# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022
##### 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] T. Caraballo, P. 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. [4] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. [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. [6] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225. [7] A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491.  doi: 10.1080/07362999708809490. [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. [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. [10] X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741. [11] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. [12] J. A. Langa, Finite-dimensional limiting dynamics of random dynamical systems, Dyn. Syst., 18 (2003), 57-68.  doi: 10.1080/1468936031000080812. [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. [14] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418. [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. [16] H. Li, Y. 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. [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. [18] T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Stat. Phys., 65 (1991), 579-616.  doi: 10.1007/BF01053745. [19] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [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. [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. [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. [23] B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations, 2009 (2009), 1-18. [24] G. Wang and Y. Tang, Fractal dimension of a random invariant set and applications, J. Appl. Math. , (2013), Art. ID 415764, 5 pp. [25] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. [26] 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. [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. [28] 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. [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. [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. [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.

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##### 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] T. Caraballo, P. 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. [4] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. [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. [6] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225. [7] A. Debussche, On the finite dimensionality of random attractors, Stochastic Anal. Appl., 15 (1997), 473-491.  doi: 10.1080/07362999708809490. [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. [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. [10] X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741. [11] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. [12] J. A. Langa, Finite-dimensional limiting dynamics of random dynamical systems, Dyn. Syst., 18 (2003), 57-68.  doi: 10.1080/1468936031000080812. [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. [14] O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569418. [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. [16] H. Li, Y. 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. [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. [18] T. Sauer, J. A. Yorke and M. Casdagli, Embedology, J. Stat. Phys., 65 (1991), 579-616.  doi: 10.1007/BF01053745. [19] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [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. [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. [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. [23] B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations, 2009 (2009), 1-18. [24] G. Wang and Y. Tang, Fractal dimension of a random invariant set and applications, J. Appl. Math. , (2013), Art. ID 415764, 5 pp. [25] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. [26] 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. [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. [28] 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. [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. [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. [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.
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