September  2017, 22(7): 2569-2586. doi: 10.3934/dcdsb.2017092

Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received  July 2016 Revised  August 2016 Published  March 2017

Fund Project: This work was supported by the Natural Science Foundation of China Grants 11571283.

Backward compact dynamics is deduced for a non-autonomous Benjamin-Bona-Mahony (BBM) equation on an unbounded 3D-channel. A backward compact attractor is defined by a time-dependent family of backward compact, invariant and pullback attracting sets. The theoretical existence result for such an attractor is derived from the backward flattening property, and this property is proved to be equivalent to the backward asymptotic compactness in a uniformly convex Banach space. Finally, it is shown that the BBM equation has a backward compact attractor in a Sobolev space under some suitable assumptions, such as, backward translation boundedness and backward small-tail. Both spectrum decomposition and cut-off technique are used to give all required backward uniform estimates.

Citation: Yangrong Li, Renhai Wang, Jinyan Yin. Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2569-2586. doi: 10.3934/dcdsb.2017092
References:
[1]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.

[2]

A. Adili and B. Wang, Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise, Discrete Contin. Dyn. Syst. SI, (2013), 1-10.  doi: 10.3934/proc.2013.2013.1.

[3]

T.B. BenjaminJ.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[4]

T. CaraballoA.N. CarvalhoJ.A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal, 72 (2010), 1967-1976.  doi: 10.1016/j.na.2009.09.037.

[5]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems Appl. Math. Sciences, Springer, 182,2013. doi: 10.1007/978-1-4614-4581-4.

[7]

A.O. CelebiV.K. Kalantarov and M. Polat, Attractors for the generalized Benjamin-Bona-Mahony equation, J. Differ. Equ., 157 (1999), 439-451.  doi: 10.1006/jdeq.1999.3634.

[8]

H.Y. CuiJ.A. Langa and Y.R. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal, 140 (2016), 208-235.  doi: 10.1016/j.na.2016.03.012.

[9]

K. Deimling, Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[10]

J.R. Kang, Attractors for autonomous and nonautonomous 3D Benjamin-Bona-Mahony equations, Appl. Math. Comput., 274 (2016), 343-352.  doi: 10.1016/j.amc.2015.10.086.

[11]

P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond, 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems 176, American Mathematical Society, Providence, 2011. doi: 10.1090/surv/176.

[13]

A. Krause and B.X. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038.  doi: 10.1016/j.jmaa.2014.03.037.

[14]

Y.R. LiA.H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.

[15]

Y.R. Li and B.L. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differ. Equ., 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.

[16]

Y.R. Li and J.Y. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Disrete Contin. Dyn. Syst. Series B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.

[17]

G. Lukaszewicz, On pullback attractors in $H^1_0$ for nonautonomous reaction-diffusion equations, International J. Bifurcation and Chaos, 20 (2010), 2637-2644.  doi: 10.1142/S0218127410027258.

[18]

Q.F. MaS.H. Wang and C.K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J, 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[19]

L.A. Medeiros and G. Perla Menzala, Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equation, SIAM J. Math. Anal, 8 (1977), 792-799.  doi: 10.1137/0508062.

[20]

L. Rosier and B.Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Differ. Equ., 254 (2013), 141-178.  doi: 10.1016/j.jde.2012.08.014.

[21]

M. Stanislavova, On the global attractor for the damped Benjamin-Bona-Mahony equation, Disrete Continu. Dyn. Syst., 35 (2005), 824-832. 

[22]

M. StanislavovaA. Stefanow and B. Wang, Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on $R^{3}$, J. Differ. Equ., 219 (2005), 451-483.  doi: 10.1016/j.jde.2005.08.004.

[23]

A.S. de Suzzoni, Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures, Disrete Continu. Dyn. Syst., 35 (2015), 2905-2920.  doi: 10.3934/dcds.2015.35.2905.

[24]

B. Wang, Random attractors for atochastic Benjamin-Bona-Mahony equation on unbounded Domains, J. Differ. Equ., 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[25]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Continu. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[26]

B. Wang, Regularity of attractors for the Benjamin-Bona-Mahony equation, J. Phys. A, 31 (1998), 7635-7645.  doi: 10.1088/0305-4470/31/37/021.

[27]

B. Wang, Strong attractors for the Benjamin-Bona-Mahony equation, Appl. Math. Lett, 10 (1997), 23-28.  doi: 10.1016/S0893-9659(97)00005-0.

[28]

B. WangD. W. Fussner and C. Bi, Existence of global attractors for the Benjamin-Bona-Mahony equation in unbounded domains, J. Phys. A, 40 (2007), 10491-10504.  doi: 10.1088/1751-8113/40/34/007.

[29]

B. Wang and W. Yang, Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A, 30 (1997), 4877-4885.  doi: 10.1088/0305-4470/30/13/035.

[30]

Y. WangC. Zhong and S. Zhou, Pullback attractors of nonautonomous dynamical systems, Disrete Continu. Dyn. Syst., 16 (2006), 587-614.  doi: 10.3934/dcds.2006.16.587.

show all references

References:
[1]

A. Adili and B. Wang, Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing, Discrete Contin. Dyn. Syst. Series B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.

[2]

A. Adili and B. Wang, Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise, Discrete Contin. Dyn. Syst. SI, (2013), 1-10.  doi: 10.3934/proc.2013.2013.1.

[3]

T.B. BenjaminJ.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.

[4]

T. CaraballoA.N. CarvalhoJ.A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes, Nonlinear Anal, 72 (2010), 1967-1976.  doi: 10.1016/j.na.2009.09.037.

[5]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal, 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems Appl. Math. Sciences, Springer, 182,2013. doi: 10.1007/978-1-4614-4581-4.

[7]

A.O. CelebiV.K. Kalantarov and M. Polat, Attractors for the generalized Benjamin-Bona-Mahony equation, J. Differ. Equ., 157 (1999), 439-451.  doi: 10.1006/jdeq.1999.3634.

[8]

H.Y. CuiJ.A. Langa and Y.R. Li, Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness, Nonlinear Anal, 140 (2016), 208-235.  doi: 10.1016/j.na.2016.03.012.

[9]

K. Deimling, Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.

[10]

J.R. Kang, Attractors for autonomous and nonautonomous 3D Benjamin-Bona-Mahony equations, Appl. Math. Comput., 274 (2016), 343-352.  doi: 10.1016/j.amc.2015.10.086.

[11]

P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond, 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems 176, American Mathematical Society, Providence, 2011. doi: 10.1090/surv/176.

[13]

A. Krause and B.X. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains, J. Math. Anal. Appl., 417 (2014), 1018-1038.  doi: 10.1016/j.jmaa.2014.03.037.

[14]

Y.R. LiA.H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differ. Equ., 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.

[15]

Y.R. Li and B.L. Guo, Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations, J. Differ. Equ., 245 (2008), 1775-1800.  doi: 10.1016/j.jde.2008.06.031.

[16]

Y.R. Li and J.Y. Yin, A modified proof of pullback attractors in a Sobolev space for stochastic FitzHugh-Nagumo equations, Disrete Contin. Dyn. Syst. Series B, 21 (2016), 1203-1223.  doi: 10.3934/dcdsb.2016.21.1203.

[17]

G. Lukaszewicz, On pullback attractors in $H^1_0$ for nonautonomous reaction-diffusion equations, International J. Bifurcation and Chaos, 20 (2010), 2637-2644.  doi: 10.1142/S0218127410027258.

[18]

Q.F. MaS.H. Wang and C.K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J, 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[19]

L.A. Medeiros and G. Perla Menzala, Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equation, SIAM J. Math. Anal, 8 (1977), 792-799.  doi: 10.1137/0508062.

[20]

L. Rosier and B.Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Differ. Equ., 254 (2013), 141-178.  doi: 10.1016/j.jde.2012.08.014.

[21]

M. Stanislavova, On the global attractor for the damped Benjamin-Bona-Mahony equation, Disrete Continu. Dyn. Syst., 35 (2005), 824-832. 

[22]

M. StanislavovaA. Stefanow and B. Wang, Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on $R^{3}$, J. Differ. Equ., 219 (2005), 451-483.  doi: 10.1016/j.jde.2005.08.004.

[23]

A.S. de Suzzoni, Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures, Disrete Continu. Dyn. Syst., 35 (2015), 2905-2920.  doi: 10.3934/dcds.2015.35.2905.

[24]

B. Wang, Random attractors for atochastic Benjamin-Bona-Mahony equation on unbounded Domains, J. Differ. Equ., 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[25]

B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Disrete Continu. Dyn. Syst., 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[26]

B. Wang, Regularity of attractors for the Benjamin-Bona-Mahony equation, J. Phys. A, 31 (1998), 7635-7645.  doi: 10.1088/0305-4470/31/37/021.

[27]

B. Wang, Strong attractors for the Benjamin-Bona-Mahony equation, Appl. Math. Lett, 10 (1997), 23-28.  doi: 10.1016/S0893-9659(97)00005-0.

[28]

B. WangD. W. Fussner and C. Bi, Existence of global attractors for the Benjamin-Bona-Mahony equation in unbounded domains, J. Phys. A, 40 (2007), 10491-10504.  doi: 10.1088/1751-8113/40/34/007.

[29]

B. Wang and W. Yang, Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A, 30 (1997), 4877-4885.  doi: 10.1088/0305-4470/30/13/035.

[30]

Y. WangC. Zhong and S. Zhou, Pullback attractors of nonautonomous dynamical systems, Disrete Continu. Dyn. Syst., 16 (2006), 587-614.  doi: 10.3934/dcds.2006.16.587.

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