February 2019, 39(2): 683-706. doi: 10.3934/dcds.2019028

Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains

1. 

School of Mathematics and Statistics, Shandong University, Weihai, Shandong Provence, 264209, China

2. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

Received  April 2017 Revised  January 2018 Published  November 2018

We study the asymptotic behavior of a class of non-autonomous non-local fractional stochastic parabolic equation driven by multiplicative white noise on the entire space $\mathbb{R}^n$. We first prove the pathwise well-posedness of the equation and define a continuous non-autonomous cocycle in $L^2({\mathbb{R}} ^n)$. We then prove the existence and uniqueness of tempered pullback attractors for the cocycle under certain dissipative conditions. The periodicity of the tempered attractors is also proved when the deterministic non-autonomous external terms are periodic in time. The pullback asymptotic compactness of the cocycle in $L^2({\mathbb{R}} ^n)$ is established by the uniform estimates on the tails of solutions for sufficiently large space and time variables.

Citation: Hong Lu, Jiangang Qi, Bixiang Wang, Mingji Zhang. Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 683-706. doi: 10.3934/dcds.2019028
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A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666. doi: 10.3934/dcdsb.2013.18.643.

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

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

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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.

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W. J. BeynB. GessP. Lescot and M. R$\ddot o$ckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469. doi: 10.1080/03605302.2010.523919.

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L. CaffarelliJ. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

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

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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.

[10]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Ser. A: Mathematical Analysis, 10 (2003), 491-513.

[11]

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

[12]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047.

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

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I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004), 127-144. doi: 10.1080/1468936042000207792.

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I. Chueshow, Monotone Random Systems - Theory and Applications, Lecture Notes in Mathematics, 1779, Springer, Berlin, 2002. doi: 10.1007/b83277.

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

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

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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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J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.

[20]

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.

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

[22]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5.

[23]

M. J. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.

[24]

M. J. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, Int. J. Bifur. Chaos, 20 (2010), 2761-2782. doi: 10.1142/S0218127410027349.

[25]

A. Garroni and S. Muller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal., 181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.

[26]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013.

[27]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dynam. Differential Equations, 25 (2013), 121-157. doi: 10.1007/s10884-013-9294-5.

[28]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559. doi: 10.1016/j.jde.2013.04.023.

[29]

Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.

[30]

Q. Guan and Z. Ma, Reflected symmetric $α$-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694. doi: 10.1007/s00440-005-0438-3.

[31]

Q. Guan and Z. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X.

[32]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst. Ser. A, 24 (2009), 855-882. doi: 10.3934/dcds.2009.24.855.

[33]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253.

[34]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Ser. A., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.

[35]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, Amer. Math. Soc., Providence, 2011. doi: 10.1090/surv/176.

[36]

M. KoslowskiA. Cuitino and M. Ortiz, A phasefield theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal, J. Mech. Phys. Solids, 50 (2002), 2597-2635. doi: 10.1016/S0022-5096(02)00037-6.

[37]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.

[38]

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

[39]

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.

[40]

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

[41]

H. LuS. Lv and M. Zhang, Fourier spectral approximation to the dynamical behavior of 3D fractional Ginzburg-Landau equation, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 2539-2564. doi: 10.3934/dcds.2017109.

[42]

H. Lu and M. Zhang, The spectral method for long-time behavior of a fractional power dissipative system, Taiwanese J. Math., 22 (2018), 453-483. doi: 10.11650/tjm/170902.

[43]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23. doi: 10.1016/j.jde.2007.10.009.

[44]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal de Mathematiques Pures et Appliquees, 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.

[45]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185–192, Dresden, 1992.

[46]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783.

[47]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105.

[48]

Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457. doi: 10.1016/j.jde.2009.10.007.

[49]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2.

[50]

B. Wang, Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains, J. Differential Equations, 246 (2009), 2506-2537. doi: 10.1016/j.jde.2008.10.012.

[51]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on ${\mathbb{R}} ^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.

[52]

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.

[53]

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

[54]

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.

show all references

References:
[1]

S. Abe and S. Thurner, Anomalous diffusion in view of Einsteins 1905 theory of Brownian motion, Physica A, 356 (2005), 403-407.

[2]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 643-666. doi: 10.3934/dcdsb.2013.18.643.

[3]

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

[4]

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

[5]

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.

[6]

W. J. BeynB. GessP. Lescot and M. R$\ddot o$ckner, The global random attractor for a class of stochastic porous media equations, Comm. Partial Differential Equations, 36 (2011), 446-469. doi: 10.1080/03605302.2010.523919.

[7]

L. CaffarelliJ. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

[8]

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

[9]

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.

[10]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dynamics of Continuous, Discrete and Impulsive Systems Ser. A: Mathematical Analysis, 10 (2003), 491-513.

[11]

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

[12]

T. CaraballoM. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion, Nonlinear Anal., 74 (2011), 3671-3684. doi: 10.1016/j.na.2011.02.047.

[13]

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

[14]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004), 127-144. doi: 10.1080/1468936042000207792.

[15]

I. Chueshow, Monotone Random Systems - Theory and Applications, Lecture Notes in Mathematics, 1779, Springer, Berlin, 2002. doi: 10.1007/b83277.

[16]

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

[17]

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

[18]

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.

[19]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.

[20]

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.

[21]

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

[22]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681. doi: 10.1007/s10884-011-9222-5.

[23]

M. J. Garrido-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.

[24]

M. J. Garrido-AtienzaB. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion, Int. J. Bifur. Chaos, 20 (2010), 2761-2782. doi: 10.1142/S0218127410027349.

[25]

A. Garroni and S. Muller, A variational model for dislocations in the line tension limit, Arch. Ration. Mech. Anal., 181 (2006), 535-578. doi: 10.1007/s00205-006-0432-7.

[26]

B. GessW. Liu and M. Rockner, Random attractors for a class of stochastic partial differential equations driven by general additive noise, J. Differential Equations, 251 (2011), 1225-1253. doi: 10.1016/j.jde.2011.02.013.

[27]

B. Gess, Random attractors for degenerate stochastic partial differential equations, J. Dynam. Differential Equations, 25 (2013), 121-157. doi: 10.1007/s10884-013-9294-5.

[28]

B. Gess, Random attractors for singular stochastic evolution equations, J. Differential Equations, 255 (2013), 524-559. doi: 10.1016/j.jde.2013.04.023.

[29]

Q. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.

[30]

Q. Guan and Z. Ma, Reflected symmetric $α$-stable processes and regional fractional Laplacian, Probab. Theory Related Fields, 134 (2006), 649-694. doi: 10.1007/s00440-005-0438-3.

[31]

Q. Guan and Z. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X.

[32]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains, Discrete Contin. Dyn. Syst. Ser. A, 24 (2009), 855-882. doi: 10.3934/dcds.2009.24.855.

[33]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253.

[34]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. Royal Soc. London Ser. A., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.

[35]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, Vol. 176, Amer. Math. Soc., Providence, 2011. doi: 10.1090/surv/176.

[36]

M. KoslowskiA. Cuitino and M. Ortiz, A phasefield theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal, J. Mech. Phys. Solids, 50 (2002), 2597-2635. doi: 10.1016/S0022-5096(02)00037-6.

[37]

J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires, Dunod, Paris, 1969.

[38]

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

[39]

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.

[40]

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

[41]

H. LuS. Lv and M. Zhang, Fourier spectral approximation to the dynamical behavior of 3D fractional Ginzburg-Landau equation, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 2539-2564. doi: 10.3934/dcds.2017109.

[42]

H. Lu and M. Zhang, The spectral method for long-time behavior of a fractional power dissipative system, Taiwanese J. Math., 22 (2018), 453-483. doi: 10.11650/tjm/170902.

[43]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23. doi: 10.1016/j.jde.2007.10.009.

[44]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, Journal de Mathematiques Pures et Appliquees, 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.

[45]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185–192, Dresden, 1992.

[46]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783.

[47]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105.

[48]

Z. ShenS. Zhou and W. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457. doi: 10.1016/j.jde.2009.10.007.

[49]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. doi: 10.1016/S0167-2789(98)00304-2.

[50]

B. Wang, Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains, J. Differential Equations, 246 (2009), 2506-2537. doi: 10.1016/j.jde.2008.10.012.

[51]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on ${\mathbb{R}} ^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.

[52]

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.

[53]

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

[54]

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.

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