# 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  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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [4] I. Chueshov, Monotone Random Systems Theory and Applications, vol.1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Int. J. Math., 19 (2008), 421-437. doi: 10.1142/S0129167X08004741. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] A. K. 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Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [4] I. Chueshov, Monotone Random Systems Theory and Applications, vol.1779, Springer Science & Business Media, 2002. doi: 10.1007/b83277. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Int. J. Math., 19 (2008), 421-437. doi: 10.1142/S0129167X08004741. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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