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July  2019, 24(7): 3395-3438. doi: 10.3934/dcdsb.2018326

Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received  May 2018 Revised  July 2018 Published  January 2019

Fund Project: The author is supported by NSF grant 11601046, CTBU Grant 1751041 and Chongqing key laboratory of social economy and applied statistics

In this article, a notion of bi-spatial continuous random dynamical system is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptotically compact and random absorbing in the initial space, then it admits a bi-spatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the non-autonomous stochastic fractional power dissipative equation on $ \mathbb{R}^N $ with additive white noise and a polynomial-like growth nonlinearity of order $ p, p\geq2 $. We prove that this equation generates a bi-spatial $ (L^2(\mathbb{R}^N), H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)) $-continuous random dynamical system, and the random dynamics for this system is captured by a bi-spatial pullback attractor which is compact and attracting in $ H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N) $, where $ H^s(\mathbb{R}^N) $ is a fractional Sobolev space with $ s\in(0,1) $. Especially, the measurability of pullback attractor is individually derived by proving the the continuity of solutions in $ H^s(\mathbb{R}^N) $ and $ L^p(\mathbb{R}^N) $ with respect to the sample. A difference estimates approach, rather than the usual truncation estimate and spectral decomposition technique, is employed to overcome the loss of Sobolev compact embedding in $ H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N),s\in(0,1),N\geq1 $.

Citation: Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326
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L. Arnold and B. Schmalfuss, Fixed points and attractors for random dynamical systems, in Advances in Nonlinear Stochastic Mechanics(eds. A. Naess and S. Krenk), Solid Mech. Appl. vol. 47, Kluwer Acad. Publ. Dordrecht, (1996), 19-28. doi: 10.1007/978-94-009-0321-0_3.

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J. P. Aubin and H. Frankowska, Set-Valued Analysis, Springer, Boston, 1990. doi: 10.1007/978-0-8176-4848-0.

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A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

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T. Bartsch and Z. Liu, On a supperlinear elliptic p-Laplacian equation, J. Differential Equations, 198 (2004), 149-175. doi: 10.1016/j.jde.2003.08.001.

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

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D. CaoC. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020.

[8]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Nonautonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.

[9]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst., 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439.

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T. CaraballoF. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference. Equ. Appl., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010.

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A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Appl. Math. Sciences, 184, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.

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C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, 580, Springer-Verlag, Berlin, 1977.

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I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

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

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H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225.

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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.

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E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

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A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on Rn, J. Differential Equations, 264 (2018), 7094-7137. doi: 10.1016/j.jde.2018.02.011.

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P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268. doi: 10.1090/proc/12735.

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P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference. Equ. Appl., 22 (2016), 513-525. doi: 10.1080/10236198.2015.1107550.

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A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038. doi: 10.1016/j.jmaa.2014.03.037.

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[28]

L. LiJ. Sun and S. Tersian, Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian, Fract. Calc. Appl. Anal., 20 (2017), 1146-1164. doi: 10.1515/fca-2017-0061.

[29]

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.

[30]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800. doi: 10.1016/j.jde.2008.06.031.

[31]

Y. LiA. Gu and J. Li, Existences and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.

[32]

H. Lu, P. W. Bates, J. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on Rn, Nonlinear Anal., 128 (2015), 176-198. doi: 10.1016/j.na.2015.06.033.

[33]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295. doi: 10.4310/CMS.2016.v14.n1.a11.

[34]

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

[35]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[36]

J. C. Robinson, Infinite-Dimensional Dyanmical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

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B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (eds. V. Reitmann, T. Riedrich and N. Koksch), Technische Universität, Dresden, (1992), 185-192.

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B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in International Conference on Differential Equations (eds. B. Fiedler, K. Gröger and J. Sprekels), World Sci. Publishing, Singapore, (2000), 684-690.

[39] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NY, 1970.
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P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680.

[41]

B. Q. Tang, Regularity of pullback random attractors for stochasitic FitzHugh-Nagumo system on unbounded domains, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 441-466. doi: 10.3934/dcds.2015.35.441.

[42]

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

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B. Wang, Random attractors for non-autonomous stochastic wave euqations with multiplicative noises, Discrete Contin. Dyn. Syst., 34 (2014), 269-330. doi: 10.3934/dcds.2014.34.269.

[44]

B. Wang, Suffcient 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.

[45]

B. Wang, Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2011-2051. doi: 10.3934/dcdsb.2017119.

[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.

[47]

Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamcal systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026.

[48]

Z. Wang and S. Zhou, Random attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems with random coupled coefficients, Taiwanese J. Math., 20 (2016), 589-616. doi: 10.11650/tjm.20.2016.6699.

[49]

J. Yin and Y. Li, Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic p-Laplacian equations on RN, Math. Meth. Appl. Sci., 40 (2017), 4863-4879.

[50]

J. YinY. 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.

[51]

K. Yosida, Functional Analysis, Springer-Verlag, New York-Heidelberg, 1974. doi: 10.1007/978-3-642-61859-8.

[52]

W. Zhao and Y. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space $\ell_\rho^p$, Appl. Math. Comput., 291 (2016), 226-243. doi: 10.1016/j.amc.2016.06.045.

[53]

W. Zhao, Continuity and random dynamics of the non-autonomous stochastic FitzHugh-Nagumo system on $\mathbb{R}^N$, Comput. Math. Appl., 75 (2018), 3801-3824. doi: 10.1016/j.camwa.2018.02.031.

[54]

W. Zhao, Random dynamics of stochastic p-Laplacian equations on $\mathbb{R}^N$ with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203. doi: 10.1016/j.jmaa.2017.06.025.

[55]

W. Zhao, Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2499-2526. doi: 10.3934/dcdsb.2018065.

[56]

W. Zhao and Y. Li, (L2, Lp)-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal., 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050.

[57]

W. Zhao, Regularity of random attractors for a degenerate parabolic equations driven by additive noises, Applied Math. Comput., 239 (2014), 358-374. doi: 10.1016/j.amc.2014.04.106.

[58]

Q. Zhou and K. Wang, Existence and multiplicity of solutions for nonlinear elliptic problems with the fractional Laplacian, Fract. Calc. Appl. Anal., 18 (2015), 133-145. doi: 10.1515/fca-2015-0009.

[59]

K. Zhu and F. Zhou, Continuity and pullback attractors for a non-autonomous reaction-diffusion equation in $\mathbb{R}^N$, Comput. Math. Appl., 71 (2016), 2089-2105. doi: 10.1016/j.camwa.2016.04.004.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer, Berlin, Heidelberg, 1998. doi: 10.1007/BFb0095238.

[2]

L. Arnold and B. Schmalfuss, Fixed points and attractors for random dynamical systems, in Advances in Nonlinear Stochastic Mechanics(eds. A. Naess and S. Krenk), Solid Mech. Appl. vol. 47, Kluwer Acad. Publ. Dordrecht, (1996), 19-28. doi: 10.1007/978-94-009-0321-0_3.

[3]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Springer, Boston, 1990. doi: 10.1007/978-0-8176-4848-0.

[4]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

[5]

T. Bartsch and Z. Liu, On a supperlinear elliptic p-Laplacian equation, J. Differential Equations, 198 (2004), 149-175. doi: 10.1016/j.jde.2003.08.001.

[6]

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

[7]

D. CaoC. Sun and M. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020.

[8]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuß and J. Valero, Nonautonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.

[9]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuß and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst., 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439.

[10]

T. CaraballoF. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference. Equ. Appl., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010.

[11]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Appl. Math. Sciences, 184, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.

[12]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, 580, Springer-Verlag, Berlin, 1977.

[13]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[14]

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

[15]

H. CrauelA. Debussche and F. Flandoli, Random attractors, J. Dyn. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225.

[16]

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.

[17]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[18]

M. D. Donsker and S. R. S. Varadhan, On laws of the iterated logarithm for local times, Comm. Pure Appl. Math., 30 (1977), 707-753. doi: 10.1002/cpa.3160300603.

[19]

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

[20]

R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101 (1961), 75-90. doi: 10.1090/S0002-9947-1961-0137148-5.

[21]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on Rn, J. Differential Equations, 264 (2018), 7094-7137. doi: 10.1016/j.jde.2018.02.011.

[22]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. Ⅰ, volume 419 of Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.

[23]

P. E. Kloeden and T. Lorenz, Pullback and forward attractors of nonautonomous difference equations, in, Difference equations, discrete dynamical systems and applications (eds. M. Bohner, Y. Ding and O. Dosly), Springer-Verlag, Heidelberg, 150 (2015), 37-48. doi: 10.1007/978-3-319-24747-2_3.

[24]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268. doi: 10.1090/proc/12735.

[25]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference. Equ. Appl., 22 (2016), 513-525. doi: 10.1080/10236198.2015.1107550.

[26]

A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038. doi: 10.1016/j.jmaa.2014.03.037.

[27]

N. S. Landkof, Foundations of Modern Potential Theory (Translated from the Russian by A. P. Doohovskoy), Die Grundlehren der mathematischen Wissenschaften, vol.180, Springer-Verlag, NewYork-Heidelberg, 1972.

[28]

L. LiJ. Sun and S. Tersian, Infinitely many sign-changing solutions for the Brézis-Nirenberg problem involving the fractional Laplacian, Fract. Calc. Appl. Anal., 20 (2017), 1146-1164. doi: 10.1515/fca-2017-0061.

[29]

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.

[30]

Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differential Equations, 245 (2008), 1775-1800. doi: 10.1016/j.jde.2008.06.031.

[31]

Y. LiA. Gu and J. Li, Existences and continuity of bi-spatial random attractors and application to stochasitic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534. doi: 10.1016/j.jde.2014.09.021.

[32]

H. Lu, P. W. Bates, J. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on Rn, Nonlinear Anal., 128 (2015), 176-198. doi: 10.1016/j.na.2015.06.033.

[33]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on an unbounded domain, Commun. Math. Sci., 14 (2016), 273-295. doi: 10.4310/CMS.2016.v14.n1.a11.

[34]

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

[35]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3.

[36]

J. C. Robinson, Infinite-Dimensional Dyanmical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001. doi: 10.1007/978-94-010-0732-0.

[37]

B. Schmalfuß, Backward cocycle and attractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (eds. V. Reitmann, T. Riedrich and N. Koksch), Technische Universität, Dresden, (1992), 185-192.

[38]

B. Schmalfuß, Attractors for the nonautonomous dynamical systems, in International Conference on Differential Equations (eds. B. Fiedler, K. Gröger and J. Sprekels), World Sci. Publishing, Singapore, (2000), 684-690.

[39] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NY, 1970.
[40]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680.

[41]

B. Q. Tang, Regularity of pullback random attractors for stochasitic FitzHugh-Nagumo system on unbounded domains, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 441-466. doi: 10.3934/dcds.2015.35.441.

[42]

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

[43]

B. Wang, Random attractors for non-autonomous stochastic wave euqations with multiplicative noises, Discrete Contin. Dyn. Syst., 34 (2014), 269-330. doi: 10.3934/dcds.2014.34.269.

[44]

B. Wang, Suffcient 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.

[45]

B. Wang, Multivalued non-autonomous random dynamical systems for wave equations without uniqueness, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2011-2051. doi: 10.3934/dcdsb.2017119.

[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.

[47]

Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamcal systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026.

[48]

Z. Wang and S. Zhou, Random attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems with random coupled coefficients, Taiwanese J. Math., 20 (2016), 589-616. doi: 10.11650/tjm.20.2016.6699.

[49]

J. Yin and Y. Li, Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic p-Laplacian equations on RN, Math. Meth. Appl. Sci., 40 (2017), 4863-4879.

[50]

J. YinY. 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.

[51]

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