# American Institute of Mathematical Sciences

August  2020, 25(8): 3153-3170. doi: 10.3934/dcdsb.2020055

## Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise

 1 School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China 2 Hunan Defense Industry Polytechnic, Xiangtan, Hunan 411207, China 3 Department of Mathematics, Columbus State University, Columbus, Georgia 31907, USA

* Corresponding author: Fuqi Yin

Received  May 2019 Revised  September 2019 Published  February 2020

Fund Project: The corresponding author is supported by The Scientific Research Foundation Funded by Hunan Provincial Education Department under grant 19A503 and 15K127; partially supported by Hunan Provincial Natural Science Foundation of China under grant 2015JJ2144, National Natural Science Foundation of People’s Republic of China under grant 11671343

We mainly consider the existence of a random exponential attractor (positive invariant compact measurable set with finite fractal dimension and attracting orbits exponentially) for stochastic discrete long wave-short wave resonance equation driven by multiplicative white noise. Firstly, we prove the existence of a random attractor of the considered equation by proving the existence of a uniformly tempered pullback absorbing set and making an estimate on the "tails" of solutions. Secondly, we show the Lipschitz property of the solution process generated by the considered equation. Finally, we prove the existence of a random exponential attractor of the considered equation, which implies the finiteness of fractal dimension of random attractor.

Citation: Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055
##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] P. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.  Google Scholar [3] H. Cui and S. Zhou, Random attractor for Schr$\mathrm{\ddot{o}}$dinger lattice system with with multiplicative white noise, Journal of Zhejiang Normal University, 40 (2017), 17-23.   Google Scholar [4] 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 [5] R. H. J. Grimshaw, The modulation of an internal gravity-wave packet and the resonance with the mean motion, Stud Appl Math, 56 (1977), 241-266.  doi: 10.1002/sapm1977563241.  Google Scholar [6] H. Li, M. Jin, F. Yin and Z. Liu, Structure and continuity properties of global attractor for the Klein-Gordon equation, Natural Science Journal of Xiangtan Universty, 41 (2019), 15-30.   Google Scholar [7] Y. Liang, Z. Zhu and M. Zhao, Finite fractal dimension of kernel sections for long-wave-short-wave resonance equations on infinite lattices, Acta Mathematica Scientia, 35 (2015), 1146-1157.   Google Scholar [8] A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. PDE: Anal. Comp, 1 (2013), 241-281.  doi: 10.1007/s40072-013-0007-1.  Google Scholar [9] F. Yin and X. Li, Fractal dimensions of random attractors for stochastic Benjamin-Bona-Mahony equation on unbounded domains, Computers and Mathematics with Applications, 75 (2018), 1595-1615.  doi: 10.1016/j.camwa.2017.11.025.  Google Scholar [10] F. Yin and L. Liu, D-pullback attractor for a non-autonomous wave equation with additive noise on unbounded domains, Computers and Mathematics with Applications, 68 (2014), 424-438.  doi: 10.1016/j.camwa.2014.06.018.  Google Scholar [11] C. Zhao and S. Zhou, Compact kernel sections of long-wave-short-wave resonance equations on infinite lattices, Nonlinear Analysis, 68 (2008), 652-670.  doi: 10.1016/j.na.2006.11.027.  Google Scholar [12] S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic damped wave equation with multiplicative noise, Discrete Contin.Dyn.Syst, 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.  Google Scholar [13] X. Zhou, F. Yin and S. Zhou, Uniform exponential attractors for second order non-autonomous lattice dynamical systems, Acta Mathematicae Applicatae Sinica, 33 (2017), 587-606.  doi: 10.1007/s10255-017-0684-z.  Google Scholar [14] S. Zhou, Random exponential attractors for cocyclethe and application to non-autonomous stochastic lattice systems with multiplicative white noise, Journal of Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.  Google Scholar [15] S. Zhou and Z. Wang, Random attractor and random exponential attractor for stochastic nonautonomous damped cubic wave equation with linear multiplicative white noise, Discrete and Continuous Dynamical System, 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210.  Google Scholar [16] S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $R^{3}$, Journal of Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.  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] P. Bates, K. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.  Google Scholar [3] H. Cui and S. Zhou, Random attractor for Schr$\mathrm{\ddot{o}}$dinger lattice system with with multiplicative white noise, Journal of Zhejiang Normal University, 40 (2017), 17-23.   Google Scholar [4] 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 [5] R. H. J. Grimshaw, The modulation of an internal gravity-wave packet and the resonance with the mean motion, Stud Appl Math, 56 (1977), 241-266.  doi: 10.1002/sapm1977563241.  Google Scholar [6] H. Li, M. Jin, F. Yin and Z. Liu, Structure and continuity properties of global attractor for the Klein-Gordon equation, Natural Science Journal of Xiangtan Universty, 41 (2019), 15-30.   Google Scholar [7] Y. Liang, Z. Zhu and M. Zhao, Finite fractal dimension of kernel sections for long-wave-short-wave resonance equations on infinite lattices, Acta Mathematica Scientia, 35 (2015), 1146-1157.   Google Scholar [8] A. Shirikyan and S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. PDE: Anal. Comp, 1 (2013), 241-281.  doi: 10.1007/s40072-013-0007-1.  Google Scholar [9] F. Yin and X. Li, Fractal dimensions of random attractors for stochastic Benjamin-Bona-Mahony equation on unbounded domains, Computers and Mathematics with Applications, 75 (2018), 1595-1615.  doi: 10.1016/j.camwa.2017.11.025.  Google Scholar [10] F. Yin and L. Liu, D-pullback attractor for a non-autonomous wave equation with additive noise on unbounded domains, Computers and Mathematics with Applications, 68 (2014), 424-438.  doi: 10.1016/j.camwa.2014.06.018.  Google Scholar [11] C. Zhao and S. Zhou, Compact kernel sections of long-wave-short-wave resonance equations on infinite lattices, Nonlinear Analysis, 68 (2008), 652-670.  doi: 10.1016/j.na.2006.11.027.  Google Scholar [12] S. Zhou and M. Zhao, Fractal dimension of random attractor for stochastic damped wave equation with multiplicative noise, Discrete Contin.Dyn.Syst, 36 (2016), 2887-2914.  doi: 10.3934/dcds.2016.36.2887.  Google Scholar [13] X. Zhou, F. Yin and S. Zhou, Uniform exponential attractors for second order non-autonomous lattice dynamical systems, Acta Mathematicae Applicatae Sinica, 33 (2017), 587-606.  doi: 10.1007/s10255-017-0684-z.  Google Scholar [14] S. Zhou, Random exponential attractors for cocyclethe and application to non-autonomous stochastic lattice systems with multiplicative white noise, Journal of Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.  Google Scholar [15] S. Zhou and Z. Wang, Random attractor and random exponential attractor for stochastic nonautonomous damped cubic wave equation with linear multiplicative white noise, Discrete and Continuous Dynamical System, 38 (2018), 4767-4817.  doi: 10.3934/dcds.2018210.  Google Scholar [16] S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $R^{3}$, Journal of Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.  Google Scholar
 [1] 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 [2] Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210 [3] Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120 [4] Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269 [5] Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165 [6] Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645 [7] Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757 [8] Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465 [9] Chi Phan. Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3233-3256. doi: 10.3934/dcdsb.2020060 [10] Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255 [11] Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939 [12] Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039 [13] Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875 [14] 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 [15] André Nachbin. Discrete and continuous random water wave dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1603-1633. doi: 10.3934/dcds.2010.28.1603 [16] Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901 [17] Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695 [18] Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305 [19] Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607 [20] Nguyen Huy Tuan, Tran Ngoc Thach, Yong Zhou. On a backward problem for two-dimensional time fractional wave equation with discrete random data. Evolution Equations & Control Theory, 2020, 9 (2) : 561-579. doi: 10.3934/eect.2020024

2019 Impact Factor: 1.27