• Previous Article
    Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion
  • DCDS Home
  • This Issue
  • Next Article
    Sinai-Ruelle-Bowen measures for piecewise hyperbolic maps with two directions of instability in three-dimensional spaces
May  2016, 36(5): 2887-2914. doi: 10.3934/dcds.2016.36.2887

Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China, China

Received  March 2015 Revised  July 2015 Published  October 2015

In this paper, we first present some conditions for bounding the fractal dimension of a random invariant set of a non-autonomous random dynamical system on a separable Banach space. Then we apply these conditions to prove the finiteness of fractal dimension of random attractor for stochastic damped wave equation with linear multiplicative white noise.
Citation: 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
References:
[1]

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

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Publishing Co., (1992).   Google Scholar

[3]

F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2591.  doi: 10.1142/S0218127410027246.  Google Scholar

[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.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[5]

T. Caraballo, J. A. Langa and J. C. Robinson, Stability and random attractors for a reaction-diffusion equation with multiplicative noise,, Discrete Contin. Dynam. Systems, 6 (2000), 875.  doi: 10.3934/dcds.2000.6.875.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).   Google Scholar

[7]

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

[8]

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

[9]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems,, J. Dynam. Differential Equations, 10 (1998), 449.  doi: 10.1023/A:1022605313961.  Google Scholar

[10]

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

[11]

A. Debussche, On the finite dimensionality of random attractors,, Stochastic Anal. Appl., 15 (1997), 473.  doi: 10.1080/07362999708809490.  Google Scholar

[12]

A. Debussche, Hausdorff dimension of a random invariant set,, J. Math. Pures Appl., 77 (1998), 967.  doi: 10.1016/S0021-7824(99)80001-4.  Google Scholar

[13]

X. Fan, Random attractor for a damped sine-Gordon equation with white noise,, Pacific J. Math., 216 (2004), 63.  doi: 10.2140/pjm.2004.216.63.  Google Scholar

[14]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise,, Internat. J. Math., 19 (2008), 421.  doi: 10.1142/S0129167X08004741.  Google Scholar

[15]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise,, Stoch. Anal. Appl., 24 (2006), 767.  doi: 10.1080/07362990600751860.  Google Scholar

[16]

H. Gao, M. J. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion,, SIAM J. Math. Anal., 46 (2014), 2281.  doi: 10.1137/130930662.  Google Scholar

[17]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).   Google Scholar

[18]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991).  doi: 10.1017/CBO9780511569418.  Google Scholar

[19]

J. A. Langa, Finite-dimensional limiting dynamics of random dynamical systems,, Dyn. Syst., 18 (2003), 57.  doi: 10.1080/1468936031000080812.  Google Scholar

[20]

J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set,, J. Math. Pures Appl., 85 (2006), 269.  doi: 10.1016/j.matpur.2005.08.001.  Google Scholar

[21]

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

[22]

T. Sauer, J. A. Yorke and M. Casdagli, Embedology,, J. Statist. Phys., 65 (1991), 579.  doi: 10.1007/BF01053745.  Google Scholar

[23]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[24]

P. Walters, Introduction to Ergodic Theory,, Springer-Verlag, (2000).   Google Scholar

[25]

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

[26]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, J. Differential Equations, 253 (2012), 1544.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[27]

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

[28]

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

[29]

G. Wang and Y. Tang, Fractal dimension of a random invariant set and applications,, J. Appl. Math., (2013).   Google Scholar

[30]

M. Yang, J. Duan and P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise,, Nonlinear Anal. Real World Appl., 12 (2011), 464.  doi: 10.1016/j.nonrwa.2010.06.032.  Google Scholar

[31]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

[32]

S. Zhou, F. Yin and Z. Ouyang, Random attractor for damped nonlinear wave equations with white noise,, SIAM J. Appl. Dyn. Syst., 4 (2005), 883.  doi: 10.1137/050623097.  Google Scholar

show all references

References:
[1]

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

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland Publishing Co., (1992).   Google Scholar

[3]

F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: Three perspectives,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2591.  doi: 10.1142/S0218127410027246.  Google Scholar

[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.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[5]

T. Caraballo, J. A. Langa and J. C. Robinson, Stability and random attractors for a reaction-diffusion equation with multiplicative noise,, Discrete Contin. Dynam. Systems, 6 (2000), 875.  doi: 10.3934/dcds.2000.6.875.  Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).   Google Scholar

[7]

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

[8]

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

[9]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems,, J. Dynam. Differential Equations, 10 (1998), 449.  doi: 10.1023/A:1022605313961.  Google Scholar

[10]

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

[11]

A. Debussche, On the finite dimensionality of random attractors,, Stochastic Anal. Appl., 15 (1997), 473.  doi: 10.1080/07362999708809490.  Google Scholar

[12]

A. Debussche, Hausdorff dimension of a random invariant set,, J. Math. Pures Appl., 77 (1998), 967.  doi: 10.1016/S0021-7824(99)80001-4.  Google Scholar

[13]

X. Fan, Random attractor for a damped sine-Gordon equation with white noise,, Pacific J. Math., 216 (2004), 63.  doi: 10.2140/pjm.2004.216.63.  Google Scholar

[14]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise,, Internat. J. Math., 19 (2008), 421.  doi: 10.1142/S0129167X08004741.  Google Scholar

[15]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise,, Stoch. Anal. Appl., 24 (2006), 767.  doi: 10.1080/07362990600751860.  Google Scholar

[16]

H. Gao, M. J. Garrido-Atienza and B. Schmalfuss, Random attractors for stochastic evolution equations driven by fractional Brownian motion,, SIAM J. Math. Anal., 46 (2014), 2281.  doi: 10.1137/130930662.  Google Scholar

[17]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, American Mathematical Society, (1988).   Google Scholar

[18]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations,, Cambridge University Press, (1991).  doi: 10.1017/CBO9780511569418.  Google Scholar

[19]

J. A. Langa, Finite-dimensional limiting dynamics of random dynamical systems,, Dyn. Syst., 18 (2003), 57.  doi: 10.1080/1468936031000080812.  Google Scholar

[20]

J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set,, J. Math. Pures Appl., 85 (2006), 269.  doi: 10.1016/j.matpur.2005.08.001.  Google Scholar

[21]

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

[22]

T. Sauer, J. A. Yorke and M. Casdagli, Embedology,, J. Statist. Phys., 65 (1991), 579.  doi: 10.1007/BF01053745.  Google Scholar

[23]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[24]

P. Walters, Introduction to Ergodic Theory,, Springer-Verlag, (2000).   Google Scholar

[25]

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

[26]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems,, J. Differential Equations, 253 (2012), 1544.  doi: 10.1016/j.jde.2012.05.015.  Google Scholar

[27]

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

[28]

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

[29]

G. Wang and Y. Tang, Fractal dimension of a random invariant set and applications,, J. Appl. Math., (2013).   Google Scholar

[30]

M. Yang, J. Duan and P. Kloeden, Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise,, Nonlinear Anal. Real World Appl., 12 (2011), 464.  doi: 10.1016/j.nonrwa.2010.06.032.  Google Scholar

[31]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent,, Commun. Pure Appl. Anal., 3 (2004), 921.  doi: 10.3934/cpaa.2004.3.921.  Google Scholar

[32]

S. Zhou, F. Yin and Z. Ouyang, Random attractor for damped nonlinear wave equations with white noise,, SIAM J. Appl. Dyn. Syst., 4 (2005), 883.  doi: 10.1137/050623097.  Google Scholar

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081

[7]

Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875

[8]

Boris P. Belinskiy, Peter Caithamer. Stochastic stability of some mechanical systems with a multiplicative white noise. Conference Publications, 2003, 2003 (Special) : 91-99. doi: 10.3934/proc.2003.2003.91

[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]

Yuncheng You. Random attractor for stochastic reversible Schnackenberg equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1347-1362. doi: 10.3934/dcdss.2014.7.1347

[11]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701

[12]

Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645

[13]

Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305

[14]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165

[15]

Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047

[16]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106

[17]

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

[18]

Xiaojun Li, Xiliang Li, Kening Lu. Random attractors for stochastic parabolic equations with additive noise in weighted spaces. Communications on Pure & Applied Analysis, 2018, 17 (3) : 729-749. doi: 10.3934/cpaa.2018038

[19]

Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303

[20]

Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]