July  2021, 26(7): 3643-3665. 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  July 2021 Early access  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 and Continuous Dynamical Systems - B, 2021, 26 (7) : 3643-3665. doi: 10.3934/dcdsb.2020250
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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.

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

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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.

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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.

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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.

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[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.

[21]

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[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.

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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.

[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.

[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.

[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.

[27]

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[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.

[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.

[30]

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[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.

[32]

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[33]

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[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[45]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[45]

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

[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.

[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.

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