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doi: 10.3934/mcrf.2020044

Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains

1. 

School of Mathematical Sciences and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, Sichuan 610068, China

2. 

School of Mathematical Sciences, Sichuan Normal University, Chengdu, Sichuan 610068, China

3. 

Department of Basic Courses, Sichuan Vocational College of Finance and Economics, Chengdu, Sichuan 610101, China

4. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

* Corresponding author: Jian Zhang

Received  May 2020 Revised  July 2020 Published  October 2020

We consider the dynamical behavior of fractional stochastic integro-differential equations with additive noise on unbounded domains. The existence and uniqueness of tempered random attractors for the equation in $ \mathbb{R}^{3} $ are proved. The upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero. The main difficulty is to show the pullback asymptotic compactness due to the lack of compactness on unbounded domains and the fact that the memory term includes the whole past history of the phenomenon. We establish such compactness by the tail-estimate method and the splitting method.

Citation: Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020044
References:
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Q. Bai, J. Shu, L. Li and H. Li, Dynamical behavior of non-autonomous fractional stochastic reaction-diffusion equations, J. Math. Anal. Appl., 485 (2020), 123833. doi: 10.1016/j.jmaa.2019.123833.  Google Scholar

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T. CaraballoJ. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

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T. CaraballoI. D. ChueshovP. Marin-Rubio and J. Real, Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst., 18 (2007), 253-270.  doi: 10.3934/dcds.2007.18.253.  Google Scholar

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T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set–Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

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T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[9]

M. ContiV. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.  doi: 10.1512/iumj.2006.55.2661.  Google Scholar

[10]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.  Google Scholar

[11]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[12]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[13]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

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E. DiNezzaG. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

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J. Dong and M. Xu, Space-time fractional Schr$\ddot{o}$dinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.  doi: 10.1016/j.jmaa.2008.03.061.  Google Scholar

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M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Progress in Nonlinear Differential Equations and Their Applications, Vol. 50, Birkh$\ddot{a}$user, Basel, 2002,155–178.  Google Scholar

[20]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.  Google Scholar

[21]

B. GuoY. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schr$\ddot{o}$dinger equation, Appl. Math. Comput., 204 (2008), 468-477.  doi: 10.1016/j.amc.2008.07.003.  Google Scholar

[22]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schr$\ddot{o}$dinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.  Google Scholar

[23]

B. Guo and M. Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138.  doi: 10.1016/j.jmaa.2009.09.009.  Google Scholar

[24]

B. Guo and G. Zhou, Ergodicity of the stochastic fractional reaction diffusion equation, Nonlinear Anal., 109 (2014), 1-22.  doi: 10.1016/j.na.2014.06.008.  Google Scholar

[25]

C. GuoJ. Shu and X. Wang, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations, Acta Math. Sin.(Engl. Ser.), 36 (2020), 318-336.  doi: 10.1007/s10114-020-8407-4.  Google Scholar

[26]

Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300.  doi: 10.1080/14689367.2018.1523368.  Google Scholar

[27]

Y. Lan and J. Shu, Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 2409-2431.  doi: 10.3934/cpaa.2019109.  Google Scholar

[28]

D. LiB. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equtions, 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.  Google Scholar

[29]

L. Li, J. Shu, Q. Bai and H. Li, Asymptotic behavior of fractional stochastic heat equations in materials with memory, Appl. Anal. (2019). doi: 10.1080/00036811.2019.1597057.  Google Scholar

[30]

J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[31]

L. Liu and T. Caraballo, Well-posedness and dynamics of a fractional stochastic integro-differential equation, Phys. D, 355 (2017), 45-57.  doi: 10.1016/j.physd.2017.05.006.  Google Scholar

[32]

H. LuP. W. BatesJ. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on $\mathbb{R}^{n}$, Nonlinear Anal., 128 (2015), 176-198.  doi: 10.1016/j.na.2015.06.033.  Google Scholar

[33]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.  Google Scholar

[34]

G. Lv and J. Duan, Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals, J. Math. Anal. Appl., 449 (2017), 176-194.  doi: 10.1016/j.jmaa.2016.12.011.  Google Scholar

[35]

G. LvH. GaoJ. Wei and J. Wu, BMO and Morrey Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.  doi: 10.1016/j.jde.2018.08.042.  Google Scholar

[36]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^{n}$ with continuous nonlinearity, Asymptot. Anal., 44 (2005), 111-130.   Google Scholar

[37]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.   Google Scholar

[38]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[39]

X. Pu and B. Guo, Global weak solutions of the fractional Landau-Lifshitz-Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98.  doi: 10.1016/j.jmaa.2010.06.035.  Google Scholar

[40]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Laudau equation, Appl. Anal., 92 (2013), 318-334.  doi: 10.1080/00036811.2011.614601.  Google Scholar

[41]

B. Schmalfuss, Backward cocycle and atttractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universit$\ddot{a}$t, (1992), 185–192. Google Scholar

[42]

T. Shen and J. Huang, Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal., 110 (2014), 33-46.  doi: 10.1016/j.na.2014.06.018.  Google Scholar

[43]

J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1587-1599.  doi: 10.3934/dcdsb.2017077.  Google Scholar

[44]

J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702. doi: 10.1063/1.4934724.  Google Scholar

[45]

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

[46]

B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006.  Google Scholar

[47]

B. Wang, Upper semicontinuity of random attractors for non-compact random systems, Electron J. Differential Equations, 139 (2009), 1-18.   Google Scholar

[48]

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

[49]

B. Wang, Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009. 1–31. doi: 10.1142/S0219493714500099.  Google Scholar

[50]

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

[51]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.  Google Scholar

[52]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[53]

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $\mathbb{R}^{3}$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

Q. Bai, J. Shu, L. Li and H. Li, Dynamical behavior of non-autonomous fractional stochastic reaction-diffusion equations, J. Math. Anal. Appl., 485 (2020), 123833. doi: 10.1016/j.jmaa.2019.123833.  Google Scholar

[3]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[4]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[5]

T. CaraballoJ. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

[6]

T. CaraballoI. D. ChueshovP. Marin-Rubio and J. Real, Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst., 18 (2007), 253-270.  doi: 10.3934/dcds.2007.18.253.  Google Scholar

[7]

T. CaraballoJ. A. LangaV. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set–Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[8]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[9]

M. ContiV. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.  doi: 10.1512/iumj.2006.55.2661.  Google Scholar

[10]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.  Google Scholar

[11]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[12]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[13]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[14]

E. DiNezzaG. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[15]

J. Dong and M. Xu, Space-time fractional Schr$\ddot{o}$dinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.  doi: 10.1016/j.jmaa.2008.03.061.  Google Scholar

[16]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[17]

C. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.  Google Scholar

[18]

C. GiorgiV. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049.  Google Scholar

[19]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Progress in Nonlinear Differential Equations and Their Applications, Vol. 50, Birkh$\ddot{a}$user, Basel, 2002,155–178.  Google Scholar

[20]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.  Google Scholar

[21]

B. GuoY. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schr$\ddot{o}$dinger equation, Appl. Math. Comput., 204 (2008), 468-477.  doi: 10.1016/j.amc.2008.07.003.  Google Scholar

[22]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schr$\ddot{o}$dinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.  Google Scholar

[23]

B. Guo and M. Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138.  doi: 10.1016/j.jmaa.2009.09.009.  Google Scholar

[24]

B. Guo and G. Zhou, Ergodicity of the stochastic fractional reaction diffusion equation, Nonlinear Anal., 109 (2014), 1-22.  doi: 10.1016/j.na.2014.06.008.  Google Scholar

[25]

C. GuoJ. Shu and X. Wang, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations, Acta Math. Sin.(Engl. Ser.), 36 (2020), 318-336.  doi: 10.1007/s10114-020-8407-4.  Google Scholar

[26]

Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300.  doi: 10.1080/14689367.2018.1523368.  Google Scholar

[27]

Y. Lan and J. Shu, Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 2409-2431.  doi: 10.3934/cpaa.2019109.  Google Scholar

[28]

D. LiB. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equtions, 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.  Google Scholar

[29]

L. Li, J. Shu, Q. Bai and H. Li, Asymptotic behavior of fractional stochastic heat equations in materials with memory, Appl. Anal. (2019). doi: 10.1080/00036811.2019.1597057.  Google Scholar

[30]

J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[31]

L. Liu and T. Caraballo, Well-posedness and dynamics of a fractional stochastic integro-differential equation, Phys. D, 355 (2017), 45-57.  doi: 10.1016/j.physd.2017.05.006.  Google Scholar

[32]

H. LuP. W. BatesJ. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on $\mathbb{R}^{n}$, Nonlinear Anal., 128 (2015), 176-198.  doi: 10.1016/j.na.2015.06.033.  Google Scholar

[33]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.  Google Scholar

[34]

G. Lv and J. Duan, Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals, J. Math. Anal. Appl., 449 (2017), 176-194.  doi: 10.1016/j.jmaa.2016.12.011.  Google Scholar

[35]

G. LvH. GaoJ. Wei and J. Wu, BMO and Morrey Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.  doi: 10.1016/j.jde.2018.08.042.  Google Scholar

[36]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^{n}$ with continuous nonlinearity, Asymptot. Anal., 44 (2005), 111-130.   Google Scholar

[37]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.   Google Scholar

[38]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[39]

X. Pu and B. Guo, Global weak solutions of the fractional Landau-Lifshitz-Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98.  doi: 10.1016/j.jmaa.2010.06.035.  Google Scholar

[40]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Laudau equation, Appl. Anal., 92 (2013), 318-334.  doi: 10.1080/00036811.2011.614601.  Google Scholar

[41]

B. Schmalfuss, Backward cocycle and atttractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universit$\ddot{a}$t, (1992), 185–192. Google Scholar

[42]

T. Shen and J. Huang, Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal., 110 (2014), 33-46.  doi: 10.1016/j.na.2014.06.018.  Google Scholar

[43]

J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1587-1599.  doi: 10.3934/dcdsb.2017077.  Google Scholar

[44]

J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702. doi: 10.1063/1.4934724.  Google Scholar

[45]

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

[46]

B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006.  Google Scholar

[47]

B. Wang, Upper semicontinuity of random attractors for non-compact random systems, Electron J. Differential Equations, 139 (2009), 1-18.   Google Scholar

[48]

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

[49]

B. Wang, Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009. 1–31. doi: 10.1142/S0219493714500099.  Google Scholar

[50]

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

[51]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.  Google Scholar

[52]

X. WangK. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[53]

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $\mathbb{R}^{3}$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.  Google Scholar

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