Random dynamics of stochastic BBM equations driven by nonlinear colored noise on unbounded channel. (English) Zbl 1501.35081
Summary: This paper is devoted to study the asymptotic behavior of solutions to a class of stochastic BBM equations driven by nonlinear or linear colored noise defined on a three-dimensional unbounded channel. We first prove existence of compact pullback random attractors of the stochastic BBM equation driven by nonlinear colored noise and then establish the upper semi-continuity of the random attractors to a special class of stochastic BBM equations driven by linear colored noise as the correlation time tends to zero. The pullback asymptotic compactness of the solutions is established by a tail-estimates method in order to overcome the difficulty introduced by the noncompactness of Sobolev embedding on unbounded domains.
MSC:
| 35B41 | Attractors |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 37L55 | Infinite-dimensional random dynamical systems; stochastic equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
Keywords:
Benjamin-Bona-Mahony (BBM) equation; pullback random attractor; unbounded channel; nonlinear colored noise; upper semi-continuityReferences:
| [1] | Astaburuaga, MA; Bisognin, E.; Bisognin, V.; Fernandez, C., Global attractors and finite dimensionality for a class of dissipative equations of BBM’s type, Electron. J. Differential Equations, 25, 1-14 (1998) · Zbl 0913.35022 |
| [2] | Avrin, J.; Goldstein, JA, Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions, Nonlinear Anal., 9, 861-865 (1985) · Zbl 0591.35012 · doi:10.1016/0362-546X(85)90023-9 |
| [3] | J. Avrin, The generalized Benjamin-Bona-Mahony equation in \({{\mathbb{R} }}^n\) with singular initial data, Nonlinear Anal. 11 (1987) 139-147. · Zbl 0645.35084 |
| [4] | Ball, JM, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci., 7, 475-502 (1997) · Zbl 0903.58020 · doi:10.1007/s003329900037 |
| [5] | Bates, PW; Lu, K.; Wang, B., Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246, 845-869 (2009) · Zbl 1155.35112 · doi:10.1016/j.jde.2008.05.017 |
| [6] | Benjamin, TB; Bona, JL; Mahony, JJ, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond., 272, 47-78 (1972) · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032 |
| [7] | Celebi, AO; Kalantarov, VK; Polat, M., Attractors for the generalized Benjamin-Bona-Mahony equation, J. Differential Equations, 157, 439-451 (1999) · Zbl 0934.35151 · doi:10.1006/jdeq.1999.3634 |
| [8] | P. Chen, B. Wang, R. Wang, X. Zhang, Multivalued random dynamics of Benjamin-Bona-Mahony equations driven by nonlinear colored noise on unbounded domains, Math. Ann., doi:10.1007/s00208-022-02400-0. · Zbl 1520.35012 |
| [9] | Chueshov, I.; Polat, M.; Siegmund, S., Gevrey regularity of global attractor for generalized Benjamin-Bona-Mahony equation, Mat. Fiz. Anal. Geom., 11, 226-242 (2004) · Zbl 1082.35042 |
| [10] | Chen, YM, Remark on the global existence for the generalized Benjamin-Bona-Mahony equations in arbitrary dimension, Appl. Anal., 30, 1-15 (1988) · Zbl 0631.35080 · doi:10.1080/00036818808839812 |
| [11] | Crauel, H.; Debussche, A.; Flandoli, F., Random attractors, J. Dynam. Differential Equations, 9, 307-341 (1997) · Zbl 0884.58064 · doi:10.1007/BF02219225 |
| [12] | Goldstein, JA; Wichnoski, BJ, On the Benjamin-Bona-Mahony equation in higher dimensions, Nonlinear Anal., 4, 665-675 (1980) · Zbl 0447.35068 · doi:10.1016/0362-546X(80)90067-X |
| [13] | Gu, A.; Wang, B., Asymptotic behavior of random FitzHugh-Nagumo systems driven by colored noise, Discrete Continuous Dyn. Syst. B, 23, 1689-1720 (2018) · Zbl 1396.35078 · doi:10.3934/dcdsb.2018072 |
| [14] | Gu, A.; Lu, K.; Wang, B., Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations, Discrete Contin. Dyn. Syst., 39, 185-218 (2019) · Zbl 1408.35123 · doi:10.3934/dcds.2019008 |
| [15] | Huang, J.; Tang, Y.; Wang, M., Singular support of the global attractor for a damped BBM equation, Discrete Continuous Dyn. Syst. B, 26, 5321-5335 (2021) · Zbl 1467.35285 · doi:10.3934/dcdsb.2020345 |
| [16] | Kang, JR, Attractors for autonomous and nonautonomous 3D Benjamin-Bona-Mahony equations, Appl. Math. Comput., 274, 343-352 (2016) · Zbl 1410.35170 |
| [17] | Y. Li, R. Wang, Random attractors for 3D Benjamin-Bona-Mahony equations derived by a Laplace-multiplier noise, Stoch. Dyn. 18 (2018) 1850004, 26 · Zbl 1386.60222 |
| [18] | L.A. Medeiros, G. Perla Menzala, Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equation, SIAM J. Math. Anal. 8 (1977) 792-799. · Zbl 0337.35004 |
| [19] | Park, JY; Park, SH, Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains, Sci. China Math., 54, 741-752 (2011) · Zbl 1219.35026 · doi:10.1007/s11425-011-4190-0 |
| [20] | M. Stanislavova, A. Stefanov, B. Wang, Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on \({\,{mathbb{R}} }^3\), J. Differential Equations 219 (2005) 451-483. · Zbl 1160.35354 |
| [21] | Wang, B., Strong attractors for the Benjamin-Bona-Mahony equation, Appl. Math. Lett., 10, 23-28 (1997) · Zbl 0882.35109 · doi:10.1016/S0893-9659(97)00005-0 |
| [22] | Wang, B.; Yang, W., Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A, 30, 4877-4885 (1997) · Zbl 0924.35139 · doi:10.1088/0305-4470/30/13/035 |
| [23] | Wang, B., Regularity of attractors for the Benjamin-Bona-Mahony equation, J. Phys. A, 31, 7635-7645 (1998) · Zbl 0931.35156 · doi:10.1088/0305-4470/31/37/021 |
| [24] | Wang, B., Attractors for reaction-diffusion equations in unbounded domains, Phys. D, 128, 41-52 (1999) · Zbl 0953.35022 · doi:10.1016/S0167-2789(98)00304-2 |
| [25] | Wang, B.; Fussner, DW; Bi, C., Existence of global attractors for the Benjamin-Bona-Mahony equation in unbounded domains, J. Phys. A, 40, 10491-10504 (2007) · Zbl 1120.35028 · doi:10.1088/1751-8113/40/34/007 |
| [26] | Wang, B., Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246, 2506-2537 (2009) · Zbl 1165.60025 · doi:10.1016/j.jde.2008.10.012 |
| [27] | R. Wang, L. Shi, B. Wang, Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on \({\mathbb{R}}^N\), Nonlinearity 32 (2019) 4524-4556. · Zbl 1423.35419 |
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.
