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Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation
Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise
1. | School of Mathematical Science, Sichuan Normal University, Chengdu, Sichuan 610068, China |
2. | School of Mathematical Science and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, Sichuan 610068, China |
This paper is concerned with the asymptotic behavior of solutions for non-autonomous stochastic fractional complex Ginzburg-Landau equations driven by multiplicative noise with $ \alpha\in(0, 1) $. We first apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions, which is slightly different from the deterministic fractional case with $ \alpha\in(\frac{1}{2}, 1) $ and the real fractional case with $ \alpha\in(0, 1) $. Consequently, we establish the existence and uniqueness of tempered pullback random attractors for the equations in a bounded domain. At last, we obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero.
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L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998.
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M. Bartuccelli, P. Constantin, C. Doering, J. Gibbon and M. Gisselfält,
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doi: 10.1016/0167-2789(90)90156-J. |
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P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21.
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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.
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Z. Brzezniak and Y. Li,
Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629.
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I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, Heidelbrg, 2002.
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H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[8] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[9] |
C. Doering, J. Gibbon and C. Levermore,
Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71 (1994), 285-318.
doi: 10.1016/0167-2789(94)90150-3. |
[10] |
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Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.
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J. Duan, P. Holme and E. S. Titi,
Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (2009), 1303-1314.
|
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Attractors for a second order nonautonomous lattice dynamical systems with nonlinear damping, Phys. Lett. A, 365 (2007), 17-27.
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Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45.
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Random dynamical systems for stochastic equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473-493.
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Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204 (2008), 458-477.
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Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differential Equations, 36 (2011), 247-255.
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Random attractors for stochastic lattice dynamical system in weighted space, J. Differential Equations, 250 (2011), 1235-1266.
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Long time behavior for generalized complex Ginzburg-Landau equation, J. Math. Anal. Appl., 330 (2007), 938-948.
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D. Li and B. Guo,
Asymptotic behavior of the 2D generalized stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Mech., 30 (2009), 883-894.
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Random attrator for fractional Ginzburg-Laudau equation with multiplicative noise, Taiwanese J. Math., 18 (2014), 435-450.
doi: 10.11650/tjm.18.2014.3053. |
[26] |
H. Lu, P. W. Bates, S. 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. |
[27] |
H. Lu, P. W. Bates, J. Xin and M. Zhang,
Asymptotic behavior of stochastic fractional power dissipative equations on Rn, Nonlinear Anal. TMA, 128 (2015), 176-198.
doi: 10.1016/j.na.2015.06.033. |
[28] |
H. Lu, P. W. Bates, S. Lu and M. Zhang,
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on a unbounded domain, Commmu. Math. Sci., 14 (2016), 273-295.
doi: 10.4310/CMS.2016.v14.n1.a11. |
[29] |
Y. Lv and J. Sun,
Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221 (2006), 157-169.
doi: 10.1016/j.physd.2006.07.023. |
[30] |
B. Maslowski and B. Schmalfuss,
Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stoch. Anal. Appl., 22 (2004), 1577-1607.
doi: 10.1081/SAP-200029498. |
[31] |
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. |
[32] |
X. Pu and B. Guo,
Global weak Soltuions of the fractional Landau-Lifshitz -Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98.
doi: 10.1016/j.jmaa.2010.06.035. |
[33] |
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. |
[34] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001.
doi: 10.1007/978-94-010-0732-0. |
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On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855.
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T. Shen and J. Huang,
Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal. TMA, 110 (2014), 33-46.
doi: 10.1016/j.na.2014.06.018. |
[39] |
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. |
[40] |
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. |
[41] |
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. |
[42] |
Vasily E. Tarasov and George M. Zaslavsky,
Fractional Ginzburg-Laudau equation for fractal media, Physica A, 354 (2005), 249-261.
|
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R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4684-0313-8. |
[44] |
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. |
[45] |
B. Wang,
Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. TMA, 71 (2009), 2811-2828.
doi: 10.1016/j.na.2009.01.131. |
[46] |
B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron J. Differential Equations, 139 (2009), 1-18. |
[47] |
B. Wang,
Asymptotic behavior of stochastic wave equations with critical exponents on R3, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[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. |
[49] |
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. |
[50] |
B. Wang,
Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2012), 1791-1798.
doi: 10.1142/S0219493714500099. |
[51] |
B. Wang,
Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal. TMA, 158 (2017), 60-82.
doi: 10.1016/j.na.2017.04.006. |
[52] |
B. Wang and X. Gao, Random attractors for stochastic wave equations on unbounded domains, Discrete Contin. Dyn. Syst. Suppl., (2009), 800–809.
doi: 10.1016/j.nonrwa.2011.06.008. |
[53] |
X. Wang, S. Li and D. Xu,
Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal. TMA, 72 (2010), 483-494.
doi: 10.1016/j.na.2009.06.094. |
[54] |
W. Yan, S. Ji and Y. Li,
Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations, Phys. Lett. A, 373 (2009), 1268-1275.
doi: 10.1016/j.physleta.2009.02.019. |
[55] |
F. Yin and L. Liu,
D-pullback attractor for a non-autonomous wave equation with additive noise on unbounded domains, Comput. Math. Appl., 68 (2014), 424-438.
doi: 10.1016/j.camwa.2014.06.018. |
[56] |
J. Yin, Y. Li and A. Gu,
Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.
doi: 10.1016/j.camwa.2017.05.015. |
[57] |
J. Zhang, C. Huang and J. Shu,
Random attractors for the stochastic discrete complex Ginzburg-Landau equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 24 (2017), 303-315.
|
[58] |
C. Zhao and S. Zhou,
Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95.
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Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal. TMA, 120 (2015), 202-226.
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show all references
References:
[1] |
L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
M. Bartuccelli, P. Constantin, C. Doering, J. Gibbon and M. Gisselfält,
On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Physica D, 44 (1990), 421-444.
doi: 10.1016/0167-2789(90)90156-J. |
[3] |
P. W. Bates, H. Lisei and K. Lu,
Attractors for stochastic lattice dynamical system, Stoch. Dyn., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[4] |
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. |
[5] |
Z. Brzezniak and Y. Li,
Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains, Trans. Amer. Math. Soc., 358 (2006), 5587-5629.
doi: 10.1090/S0002-9947-06-03923-7. |
[6] |
I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, Heidelbrg, 2002.
doi: 10.1007/b83277. |
[7] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[8] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[9] |
C. Doering, J. Gibbon and C. Levermore,
Weak and strong solutions of the complex Ginzburg-Landau equation, Physica D, 71 (1994), 285-318.
doi: 10.1016/0167-2789(94)90150-3. |
[10] |
J. Dong and M. Xu,
Space-time fractional Schrödinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.
doi: 10.1016/j.jmaa.2008.03.061. |
[11] |
J. Duan, P. Holme and E. S. Titi,
Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (2009), 1303-1314.
|
[12] |
X. Fan and Y. Wang,
Attractors for a second order nonautonomous lattice dynamical systems with nonlinear damping, Phys. Lett. A, 365 (2007), 17-27.
doi: 10.1016/j.physleta.2006.12.045. |
[13] |
F. Flandoli and B. Schmalfuss,
Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45.
doi: 10.1080/17442509608834083. |
[14] |
M. Garrido-Atienza, K. Lu and B. Schmalfuss,
Random dynamical systems for stochastic equations driven by a fractional Brownian motion, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 473-493.
doi: 10.3934/dcdsb.2010.14.473. |
[15] |
B. Guo, Y. Han and J. Xin,
Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schrödinger equation, Appl. Math. Comput., 204 (2008), 458-477.
doi: 10.1016/j.amc.2008.07.003. |
[16] |
B. Guo and Z. Huo,
Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differential Equations, 36 (2011), 247-255.
doi: 10.1080/03605302.2010.503769. |
[17] |
B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and their Numerical Solutions, Science Press, Beijing, 2011.
doi: 10.1142/9543. |
[18] |
B. Guo and X. Wang,
Finite dimensional behavior for the derivative Ginzburg-Landau equation in two soatial dimensions, Physica D, 89 (1995), 83-99.
doi: 10.1016/0167-2789(95)00216-2. |
[19] |
B. Guo and M. Zeng,
Soltuions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138.
doi: 10.1016/j.jmaa.2009.09.009. |
[20] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, in Math. Surveys Monogr., vol. 25, AMS, Providence, 1988. |
[21] |
X. Han, W. Shen and S. Zhou,
Random attractors for stochastic lattice dynamical system in weighted space, J. Differential Equations, 250 (2011), 1235-1266.
doi: 10.1016/j.jde.2010.10.018. |
[22] |
D. Li, Z. Dai and X. Liu,
Long time behavior for generalized complex Ginzburg-Landau equation, J. Math. Anal. Appl., 330 (2007), 938-948.
doi: 10.1016/j.jmaa.2006.07.095. |
[23] |
D. Li and B. Guo,
Asymptotic behavior of the 2D generalized stochastic Ginzburg-Landau equation with additive noise, Appl. Math. Mech., 30 (2009), 883-894.
doi: 10.1007/s10483-009-0801-x. |
[24] |
J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non Linearires, Dunod, Paris, 1969. |
[25] |
H. Lu and S. Lv,
Random attrator for fractional Ginzburg-Laudau equation with multiplicative noise, Taiwanese J. Math., 18 (2014), 435-450.
doi: 10.11650/tjm.18.2014.3053. |
[26] |
H. Lu, P. W. Bates, S. 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. |
[27] |
H. Lu, P. W. Bates, J. Xin and M. Zhang,
Asymptotic behavior of stochastic fractional power dissipative equations on Rn, Nonlinear Anal. TMA, 128 (2015), 176-198.
doi: 10.1016/j.na.2015.06.033. |
[28] |
H. Lu, P. W. Bates, S. Lu and M. Zhang,
Dynamics of the 3D fractional Ginzburg-Landau equation with multiplicative noise on a unbounded domain, Commmu. Math. Sci., 14 (2016), 273-295.
doi: 10.4310/CMS.2016.v14.n1.a11. |
[29] |
Y. Lv and J. Sun,
Asymptotic behavior of stochastic discrete complex Ginzburg-Landau equations, Physica D, 221 (2006), 157-169.
doi: 10.1016/j.physd.2006.07.023. |
[30] |
B. Maslowski and B. Schmalfuss,
Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion, Stoch. Anal. Appl., 22 (2004), 1577-1607.
doi: 10.1081/SAP-200029498. |
[31] |
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. |
[32] |
X. Pu and B. Guo,
Global weak Soltuions of the fractional Landau-Lifshitz -Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98.
doi: 10.1016/j.jmaa.2010.06.035. |
[33] |
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. |
[34] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001.
doi: 10.1007/978-94-010-0732-0. |
[35] |
B. Schmalfuss, Backward cocycle and atttractors of stochastic differential equations, in International Semilar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior (ed. V. Reitmann, T. Riedrich and N. Koksch), Technishe Universität, Dresden, 1992, pp.185–192. |
[36] |
G. Sell and Y. You, Dynamics of Evolutional Equations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[37] |
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. |
[38] |
T. Shen and J. Huang,
Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal. TMA, 110 (2014), 33-46.
doi: 10.1016/j.na.2014.06.018. |
[39] |
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. |
[40] |
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. |
[41] |
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. |
[42] |
Vasily E. Tarasov and George M. Zaslavsky,
Fractional Ginzburg-Laudau equation for fractal media, Physica A, 354 (2005), 249-261.
|
[43] |
R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4684-0313-8. |
[44] |
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. |
[45] |
B. Wang,
Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Anal. TMA, 71 (2009), 2811-2828.
doi: 10.1016/j.na.2009.01.131. |
[46] |
B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron J. Differential Equations, 139 (2009), 1-18. |
[47] |
B. Wang,
Asymptotic behavior of stochastic wave equations with critical exponents on R3, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[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. |
[49] |
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. |
[50] |
B. Wang,
Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2012), 1791-1798.
doi: 10.1142/S0219493714500099. |
[51] |
B. Wang,
Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal. TMA, 158 (2017), 60-82.
doi: 10.1016/j.na.2017.04.006. |
[52] |
B. Wang and X. Gao, Random attractors for stochastic wave equations on unbounded domains, Discrete Contin. Dyn. Syst. Suppl., (2009), 800–809.
doi: 10.1016/j.nonrwa.2011.06.008. |
[53] |
X. Wang, S. Li and D. Xu,
Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal. TMA, 72 (2010), 483-494.
doi: 10.1016/j.na.2009.06.094. |
[54] |
W. Yan, S. Ji and Y. Li,
Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations, Phys. Lett. A, 373 (2009), 1268-1275.
doi: 10.1016/j.physleta.2009.02.019. |
[55] |
F. Yin and L. Liu,
D-pullback attractor for a non-autonomous wave equation with additive noise on unbounded domains, Comput. Math. Appl., 68 (2014), 424-438.
doi: 10.1016/j.camwa.2014.06.018. |
[56] |
J. Yin, Y. Li and A. Gu,
Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.
doi: 10.1016/j.camwa.2017.05.015. |
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