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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

Shengfan Zhou 1, and  Min Zhao 1,

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.
Keywords: fractal dimension, random attractor, multiplicative white noise, Stochastic damped wave equation, random dynamical system..
Mathematics Subject Classification: 37L55, 35B41, 35B4.
Citation: Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887
References:
[1]

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

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.

[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), 2591-2636. doi: 10.1142/S0218127410027246.

[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]

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), 875-892. doi: 10.3934/dcds.2000.6.875.

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.

[7]

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

[8]

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

[9]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474. doi: 10.1023/A:1022605313961.

[10]

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

[11]

A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491. doi: 10.1080/07362999708809490.

[12]

A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl., 77 (1998), 967-988. doi: 10.1016/S0021-7824(99)80001-4.

[13]

X. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76. doi: 10.2140/pjm.2004.216.63.

[14]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437. doi: 10.1142/S0129167X08004741.

[15]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793. doi: 10.1080/07362990600751860.

[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), 2281-2309. doi: 10.1137/130930662.

[17]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.

[18]

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

[19]

J. A. Langa, Finite-dimensional limiting dynamics of random dynamical systems, Dyn. Syst., 18 (2003), 57-68. doi: 10.1080/1468936031000080812.

[20]

J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl., 85 (2006), 269-294. doi: 10.1016/j.matpur.2005.08.001.

[21]

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.

[22]

T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Statist. Phys., 65 (1991), 579-616. doi: 10.1007/BF01053745.

[23]

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

[24]

P. Walters, Introduction to Ergodic Theory, Springer-Verlag, New York, 2000.

[25]

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

[26]

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.

[27]

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

[28]

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

[29]

G. Wang and Y. Tang, Fractal dimension of a random invariant set and applications, J. Appl. Math., (2013), Art. ID 415764, 5 pp.

[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), 464-478. doi: 10.1016/j.nonrwa.2010.06.032.

[31]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934. doi: 10.3934/cpaa.2004.3.921.

[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), 883-903. doi: 10.1137/050623097.

show all references

References:
[1]

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

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland Publishing Co., Amsterdam, 1992.

[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), 2591-2636. doi: 10.1142/S0218127410027246.

[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]

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), 875-892. doi: 10.3934/dcds.2000.6.875.

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002.

[7]

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

[8]

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

[9]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 449-474. doi: 10.1023/A:1022605313961.

[10]

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

[11]

A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491. doi: 10.1080/07362999708809490.

[12]

A. Debussche, Hausdorff dimension of a random invariant set, J. Math. Pures Appl., 77 (1998), 967-988. doi: 10.1016/S0021-7824(99)80001-4.

[13]

X. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76. doi: 10.2140/pjm.2004.216.63.

[14]

X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437. doi: 10.1142/S0129167X08004741.

[15]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793. doi: 10.1080/07362990600751860.

[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), 2281-2309. doi: 10.1137/130930662.

[17]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.

[18]

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

[19]

J. A. Langa, Finite-dimensional limiting dynamics of random dynamical systems, Dyn. Syst., 18 (2003), 57-68. doi: 10.1080/1468936031000080812.

[20]

J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set, J. Math. Pures Appl., 85 (2006), 269-294. doi: 10.1016/j.matpur.2005.08.001.

[21]

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.

[22]

T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Statist. Phys., 65 (1991), 579-616. doi: 10.1007/BF01053745.

[23]

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

[24]

P. Walters, Introduction to Ergodic Theory, Springer-Verlag, New York, 2000.

[25]

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

[26]

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.

[27]

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

[28]

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

[29]

G. Wang and Y. Tang, Fractal dimension of a random invariant set and applications, J. Appl. Math., (2013), Art. ID 415764, 5 pp.

[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), 464-478. doi: 10.1016/j.nonrwa.2010.06.032.

[31]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934. doi: 10.3934/cpaa.2004.3.921.

[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), 883-903. doi: 10.1137/050623097.

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