×

Fractal dimension for the nonautonomous stochastic fifth-order Swift-Hohenberg equation. (English) Zbl 1455.35307

Summary: Some dynamics behaviors for the nonautonomous stochastic fifth-order Swift-Hohenberg equation with additive white noise are considered. The existence of pullback random attractors for the nonautonomous stochastic fifth-order Swift-Hohenberg equation with some properties is mainly investigated on the bounded domain and unbounded domain, through the Ornstein-Uhlenbeck transformation and tail-term estimates. Furthermore, on the basis of some conditions, the finiteness of fractal dimension of random attractor is proved.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35B41 Attractors
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35K35 Initial-boundary value problems for higher-order parabolic equations
35K58 Semilinear parabolic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Swift, J.; Hohenberg, P. C., Hydrodynamic fluctuations at the convective instability, Physical Review A, 15, 1, 319-328 (1977) · doi:10.1103/physreva.15.319
[2] Blömker, D.; Hairer, M.; Pavliotis, G. A., Stochastic Swift-Hohenberg equation near a change of stability, Proceedings of the International Conference on Differential Equations, Comenius University
[3] Hilali, M. F.; Métens, S.; Borckmans, P.; Dewel, G., Pattern selection in the generalized Swift-Hohenberg model, Physical Review E, 51, 3, 2046-2052 (1995) · doi:10.1103/physreve.51.2046
[4] Lin, G.; Gao, H.; Duan, J.; Ervin, V. J., Asymptotic dynamical difference between the nonlocal and local Swift-Hohenberg models, Journal of Mathematical Physics, 41, 4, 2077-2089 (2000) · Zbl 0983.35026 · doi:10.1063/1.533228
[5] Lega, J.; Moloney, J. V.; Newell, A. C., Swift-Hohenberg equation for lasers, Physical Review Letters, 73, 22, 2978-2981 (1994) · doi:10.1103/physrevlett.73.2978
[6] Song, L.; Zhang, Y.; Ma, T., Global attractor of a modified Swift-Hohenberg equation in spaces, Nonlinear Analysis: Theory, Methods & Applications, 72, 1, 183-191 (2010) · Zbl 1180.35126 · doi:10.1016/j.na.2009.06.103
[7] Peletier, L. A.; Rottschäfer, V., Pattern selection of solutions of the Swift-Hohenberg equation, Physica D: Nonlinear Phenomena, 194, 1-2, 95-126 (2004) · Zbl 1052.35076 · doi:10.1016/j.physd.2004.01.043
[8] Peletier, L. A.; Rottschäfer, V., Large time behaviour of solutions of the Swift-Hohenberg equation, Comptes Rendus Mathematique, 336, 3, 225-230 (2003) · Zbl 1031.35072 · doi:10.1016/s1631-073x(03)00021-9
[9] Peletier, L. A.; Williams, J. F., Some canonical bifurcations in the Swift-Hohenberg equation, SIAM Journal on Applied Dynamical Systems, 6, 1, 208-235 (2007) · Zbl 1210.34028 · doi:10.1137/050647232
[10] Park, S. H.; Park, J. Y., Pullback attractor for a non-autonomous modified swift-hohenberg equation, Computers & Mathematics with Applications, 67, 3, 542-548 (2014) · Zbl 1381.35071 · doi:10.1016/j.camwa.2013.11.011
[11] Wang, Z.; Du, X., Pullback attractors for modified Swift-Hohenberg equation on unbounded domains with non-autonomous deterministic and stochastic forcing terms, Journal of Applied Analysis and Computation, 7, 207-223 (2017) · Zbl 1474.35705
[12] Xu, L.; Ma, Q., Existence of the uniform attractors for a non-autonomous modified Swift-Hohenberg equation, Advances in Differential Equations, 2015, 1-11 (2015) · Zbl 1422.35138 · doi:10.1186/s13662-015-0492-9
[13] Robinson, J. C., Infinite-Dimensional Dynamical Systems (2001), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1026.37500
[14] Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics (1997), New York, NY, USA: Springer-Verlag, New York, NY, USA · Zbl 0871.35001
[15] Foias, C.; Olson, E. J., Finite fractal dimension and Hölder-Lipschitz parameterization, Indiana University Journal, 45, 603-616 (1996) · Zbl 0887.54034 · doi:10.1512/iumj.1996.45.1326
[16] Langa, J. A.; Robinson, J. C., Fractal dimension of a random invariant set, Journal de Mathématiques Pures et Appliquées, 85, 2, 269-294 (2006) · Zbl 1134.37364 · doi:10.1016/j.matpur.2005.08.001
[17] Wang, G.; Tang, Y., Fractal dimension of a random invariant set and applications, Journal of Applied Mathematics, 2013 (2013) · Zbl 1397.37087 · doi:10.1155/2013/415764
[18] Debussche, A., On the finite dimensionality of random attractors, Stochastic Analysis and Applications, 15, 4, 473-491 (1997) · Zbl 0888.60051 · doi:10.1080/07362999708809490
[19] Langa, J. A., Finite-dimensional limiting dynamics of random dynamical systems, Dynamical Systems, 18, 1, 57-68 (2003) · Zbl 1038.37041 · doi:10.1080/1468936031000080812
[20] Guo, Y., Dynamics and invariant manifolds for a nonlocal stochastic Swift-Hohenberg equation, Journal of Inequalities and Applications, 2015, 366 (2015) · Zbl 1335.60112 · doi:10.1186/s13660-015-0889-8
[21] Guo, C.; Chen, Y.; Guo, Y., Random attractors of stochastic local Swift-Hohenberg equation with additive noise, Journal of Inequalities and Applications, 2016, 228 (2016) · Zbl 1346.60095
[22] Bates, P. W.; Lu, K.; Wang, B., Random attractors for stochastic reaction-diffusion equations on unbounded domains, Journal of Differential Equations, 246, 2, 845-869 (2009) · Zbl 1155.35112 · doi:10.1016/j.jde.2008.05.017
[23] Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, Journal of Dynamics and Differential Equations, 9, 2, 307-341 (1997) · Zbl 0884.58064 · doi:10.1007/bf02219225
[24] Crauel, H.; Flandoli, F., Attractors for random dynamical systems, Probability Theory and Related Fields, 100, 3, 365-393 (1994) · Zbl 0819.58023 · doi:10.1007/bf01193705
[25] Zhou, S.; Tian, Y.; Wang, Z., Fractal dimension of random attractors for stochastic non-autonomous reaction-diffusion equations, Applied Mathematics and Computation, 276, 80-95 (2016) · Zbl 1410.37068 · doi:10.1016/j.amc.2015.12.009
[26] Zhou, S.; Wang, Z., Finite fractal dimensions of random attractors for stochastic FitzHugh-Nagumo system with multiplicative white noise, Journal of Mathematical Analysis and Applications, 441, 2, 648-667 (2016) · Zbl 1359.37115 · doi:10.1016/j.jmaa.2016.04.038
[27] Polat, M., Global attractor for a modified Swift-Hohenberg equation, Computers & Mathematics with Applications, 57, 1, 62-66 (2009) · Zbl 1165.35450 · doi:10.1016/j.camwa.2008.09.028
[28] Wang, B., Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Analysis, 158, 60-82 (2017) · Zbl 1366.35229 · doi:10.1016/j.na.2017.04.006
[29] Arnold, L., Random Dynamical Systems (1998), New York, NY, USA: Springer-Verlag, New York, NY, USA · Zbl 0906.34001
[30] Wang, B., Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, Nonlinear Analysis, 71, 7-8, 2811-2828 (2009) · Zbl 1173.37065 · doi:10.1016/j.na.2009.01.131
[31] Marion, M., Finite-dimensional attractors associated with partly dissipative reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 20, 4, 816-844 (1989) · Zbl 0684.35055 · doi:10.1137/0520057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.