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

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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

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

Received March 2015 Revised July 2015 Published October 2015

Abstract
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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 & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

References:

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[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
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[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

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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

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