Random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice systems in weighted space

Rou Lin; Min Zhao; Jinlu Zhang; Rou Lin; Min Zhao; Jinlu Zhang

doi:10.3934/math.2023150

AIMS Mathematics

2023, Volume 8, Issue 2: 2871-2890. doi: 10.3934/math.2023150

Previous Article Next Article

Research article

Random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice systems in weighted space

Department of Mathematics, Wenzhou University, Wenzhou 325035, China

Received: 16 September 2022 Revised: 28 October 2022 Accepted: 04 November 2022 Published: 11 November 2022
MSC : 34F05, 37L60, 60H10

We mainly study the existence of random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice system with multiplicative white noise and quasi-periodic forces in weighted spaces. Firstly, the stochastic Schrödinger system is transformed into a random system without white noise by the Ornstein-Uhlenbeck process, whose solution generates a jointly continuous non-autonomous random dynamical system $ \Phi $. Secondly, we prove the existence of a uniform absorbing random set for $ \Phi $ in weighted spaces. Finally, we obtain the existence of a random uniform exponential attractor for the considered system $ \Phi $ in weighted space.
- non-autonomous Schrödinger lattice system,
- random uniform exponential attractor,
- weighted space,
- quasi-periodic force,
- multiplicative white noise
Citation: Rou Lin, Min Zhao, Jinlu Zhang. Random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice systems in weighted space[J]. AIMS Mathematics, 2023, 8(2): 2871-2890. doi: 10.3934/math.2023150

Related Papers:

Abstract

We mainly study the existence of random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice system with multiplicative white noise and quasi-periodic forces in weighted spaces. Firstly, the stochastic Schrödinger system is transformed into a random system without white noise by the Ornstein-Uhlenbeck process, whose solution generates a jointly continuous non-autonomous random dynamical system $ \Phi $. Secondly, we prove the existence of a uniform absorbing random set for $ \Phi $ in weighted spaces. Finally, we obtain the existence of a random uniform exponential attractor for the considered system $ \Phi $ in weighted space.

References

[1]	L. Arnold, Random dynamical systems, Berlin: Springer-Verlag, 1998. http://dx.doi.org/10.1007/978-3-662-12878-7
[2]	P. Bates, H. Lisei, K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dynam., 6 (2006), 1–21. http://dx.doi.org/10.1142/S0219493706001621 doi: 10.1142/S0219493706001621
[3]	P. Bates, K. Lu, B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Physica D, 289 (2014), 32–50. http://dx.doi.org/10.1016/j.physd.2014.08.004 doi: 10.1016/j.physd.2014.08.004
[4]	A. Carvalho, S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pur. Appl. Anal., 12 (2013), 3047–3071. http://dx.doi.org/10.3934/cpaa.2013.12.3047 doi: 10.3934/cpaa.2013.12.3047
[5]	A. Carvalho, S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pur. Appl. Anal., 13 (2014), 1141–1165. http://dx.doi.org/10.3934/cpaa.2014.13.1141 doi: 10.3934/cpaa.2014.13.1141
[6]	H. Cui, J. Langa, Uniform attractors for nonautonomous random dynamical systems, J. Differ. Equations, 263 (2017), 1225–1268. http://dx.doi.org/10.1016/j.jde.2017.03.018 doi: 10.1016/j.jde.2017.03.018
[7]	H. Cui, S. Zhou, Random attractor for Schrödinger lattice system with multiplicative white noise (Chinese), Journal of Zhejiang Normal University, 40 (2017), 17–23. http://dx.doi.org/10.16218/j.issn.1001-5051.2017.01.003 doi: 10.16218/j.issn.1001-5051.2017.01.003
[8]	T. Chen, S. Zhou, C. Zhao, Attractors for discrete nonlinear Schrödinger equation with delay, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 633–642. http://dx.doi.org/10.1007/s10255-007-7101-y doi: 10.1007/s10255-007-7101-y
[9]	R. Czaja, M. Efendiev, Pullback exponential attractors for nonautonomous equations part Ⅰ: semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748–765. http://dx.doi.org/10.1016/j.jmaa.2011.03.053 doi: 10.1016/j.jmaa.2011.03.053
[10]	T. Caraballo, S. Sonner, Random pullback exponential attractors: general existence results for random dynamical systems in Banach spaces, Discrete Cont. Dyn., 37 (2017), 6383–6403. http://dx.doi.org/10.3934/dcds.2017277 doi: 10.3934/dcds.2017277
[11]	V. Chepyzhov, M. Vishik, Attractors for equations of mathematical physics, Providence: American Mathematical Society, 2002.
[12]	A. Eden, C. Foias, B. Nicolaenko, R. Temam, Exponential attractors for dissipative evolution equations, Chichester: Wiley, 1994.
[13]	M. Efendiev, A. Miranville, S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\mathbb{R}^{3}$, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 330 (2000), 713–718. http://dx.doi.org/10.1016/S0764-4442(00)00259-7 doi: 10.1016/S0764-4442(00)00259-7
[14]	X. Han, W. Shen, S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differ. Equations, 250 (2011), 1235–1266. http://dx.doi.org/10.1016/j.jde.2010.10.018 doi: 10.1016/j.jde.2010.10.018
[15]	X. Han, Exponential attractors for lattice dynamical systems in weighted spaces, Discrete Cont. Dyn., 31 (2011), 445–467. http://dx.doi.org/10.3934/dcds.2011.31.445 doi: 10.3934/dcds.2011.31.445
[16]	Z. Han, S. Zhou, Random uniform exponential attractors for non-autonomous stochastic lattice systems and FitzHugh-Nagumo lattice systems with quasi-periodic forces and multiplicative noise, Stoch. Dynam., 20 (2020), 2050036. http://dx.doi.org/10.1142/S0219493720500367 doi: 10.1142/S0219493720500367
[17]	X. Jiang, S. Zhou, Z. Han, Random exponential attractor for Schrödinger lattice system with multiplicative white noise (Chinese), Journal of Zhejiang Normal University, 43 (2020), 251–258. http://dx.doi.org/10.16218/j.issn.1001-5051.2020.03.002 doi: 10.16218/j.issn.1001-5051.2020.03.002
[18]	N. Karachalios, A. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differ. Equations, 217 (2005), 88–123. http://dx.doi.org/10.1016/j.jde.2005.06.002 doi: 10.1016/j.jde.2005.06.002
[19]	P. Kevrekidis, K. Rasmussen, A. Bishop, The discrete nonlinear Schrödinger equation: a survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2833–2900. http://dx.doi.org/10.1142/S0217979201007105 doi: 10.1142/S0217979201007105
[20]	A. Shirikyan, S. Zelik, Exponential attractors for random dynamical systems and applications, Stoch. PDE: Anal. Comp., 1 (2013), 241–281. http://dx.doi.org/10.1007/s40072-013-0007-1 doi: 10.1007/s40072-013-0007-1
[21]	R. Temam, Infinite dimensional dynamical systems in mechanics and physics, New York: Springer-Verlag, 1997. http://dx.doi.org/10.1007/978-1-4612-0645-3
[22]	X. Tan, F. Yin, G. Fan, Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise, Discrete Cont. Dyn.-B, 25 (2020), 3153–3170. http://dx.doi.org/10.3934/dcdsb.2020055 doi: 10.3934/dcdsb.2020055
[23]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equations, 253 (2012), 1544–1583. http://dx.doi.org/10.1016/j.jde.2012.05.015 doi: 10.1016/j.jde.2012.05.015
[24]	B. Wang, R. Wang, Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise, Stoch. Anal. Appl., 38 (2020), 213–237. http://dx.doi.org/10.1080/07362994.2019.1679646 doi: 10.1080/07362994.2019.1679646
[25]	Z. Wang, S. Zhou, Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise, Discrete Cont. Dyn., 38 (2018), 4767–4817. http://dx.doi.org/10.3934/dcds.2018210 doi: 10.3934/dcds.2018210
[26]	S. Zhou, W. Shi, Attractors and dimension of dissipative lattice systems, J. Differ. Equations, 224 (2006), 172–204. http://dx.doi.org/10.1016/j.jde.2005.06.024 doi: 10.1016/j.jde.2005.06.024
[27]	S. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differ. Equations, 263 (2017), 2247–2279. http://dx.doi.org/10.1016/j.jde.2017.03.044 doi: 10.1016/j.jde.2017.03.044
[28]	S. Zhou, H. Tan, Exponential attractor for nonlinear Schrödinger lattice equation (Chinese), Journal of Zhejiang Normal University, 38 (2015), 361–365. http://dx.doi.org/10.16218/j.issn.1001-5051.2015.04.001 doi: 10.16218/j.issn.1001-5051.2015.04.001
[29]	S. Zhou, M. Zhao, H. Tan, Pullback and uniform exponential attractor for non-autonomous Schrödinger lattice equation (Chinese), Acta Math. Appl. Sin., 42 (2019), 145–161.
[30]	S. Zhang, S. Zhou, Random uniform exponential attractors for Schrödinger lattice systems with quasi-periodic forces and multiplicative white noise, Discrete Cont. Dyn.-S, in press. http://dx.doi.org/10.3934/dcdss.2022056

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)