RANDOM PULLBACK ATTRACTOR FOR A NON-AUTONOMOUS MODIFIED SWIFT-HOHENBERG EQUATION WITH MULTIPLICATIVE NOISE

Yongjun Li; Jinying Wei; Zhengzhi Lu

doi:10.11948/20200065

2021 Volume 11 Issue 1

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Yongjun Li, Jinying Wei, Zhengzhi Lu. RANDOM PULLBACK ATTRACTOR FOR A NON-AUTONOMOUS MODIFIED SWIFT-HOHENBERG EQUATION WITH MULTIPLICATIVE NOISE[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 464-476. doi: 10.11948/20200065

Citation:

Yongjun Li, Jinying Wei, Zhengzhi Lu. RANDOM PULLBACK ATTRACTOR FOR A NON-AUTONOMOUS MODIFIED SWIFT-HOHENBERG EQUATION WITH MULTIPLICATIVE NOISE[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 464-476. doi: 10.11948/20200065

RANDOM PULLBACK ATTRACTOR FOR A NON-AUTONOMOUS MODIFIED SWIFT-HOHENBERG EQUATION WITH MULTIPLICATIVE NOISE

1.
School of Mathematics, Lanzhou City University, No.11, Jiefang Road, 730070, China

Corresponding author: Email address:li_liyong120@163.com(Y. Li)
Fund Project: The authors are supported by National Natural Science Foundation of China (11761044, 11661048) and the key constructive discipline of Lanzhou City University(LZCU-ZDJSXK-201706)

Abstract

Abstract

In this paper, we study long time behavior of a non-autonomous stochastic modified Swift-Hohenberg equation with multiplicative noise in stra-tonovich sense. We show that a random $ \mathcal{D} $-pullback attractor exists in $ H_0^2 $ for the corresponding non-autonomous random dynamical system. Due to the stochastic term, the estimates are delicate, the Ornstein-Uhlenbeck(O-U) transformation and its properties are used to overcome the difficulty that the stochastic term brings to us.
- Swift-Hohenberg equation /
- random $ \mathcal{D} $-pullback attractor /
- nonautonomous random dynamical system
MSC: 35B40, 35B41, 60H15, 60H30, 60H40

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References

[1]	L. Arnold, Random Dynamical System, Springer-Verlag, 1998. Google Scholar
[2]	G. S. Aragäo and F. D. M. Bezerra, Upper semicontinuity of the pullback attractors of non-autonomous damped wave equations with terms concentrating on the boundary, J. Mathematical Analysis and Applications, 2018, 452, 871-899. Google Scholar
[3]	P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 2009, 246, 845-869. doi: 10.1016/j.jde.2008.05.017 CrossRef Google Scholar
[4]	J. Duan, An introduction to Stochastic Dynamics, Science Press, Beijing, 2014. Google Scholar
[5]	A. Doelman, B. Standstede, A. Scheel and G. Schneider, Propagation of hexagonal patterens near onset, European J. Appl. Math., 2003, 14, 85-110. doi: 10.1017/S095679250200503X CrossRef Google Scholar
[6]	B. Gess, W. Liu and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differential Equations, 2020, 269, 3414-3455. doi: 10.1016/j.jde.2020.03.002 CrossRef Google Scholar
[7]	A. Khanmamedov, Long-time dynamics of the Swift-Hohenberg equations, J. Mathematical Analysis and Applications, 2020, 483, Article: 123626. Google Scholar
[8]	P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. A, 2007, 463, 163-181. doi: 10.1098/rspa.2006.1753 CrossRef Google Scholar
[9]	S. Lu, Y. Zhang and T. Ma, Global attractor of a modified Swift-Hohenberg equation in $H.k$ spaces, Nonlinear Anal., 2006, 64, 483-498. $H.k$ spaces" target="_blank">Google Scholar
[10]	Y. Li and C. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the non-autonomo us reaction-diffusion equations, Applied Mathematics and Computation, 2007, 190, 1020-1029. doi: 10.1016/j.amc.2006.11.187 CrossRef Google Scholar
[11]	Y. Li and B. Guo, Random attractors for quasi-continuous random dynamical systems and application to stochastic reaction-diffusion equation, J. Differential Equations, 2008, 245, 1775-1800. doi: 10.1016/j.jde.2008.06.031 CrossRef Google Scholar
[12]	Y. Li, J. Wei and T. Zhao, The existence of random $\mathcal{D}$-pullback attractors for a random dynamical system and its applicatin}, J. Applied Analysis and Computaiton, 2019, 9, 1571-1588. $\mathcal{D}$-pullback attractors for a random dynamical system and its applicatin}" target="_blank">Google Scholar
[13]	A. Miranville and A. J. Ntsokongo, On anisotropic caginalp phase-field type models with singular nonlinear terms, J. Applied Analysis and Computation, 2018, 8, 655-674. doi: 10.11948/2018.655 CrossRef Google Scholar
[14]	L. Peletier and V. Rottschäfer, Large time behavior of solution of the Swift-Hohenberg equation, Comptes Rendus Mathematique, 2003, 336, 225-230. doi: 10.1016/S1631-073X(03)00021-9 CrossRef Google Scholar
[15]	L. Peletier and V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D, 2004, 194, 95-126. doi: 10.1016/j.physd.2004.01.043 CrossRef Google Scholar
[16]	L. Peletier and J. F. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst., 2007, 6, 208-235. doi: 10.1137/050647232 CrossRef Google Scholar
[17]	S. H. Park and J. Y. Park, Pullback attractors for a non-autonomous modified Swift-Hohenberg equation, Comput. Math. Appl., 2014, 67, 542-548. doi: 10.1016/j.camwa.2013.11.011 CrossRef Google Scholar
[18]	M. Polat, Global attractor for a modified Swift-Hohenbergequation, Comput. Math. Appl., 2009, 57, 62-66. doi: 10.1016/j.camwa.2008.09.028 CrossRef Google Scholar
[19]	N. Roidos, The Swift-Hohenberg equation on conic manifolds, J. Mathematical Analysis and Applications, 2020, 481, Article: 123491. Google Scholar
[20]	J. Swift and P. C. Hohenberg, Hydrodynamics fluctuations at the convective instability, Phys. Rev. A, 1977, 15, 319-328. doi: 10.1103/PhysRevA.15.319 CrossRef Google Scholar
[21]	R. Samprogna and J. Simsen, A selected pullback attractor, J. Mathematical Analysis and Applications, 2018, 468, 364-375. doi: 10.1016/j.jmaa.2018.08.027 CrossRef Google Scholar
[22]	R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., Springer-Verlag, 1997. Google Scholar
[23]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical system, J. Differential Equations, 2009, 253, 544-1583. Google Scholar
[24]	Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 2015, 259, 728-776. doi: 10.1016/j.jde.2015.02.026 CrossRef Google Scholar
[25]	Z. Wang and S. Zhou, Random attractor for stochastic reaction-diffusion equation with multiplicative noise on unbounded domains, J. Mathematical Analysis and Applications, 2011, 384, 160-172. doi: 10.1016/j.jmaa.2011.02.082 CrossRef Google Scholar
[26]	Z. Wang and X. Du, Pullback attractors for modified Swift-Hohenberg equation on unbounded domain with non-autonomous derterministic and stochastic forcing terms, J. Applied Analysis and Computation, 2017, 7, 207-223. doi: 10.11948/2017014 CrossRef Google Scholar
[27]	L. Xu and Q. Ma, Existence of the uniform attractors for a non-autonomous modified Swift-Hohenberg equation, Advances in Difference Equations, 2015, 153, DOI10.1186/s13662-015-0492-9. doi: 10.1186/s13662-015-0492-9 CrossRef Google Scholar