# American Institute of Mathematical Sciences

August  2019, 24(8): 4145-4167. doi: 10.3934/dcdsb.2019054

## Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients

 School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: liyr@swu.edu.cn (Yangrong Li)

Dedicated to Professor Peter Kloeden on his 70th birthday

Received  June 2018 Revised  August 2018 Published  August 2019 Early access  February 2019

Fund Project: Li was supported by Natural Science Foundation of China grant 11571283, Wang was supported by Postgraduate Research and Innovation Project of Chongqing grant CYB18115.

We study some time-related properties of the random attractor for the stochastic wave equation on an unbounded domain with time-varying coefficient and force. We assume that the coefficient is bounded and the time-dependent force is backward tempered, backward complement-small, backward tail-small, and then prove both existence and backward compactness of a random attractor on the universe of all backward tempered sets. By using the Egoroff and Lusin theorems, we show the measurability of the absorbing set although it is the union of some random sets over an uncountable index set. Moreover, we obtain the backward compactness of the attractor if the force is periodic, and obtain the periodicity of the attractor if both force and coefficient are periodic.

Citation: Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054
##### References:
 [1] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31. [2] P. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004. [3] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2012), 439-455.  doi: 10.3934/dcdsb.2010.14.439. [4] I. Chueshov, Monotone Random Systems Theory and Applications, vol.1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277. [5] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of Amer Math Soc, 195 (2008), ⅷ+183 pp. doi: 10.1090/memo/0912. [6] I. Chueshov, I. Lasiecka and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam Differ. Equ., 21 (2009), 269-314.  doi: 10.1007/s10884-009-9132-y. [7] H. Cui, J. A. Langa and Y. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235.  doi: 10.1016/j.na.2016.03.012. [8] H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differ. Equ., 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z. [9] H. Cui, M. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3379-3407.  doi: 10.3934/dcdsb.2017142. [10] H. Cui, P. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Physica D, 374/375 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002. [11] X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Int. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741. [12] A. Gu and P. E. Kloeden, Asymptotic behavior of a nonautonomous $p$-Laplacian lattice system, Intern. J. Bifur. Chaos, 26 (2016), 1650174, 9pp. doi: 10.1142/S0218127416501741. [13] X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differ. Equ., 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015. [14] R. Jones and B. Wang, Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. RWA, 14 (2013), 1308-1322.  doi: 10.1016/j.nonrwa.2012.09.019. [15] A. K. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differ. Equ., 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001. [16] P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735. [17] P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069. [18] P. E. Kloeden, J. Simsen and M. S. Simsen, Asymptotically autonomous multivalued cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004. [19] P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Differ. Equ. Appl., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550. [20] D. Li, K. Lu, B. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009. [21] D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reactiondiffusion equations on thin domains, J. Differ. Equ., 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024. [22] F. Li, Y. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Cont. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158. [23] Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021. [24] Y. Li, L. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033. [25] Y. Li, R. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092. [26] Y. Li, L. She and J. Yin, Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equation, Acta. Math. Sci., 38 (2018), 591-609.  doi: 10.1016/S0252-9602(18)30768-9. [27] Y. Li, L. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058. [28] Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203. [29] M. Prizzi, Regularity of invariant sets in semilinear damped wave equations, J. Differ. Equ., 247 (2009), 3315-3337.  doi: 10.1016/j.jde.2009.08.011. [30] C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.  doi: 10.1088/0951-7715/19/11/008. [31] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015. [32] B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5. [33] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269. [34] R. Wang, Y. Li and F. Li, Probabilistic robustness for dispersive-dissipative wave equations driven by small Laplace-multiplier noise, Dynam. Syst. Appl., 27 (2018), 165-183.  doi: 10.12732/dsa.v27i1.9. [35] Z. Wang and L. Zhang, Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation, Comput. Math. Appl., 75 (2018), 3343-3357.  doi: 10.1016/j.camwa.2018.02.002. [36] Z. Wang and S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Disrete Contin. Dyn. Syst., 37 (2017), 545-573.  doi: 10.3934/dcds.2017022. [37] J. Yin, A. Gu and Y. Li, Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dyn. Partial Differ. Equ., 14 (2017), 201-218.  doi: 10.4310/DPDE.2017.v14.n2.a4. [38] J. Yin, Y. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.  doi: 10.1016/j.camwa.2017.05.015. [39] 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. [40] S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal., 120 (2015), 202-226.  doi: 10.1016/j.na.2015.03.009.

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##### References:
 [1] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Cont. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31. [2] P. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004. [3] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions, Discrete Contin. Dyn. Syst. Ser. B, 14 (2012), 439-455.  doi: 10.3934/dcdsb.2010.14.439. [4] I. Chueshov, Monotone Random Systems Theory and Applications, vol.1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277. [5] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of Amer Math Soc, 195 (2008), ⅷ+183 pp. doi: 10.1090/memo/0912. [6] I. Chueshov, I. Lasiecka and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam Differ. Equ., 21 (2009), 269-314.  doi: 10.1007/s10884-009-9132-y. [7] H. Cui, J. A. Langa and Y. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal., 140 (2016), 208-235.  doi: 10.1016/j.na.2016.03.012. [8] H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differ. Equ., 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z. [9] H. Cui, M. M. Freitas and J. A. Langa, On random cocycle attractors with autonomous attraction universes, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3379-3407.  doi: 10.3934/dcdsb.2017142. [10] H. Cui, P. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Physica D, 374/375 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002. [11] X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Int. J. Math., 19 (2008), 421-437.  doi: 10.1142/S0129167X08004741. [12] A. Gu and P. E. Kloeden, Asymptotic behavior of a nonautonomous $p$-Laplacian lattice system, Intern. J. Bifur. Chaos, 26 (2016), 1650174, 9pp. doi: 10.1142/S0218127416501741. [13] X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differ. Equ., 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015. [14] R. Jones and B. Wang, Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. RWA, 14 (2013), 1308-1322.  doi: 10.1016/j.nonrwa.2012.09.019. [15] A. K. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differ. Equ., 230 (2006), 702-719.  doi: 10.1016/j.jde.2006.06.001. [16] P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735. [17] P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069. [18] P. E. Kloeden, J. Simsen and M. S. Simsen, Asymptotically autonomous multivalued cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004. [19] P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Differ. Equ. Appl., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550. [20] D. Li, K. Lu, B. Wang and X. Wang, Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains, Discrete Contin. Dyn. Syst., 38 (2018), 187-208.  doi: 10.3934/dcds.2018009. [21] D. Li, B. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reactiondiffusion equations on thin domains, J. Differ. Equ., 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024. [22] F. Li, Y. Li and R. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Cont. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158. [23] Y. Li, A. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021. [24] Y. Li, L. She and R. Wang, Asymptotically autonomous dynamics for parabolic equations, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033. [25] Y. Li, R. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092. [26] Y. Li, L. She and J. Yin, Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equation, Acta. Math. Sci., 38 (2018), 591-609.  doi: 10.1016/S0252-9602(18)30768-9. [27] Y. Li, L. She and J. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058. [28] Y. Li and J. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic Fitzhugh-Nagumo equations, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203. [29] M. Prizzi, Regularity of invariant sets in semilinear damped wave equations, J. Differ. Equ., 247 (2009), 3315-3337.  doi: 10.1016/j.jde.2009.08.011. [30] C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity, Nonlinearity, 19 (2006), 2645-2665.  doi: 10.1088/0951-7715/19/11/008. [31] B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015. [32] B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5. [33] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Contin. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269. [34] R. Wang, Y. Li and F. Li, Probabilistic robustness for dispersive-dissipative wave equations driven by small Laplace-multiplier noise, Dynam. Syst. Appl., 27 (2018), 165-183.  doi: 10.12732/dsa.v27i1.9. [35] Z. Wang and L. Zhang, Finite fractal dimension of random attractor for stochastic non-autonomous strongly damped wave equation, Comput. Math. Appl., 75 (2018), 3343-3357.  doi: 10.1016/j.camwa.2018.02.002. [36] Z. Wang and S. Zhou, Random attractor for stochastic non-autonomous damped wave equation with critical exponent, Disrete Contin. Dyn. Syst., 37 (2017), 545-573.  doi: 10.3934/dcds.2017022. [37] J. Yin, A. Gu and Y. Li, Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dyn. Partial Differ. Equ., 14 (2017), 201-218.  doi: 10.4310/DPDE.2017.v14.n2.a4. [38] J. Yin, Y. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.  doi: 10.1016/j.camwa.2017.05.015. [39] 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. [40] S. Zhou and M. Zhao, Random attractors for damped non-autonomous wave equations with memory and white noise, Nonlinear Anal., 120 (2015), 202-226.  doi: 10.1016/j.na.2015.03.009.
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