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Stochastic dominance for shift-invariant measures
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 |
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.
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. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[5] |
P. W. Bates, K. 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. Beyn, B. Gess, P. 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. Caffarelli, J. 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. Caraballo, M. J. Garrido-Atienza, B. 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. Caraballo, J. 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. 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. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[12] |
T. Caraballo, M. 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. Caraballo, J. A. Langa, V. 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. Crauel, A. 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 Nezza, G. 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-Atienza, A. Ogrowsky and B. Schmalfuss,
Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.
doi: 10.1142/S0219493711003358. |
[24] |
M. J. Garrido-Atienza, B. 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. Gess, W. 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. Koslowski, A. 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. Lu, P. W. Bates, S. Lu and M. Zhang,
Dynamics of 3D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.
|
[39] |
H. Lu, P. W. Bates, J. 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. Lu, P. W. Bates, S. 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. Lu, S. 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. Shen, S. 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. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[5] |
P. W. Bates, K. 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. Beyn, B. Gess, P. 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. Caffarelli, J. 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. Caraballo, M. J. Garrido-Atienza, B. 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. Caraballo, J. 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. 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. Ser. B, 14 (2010), 439-455.
doi: 10.3934/dcdsb.2010.14.439. |
[12] |
T. Caraballo, M. 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. Caraballo, J. A. Langa, V. 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. Crauel, A. 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 Nezza, G. 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-Atienza, A. Ogrowsky and B. Schmalfuss,
Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388.
doi: 10.1142/S0219493711003358. |
[24] |
M. J. Garrido-Atienza, B. 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. Gess, W. 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. Koslowski, A. 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. Lu, P. W. Bates, S. Lu and M. Zhang,
Dynamics of 3D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.
|
[39] |
H. Lu, P. W. Bates, J. 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. Lu, P. W. Bates, S. 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. Lu, S. 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. Shen, S. 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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