsb.2018181.  Google Scholar

[24]

Y. Li and S. Yang, Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 1155-1175.  doi: 10.3934/cpaa.2019056.  Google Scholar

[25]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.  Google Scholar

[26]

Y. Li and Q. Zhang, Backward stability and divided invariance of an attractor for the delayed Navier-Stokes equation, Taiwanese J. Math., 24 (2020), 575-601.  doi: 10.11650/tjm/190603.  Google Scholar

[27]

L. LiuT. Caraballo and P. Marin-Rubio, Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008.  Google Scholar

[28]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.  Google Scholar

[29]

C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, J. Amer. Math. Soc., 9 (1996), 1095-1133.  doi: 10.1090/S0894-0347-96-00207-X.  Google Scholar

[30]

J. C. Robinson, Stability of random attractors under perturbation and approximation, J. Differential Equations, 186 (2002), 652-669.  doi: 10.1016/S0022-0396(02)00038-4.  Google Scholar

[31]

J. Simsen and M. S. Simsen, On asymptotically autonomous dynamics for multivalued evolution problems, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3557-3567.  doi: 10.3934/dcdsb.2018278.  Google Scholar

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[33]

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

[34]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.  Google Scholar

[35]

G. WangB. Guo and Y. Li, The asymptotic behavior of the stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Comput., 198 (2008), 849-857.  doi: 10.1016/j.amc.2007.09.029.  Google Scholar

[36]

R. Wang and Y. Li, Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4145-4167.  doi: 10.3934/dcdsb.2019054.  Google Scholar

[37]

S. Wang and Y. Li, Probabilistic continuity of a pullback random attractor in time-sample, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2699-2722.  doi: 10.3934/dcdsb.2020028.  Google Scholar

[38]

F. Wang, J. Li and Y. Li, Random attractors for Ginzburg-Landau equations driven by difference noise of a Wiener-like process, Adv. Difference Equ., 2019 (2019), Paper No. 224, 17 pp. doi: 10.1186/s13662-019-2165-6.  Google Scholar

[39]

X. WangK. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.  Google Scholar

[40]

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

[41]

S. Wang and Q. Ma, Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1299-1316.  doi: 10.3934/dcdsb.2019221.  Google Scholar

[42]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734.  doi: 10.3934/dcdsb.2013.18.1715.  Google Scholar

[43]

D. Yang, The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise, J. Math. Phys., 45 (2004), 4064-4076.  doi: 10.1063/1.1794365.  Google Scholar

[44]

J. Yin, Y. Li and H. Cui, Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207. doi: 10.1016/j.jmaa.2017.01.064.  Google Scholar

[45]

K. Yosida, Functional Analysis, Fifth Edition, Springer-Verlag, Berlin Heidelberg, New York, 1978.  Google Scholar

[46]

Q. Zhang and Y. Li, Backward controller of a pullback attractor for delay Benjamin-Bona-Mahony equations, J. Dyn. Control Syst., 26 (2020), 423-441.  doi: 10.1007/s10883-019-09450-9.  Google Scholar

[47]

W. Zhao, Smoothing dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on $\Bbb R^N$ driven by multiplicative noises, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3453-3474.  doi: 10.3934/dcdsb.2018251.  Google Scholar

show all references

References:
[1]

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

[2]

H. BessaihM. J. Garrido-Atienza and B. Schmalfuss, On 3D Navier-Stokes equations: Regularization and uniqueness by delays, Phys. D, 376/377 (2018), 228-237.  doi: 10.1016/j.physd.2018.03.004.  Google Scholar

[3]

M. Böehm and B. Schmalfuss, Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3115-3138.  doi: 10.3934/dcdsb.2018303.  Google Scholar

[4]

T. CaraballoM.-J. Garrido-AtienzaJ. López-de-la-Cruz and A. Rapaport, Modeling and analysis of random and stochastic input flows in the chemostat model, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3591-3614.  doi: 10.3934/dcdsb.2018280.  Google Scholar

[5]

T. CaraballoX. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.  doi: 10.1137/14099930X.  Google Scholar

[6]

T. CaraballoJ. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581.  doi: 10.1080/03605309808821394.  Google Scholar

[7]

G. A. ChechkinV. V. Chepyzhov and L. S. Pankratov, Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1133-1154.  doi: 10.3934/dcdsb.2018145.  Google Scholar

[8]

H. CrauelP. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[9]

H. CuiM. M. Freitas and J. A. Langa, Squeezing and finite dimensionality of cocycle attractors for 2D stochastic Navier-Stokes equation with non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1297-1324.  doi: 10.3934/dcdsb.2018152.  Google Scholar

[10]

H. CuiM. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3379-3407.  doi: 10.3934/dcdsb.2017142.  Google Scholar

[11]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374/375 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.  Google Scholar

[12]

H. CuiJ. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar

[13]

J. García-Luengo and P. Marín-Rubio, Reaction-diffusion equations with non-autonomous force in $H^{-1}$ and delays under measurability conditions on the driving delay term, J. Math. Anal. Appl., 417 (2014), 80-95.  doi: 10.1016/j.jmaa.2014.03.026.  Google Scholar

[14]

A. Gu, Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations, Discrete Contin. Dyn. Syst. B, 24 (2019), 5737-5767.  doi: 10.3934/dcdsb.2019104.  Google Scholar

[15]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.  Google Scholar

[16]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.  doi: 10.3934/cpaa.2014.13.2543.  Google Scholar

[17]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[18]

Y. LiA. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[19]

D. LiK. LuB. Wang and X. Wang, Limiting dynamics for non-autonomous stochastic retarded reaction-diffusion equations on thin domains, Discrete Contin. Dyn. Syst., 39 (2019), 3717-3747.  doi: 10.3934/dcds.2019151.  Google Scholar

[20]

Y. LiL. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058.  Google Scholar

[21]

D. Li and L. Shi, Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay, J. Math. Phys., 59 (2018), 032703, 35 pp. doi: 10.1063/1.4994869.  Google Scholar

[22]

D. Li and L. Shi, Upper semicontinuity of random attractors of stochastic discrete complex Ginzburg-Landau equations with time-varying delays in the delay, J. Difference Equ. Appl., 24 (2018), 872-897.  doi: 10.1080/10236198.2018.1437913.  Google Scholar

[23]

D. Li and X. Wang, Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 449-465.  doi: 10.3934/dcdsb.2018181.  Google Scholar

[24]

Y. Li and S. Yang, Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 1155-1175.  doi: 10.3934/cpaa.2019056.  Google Scholar

[25]

Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.  Google Scholar

[26]

Y. Li and Q. Zhang, Backward stability and divided invariance of an attractor for the delayed Navier-Stokes equation, Taiwanese J. Math., 24 (2020), 575-601.  doi: 10.11650/tjm/190603.  Google Scholar

[27]

L. LiuT. Caraballo and P. Marin-Rubio, Stability results for 2D Navier-Stokes equations with unbounded delay, J. Differential Equations, 265 (2018), 5685-5708.  doi: 10.1016/j.jde.2018.07.008.  Google Scholar

[28]

K. Lu and B. Wang, Wong-Zakai approximations and long term behavior of stochastic partial differential equations, J. Dynam. Differential Equations, 31 (2019), 1341-1371.  doi: 10.1007/s10884-017-9626-y.  Google Scholar

[29]

C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, J. Amer. Math. Soc., 9 (1996), 1095-1133.  doi: 10.1090/S0894-0347-96-00207-X.  Google Scholar

[30]

J. C. Robinson, Stability of random attractors under perturbation and approximation, J. Differential Equations, 186 (2002), 652-669.  doi: 10.1016/S0022-0396(02)00038-4.  Google Scholar

[31]

J. Simsen and M. S. Simsen, On asymptotically autonomous dynamics for multivalued evolution problems, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3557-3567.  doi: 10.3934/dcdsb.2018278.  Google Scholar

[32]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[33]

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

[34]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.  Google Scholar

[35]

G. WangB. Guo and Y. Li, The asymptotic behavior of the stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Comput., 198 (2008), 849-857.  doi: 10.1016/j.amc.2007.09.029.  Google Scholar

[36]

R. Wang and Y. Li, Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 4145-4167.  doi: 10.3934/dcdsb.2019054.  Google Scholar

[37]

S. Wang and Y. Li, Probabilistic continuity of a pullback random attractor in time-sample, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2699-2722.  doi: 10.3934/dcdsb.2020028.  Google Scholar

[38]

F. Wang, J. Li and Y. Li, Random attractors for Ginzburg-Landau equations driven by difference noise of a Wiener-like process, Adv. Difference Equ., 2019 (2019), Paper No. 224, 17 pp. doi: 10.1186/s13662-019-2165-6.  Google Scholar

[39]

X. WangK. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.  Google Scholar

[40]

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

[41]

S. Wang and Q. Ma, Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1299-1316.  doi: 10.3934/dcdsb.2019221.  Google Scholar

[42]

F. Wu and P. E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1715-1734.  doi: 10.3934/dcdsb.2013.18.1715.  Google Scholar

[43]

D. Yang, The asymptotic behavior of the stochastic Ginzburg-Landau equation with multiplicative noise, J. Math. Phys., 45 (2004), 4064-4076.  doi: 10.1063/1.1794365.  Google Scholar

[44]

J. Yin, Y. Li and H. Cui, Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207. doi: 10.1016/j.jmaa.2017.01.064.  Google Scholar

[45]

K. Yosida, Functional Analysis, Fifth Edition, Springer-Verlag, Berlin Heidelberg, New York, 1978.  Google Scholar

[46]

Q. Zhang and Y. Li, Backward controller of a pullback attractor for delay Benjamin-Bona-Mahony equations, J. Dyn. Control Syst., 26 (2020), 423-441.  doi: 10.1007/s10883-019-09450-9.  Google Scholar

[47]

W. Zhao, Smoothing dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on $\Bbb R^N$ driven by multiplicative noises, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3453-3474.  doi: 10.3934/dcdsb.2018251.  Google Scholar

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