• Previous Article
    Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation
  • CPAA Home
  • This Issue
  • Next Article
    Existence and decay property of ground state solutions for Hamiltonian elliptic system
September  2019, 18(5): 2409-2431. doi: 10.3934/cpaa.2019109

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

* Corresponding author

Received  April 2018 Revised  July 2018 Published  April 2019

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.

Citation: Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109
References:
[1]

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

[2]

M. BartuccelliP. ConstantinC. DoeringJ. 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. Google Scholar

[3]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, 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]

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. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

C. DoeringJ. 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. Google Scholar

[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. Google Scholar

[11]

J. DuanP. Holme and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (2009), 1303-1314. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

M. Garrido-AtienzaK. 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. Google Scholar

[15]

B. GuoY. 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. Google Scholar

[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. Google Scholar

[17]

B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and their Numerical Solutions, Science Press, Beijing, 2011. doi: 10.1142/9543. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, in Math. Surveys Monogr., vol. 25, AMS, Providence, 1988. Google Scholar

[21]

X. HanW. 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. Google Scholar

[22]

D. LiZ. 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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[26]

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

[27]

H. LuP. W. BatesJ. 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. Google Scholar

[28]

H. LuP. W. BatesS. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar

[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.Google Scholar

[36]

G. Sell and Y. You, Dynamics of Evolutional Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

Vasily E. Tarasov and George M. Zaslavsky, Fractional Ginzburg-Laudau equation for fractal media, Physica A, 354 (2005), 249-261. Google Scholar

[43]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4684-0313-8. Google Scholar

[44]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. Google Scholar

[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. Google Scholar

[46]

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

[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. 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, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[53]

X. WangS. 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. Google Scholar

[54]

W. YanS. 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. Google Scholar

[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. Google Scholar

[56]

J. YinY. 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. Google Scholar

[57]

J. ZhangC. 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. Google Scholar

[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. doi: 10.1016/j.jmaa.2008.12.036. Google Scholar

[59]

S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal. TMA, 120 (2015), 202-226. doi: 10.1016/j.na.2015.03.009. Google Scholar

show all references

References:
[1]

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

[2]

M. BartuccelliP. ConstantinC. DoeringJ. 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. Google Scholar

[3]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical system, 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]

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. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

C. DoeringJ. 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. Google Scholar

[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. Google Scholar

[11]

J. DuanP. Holme and E. S. Titi, Global existence theory for a generalized Ginzburg-Landau equation, Nonlinearity, 5 (2009), 1303-1314. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

M. Garrido-AtienzaK. 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. Google Scholar

[15]

B. GuoY. 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. Google Scholar

[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. Google Scholar

[17]

B. Guo, X. Pu and F. Huang, Fractional Partial Differential Equations and their Numerical Solutions, Science Press, Beijing, 2011. doi: 10.1142/9543. Google Scholar

[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. Google Scholar

[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. Google Scholar

[20]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, in Math. Surveys Monogr., vol. 25, AMS, Providence, 1988. Google Scholar

[21]

X. HanW. 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. Google Scholar

[22]

D. LiZ. 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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[26]

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

[27]

H. LuP. W. BatesJ. 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. Google Scholar

[28]

H. LuP. W. BatesS. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[31]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, UK, 2001. doi: 10.1007/978-94-010-0732-0. Google Scholar

[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.Google Scholar

[36]

G. Sell and Y. You, Dynamics of Evolutional Equations, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[39]

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[42]

Vasily E. Tarasov and George M. Zaslavsky, Fractional Ginzburg-Laudau equation for fractal media, Physica A, 354 (2005), 249-261. Google Scholar

[43]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4684-0313-8. Google Scholar

[44]

P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. Google Scholar

[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. Google Scholar

[46]

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

[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. 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, 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[53]

X. WangS. 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. Google Scholar

[54]

W. YanS. 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. Google Scholar

[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. Google Scholar

[56]

J. YinY. 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. Google Scholar

[57]

J. ZhangC. 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. Google Scholar

[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. doi: 10.1016/j.jmaa.2008.12.036. Google Scholar

[59]

S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal. TMA, 120 (2015), 202-226. doi: 10.1016/j.na.2015.03.009. Google Scholar

[1]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[2]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

[3]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210

[4]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115

[5]

Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269

[6]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757

[7]

Shujuan Lü, Hong Lu, Zhaosheng Feng. Stochastic dynamics of 2D fractional Ginzburg-Landau equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 575-590. doi: 10.3934/dcdsb.2016.21.575

[8]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035

[9]

Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1

[10]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

[11]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181

[12]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

[13]

Yangrong Li, Shuang Yang. Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1155-1175. doi: 10.3934/cpaa.2019056

[14]

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

[15]

Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523

[16]

Tianlong Shen, Jianhua Huang. Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by α-stable noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 605-625. doi: 10.3934/dcdsb.2017029

[17]

Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075

[18]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[19]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

[20]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (31)
  • HTML views (190)
  • Cited by (0)

Other articles
by authors

[Back to Top]