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

Xingni Tan; Fuqi Yin; Guihong Fan

doi:10.3934/dcdsb.2020055

Article Contents

2020, Volume 25, Issue 8: 3153-3170. Doi: 10.3934/dcdsb.2020055

This issue Previous Article Attractors for a class of delayed reaction-diffusion equations with dynamic boundary conditions Next Article Global attraction in a system of delay differential equations via compact and convex sets

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
^* Corresponding author: Fuqi Yin

Received: May 2019

Revised: September 2019

Early access: February 2020

Published: August 2020

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Random exponential attractor,
fractal dimension,
multiplicative white noise,
discrete long wave-short wave resonance equation.

Mathematics Subject Classification: 37L55; 35B40; 60H15.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[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.
[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.
[4]	X. Fan, Random attractors for damped stochastic wave equations with multiplicative noise, Int. J. Math, 19 (2008), 421-437. doi: 10.1142/S0129167X08004741.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.