January  2017, 37(1): 545-573. doi: 10.3934/dcds.2017022

Random attractor for stochastic non-autonomous damped wave equation with critical exponent

1. 

School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, China

2. 

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

*Corresponding author: Shengfan Zhou

Received  April 2016 Revised  July 2016 Published  November 2016

Fund Project: The authors is supported by NSFC grant No. 11471290,61271396,11326114,11401244. Zhejiang Natural Science Foundation grant No. LY14A010012 and Zhejiang Normal University Foundation grant No. ZC304014012

In this paper, we prove the existence of random attractor and obtainan upper bound of fractal dimension of random attractor forstochastic non-autonomous damped wave equation with criticalexponent and additive white noise. We first prove the existence of arandom attractor by carefully splitting the positivity of the linearoperator in the corresponding random evolution equation of the firstorder in time and by carefully decomposing the solutions of systemthrough two different modes, and we show the boundedness of randomattractor in a higher regular space by a recurrence method. Then weestablish a criterion to bound the fractal dimension of a randominvariant set for a cocycle and applied these conditions to get anupper bound of fractal dimension of the random attractor ofconsidered system.

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

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

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T. CaraballoP. E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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

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

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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.  Google Scholar

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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.  Google Scholar

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

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

[26]

M. YangJ. 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.  Google Scholar

[27]

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.  Google Scholar

[28]

S. ZhouF. 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.  Google Scholar

[29]

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

[30]

S. Zhou and M. Zhao, Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonl. Anal., 133 (2016), 292-318.  doi: 10.1016/j.na.2015.12.013.  Google Scholar

[31]

S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.  Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 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. , Amsterdam, 1992.  Google Scholar

[3]

T. CaraballoP. E. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

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

[7]

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

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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.  Google Scholar

[14] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418.  Google Scholar
[15]

P. Li and S. T. Yau, Estimate of the first eigenvalue of a compact Riemann manifold, Proceeding of Symposition in Pure Math., 36 (1980), 205-239.   Google Scholar

[16]

H. LiY. You and J. Tu, Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.  doi: 10.1016/j.jde.2014.09.007.  Google Scholar

[17]

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.  Google Scholar

[18]

T. SauerJ. A. Yorke and M. Casdagli, Embedology, J. Stat. Phys., 65 (1991), 579-616.  doi: 10.1007/BF01053745.  Google Scholar

[19]

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

[20]

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.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

M. YangJ. 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.  Google Scholar

[27]

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.  Google Scholar

[28]

S. ZhouF. 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.  Google Scholar

[29]

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

[30]

S. Zhou and M. Zhao, Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonl. Anal., 133 (2016), 292-318.  doi: 10.1016/j.na.2015.12.013.  Google Scholar

[31]

S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.  Google Scholar

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