doi: 10.3934/dcdsb.2021293
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Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels

School of Mathematics and Statistics, Heze University, Heze 274015, China

Received  June 2021 Revised  October 2021 Early access December 2021

This article is devoted to the asymptotic behaviour of solutions for stochastic Benjamin-Bona-Mahony (BBM) equations with distributed delay defined on unbounded channels. We first prove the existence, uniqueness and forward compactness of pullback random attractors (PRAs). We then establish the forward asymptotic autonomy of this PRA. Finally, we study the non-delay stability of this PRA. Due to the loss of usual compact Sobolev embeddings on unbounded domains, the forward uniform tail-estimates and forward flattening of solutions are used to prove the forward asymptotic compactness of solutions.

Citation: Qiangheng Zhang. Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021293
References:
[1]

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

[2]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-433.  doi: 10.3934/dcds.2008.21.415.

[3]

T. Caraballo, B. Guo, N. H. Tuan and R. Wang, Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 2020. doi: 10.1017/prm.2020.77.

[4]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.

[5]

T. CaraballoJ. Real and A. M. Márquez-Durán, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883.  doi: 10.1142/S0218127410027428.

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, vol. 182, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.

[7]

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

[8]

H. Cui, Convergences of asymptotically autonomous pullback attractors towards semigroup attractors, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3525-3535.  doi: 10.3934/dcdsb.2018276.

[9]

H. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, Asymptot. Anal., 112 (2019), 165-184.  doi: 10.3233/ASY-181501.

[10]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374/375 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.

[11]

H. CuiY. Li and J. Yin, Long time behavior of stochastic MHD equations perturbed by multiplicative noises, J. Appl. Anal. Comput., 6 (2016), 1081-1104.  doi: 10.11948/2016071.

[12]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357.  doi: 10.1515/ans-2013-0205.

[13]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.

[14]

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.

[15]

P. E. Kloeden, Upper semi continuity of attractors of retarded delay differential equations in the delay, Bull. Austral. Math. Soc., 73 (2006), 299-306.  doi: 10.1017/S0004972700038880.

[16]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.

[17]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.

[18]

A. KrauseM. Lewis and B. Wang, Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376.  doi: 10.1016/j.amc.2014.08.033.

[19]

D. Li and L. Shi, Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay, J. Math. Phys., 59 (2018), 032703, 35 pp. doi: 10.1063/1.5031770.

[20]

D. LiB. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.

[21]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.

[22]

Y. Li and R. Wang, Random attractors for 3D Benjamin-Bona-Mahony equations derived by a Laplace-multiplier noise, Stoch. Dyn., 18 (2018), 1850004, 26 pp. doi: 10.1142/S0219493718500041.

[23]

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

[24]

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.

[25]

J. Y. Park and S. H. Park, Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains, Sci. China Math., 54 (2011), 741-752.  doi: 10.1007/s11425-011-4190-0.

[26]

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

[27]

B. Wang, Random attractors for a stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[28]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[29]

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

[30]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

[31]

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.

[32]

M. Wang, Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity spaces, Nonlinear Anal., 105 (2014), 134-144.  doi: 10.1016/j.na.2014.04.013.

[33]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382/383 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.

[34]

X. WangK. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.

[35]

S. Yang and Y. Li, Asymptotic autonomous attractors for a stochastic lattice model with random viscosity, J. Difference Equ. Appl., 26 (2020), 540-560.  doi: 10.1080/10236198.2020.1755277.

[36]

F. Yin and X. Li, Fractal dimensions of random attractors for stochastic Benjamin-Bona-Mahony equation on unbounded domains, Comput. Math. Appl., 75 (2018), 1595-1615.  doi: 10.1016/j.camwa.2017.11.025.

[37]

Q. Zhang and Y. Li, Backward controller of a pullback attractor for delay Benjamin-Bona-Mahony equations, J. Dyn. Control Syst., 26 (2020), 423-441.  doi: 10.1007/s10883-019-09450-9.

[38]

Q. Zhang and Y. Li, Double stabilities of pullback random attractors for stochastic delayed p-Laplacian equations, Math. Meth. Appl. Sci., 43 (2020), 8406-8433.  doi: 10.1002/mma.6495.

[39]

M. ZhaoX.-G. YangX. Yan and X. Cui, Dynamics of a 3D Benjamin-Bona-Mahony equations with sublinear operator, Asymptot. Anal., 121 (2021), 75-100.  doi: 10.3233/ASY-201601.

show all references

References:
[1]

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

[2]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-433.  doi: 10.3934/dcds.2008.21.415.

[3]

T. Caraballo, B. Guo, N. H. Tuan and R. Wang, Asymptotically autonomous robustness of random attractors for a class of weakly dissipative stochastic wave equations on unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 2020. doi: 10.1017/prm.2020.77.

[4]

T. Caraballo and J. Real, Attractors for 2D-Navier-Stokes models with delays, J. Differential Equations, 205 (2004), 271-297.  doi: 10.1016/j.jde.2004.04.012.

[5]

T. CaraballoJ. Real and A. M. Márquez-Durán, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883.  doi: 10.1142/S0218127410027428.

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, vol. 182, Springer, 2013. doi: 10.1007/978-1-4614-4581-4.

[7]

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

[8]

H. Cui, Convergences of asymptotically autonomous pullback attractors towards semigroup attractors, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 3525-3535.  doi: 10.3934/dcdsb.2018276.

[9]

H. Cui and P. E. Kloeden, Tail convergences of pullback attractors for asymptotically converging multi-valued dynamical systems, Asymptot. Anal., 112 (2019), 165-184.  doi: 10.3233/ASY-181501.

[10]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374/375 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.

[11]

H. CuiY. Li and J. Yin, Long time behavior of stochastic MHD equations perturbed by multiplicative noises, J. Appl. Anal. Comput., 6 (2016), 1081-1104.  doi: 10.11948/2016071.

[12]

J. García-LuengoP. Marín-Rubio and J. Real, Pullback attractors for 2D Navier-Stokes equations with delays and their regularity, Adv. Nonlinear Stud., 13 (2013), 331-357.  doi: 10.1515/ans-2013-0205.

[13]

A. GuD. LiB. Wang and H. Yang, Regularity of random attractors for fractional stochastic reaction-diffusion equations on $\mathbb{R}^{n}$, J. Differential Equations, 264 (2018), 7094-7137.  doi: 10.1016/j.jde.2018.02.011.

[14]

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.

[15]

P. E. Kloeden, Upper semi continuity of attractors of retarded delay differential equations in the delay, Bull. Austral. Math. Soc., 73 (2006), 299-306.  doi: 10.1017/S0004972700038880.

[16]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.

[17]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.

[18]

A. KrauseM. Lewis and B. Wang, Dynamics of the non-autonomous stochastic $p$-Laplace equation driven by multiplicative noise, Appl. Math. Comput., 246 (2014), 365-376.  doi: 10.1016/j.amc.2014.08.033.

[19]

D. Li and L. Shi, Upper semicontinuity of attractors of stochastic delay reaction-diffusion equations in the delay, J. Math. Phys., 59 (2018), 032703, 35 pp. doi: 10.1063/1.5031770.

[20]

D. LiB. Wang and X. Wang, Limiting behavior of non-autonomous stochastic reaction-diffusion equations on thin domains, J. Differential Equations, 262 (2017), 1575-1602.  doi: 10.1016/j.jde.2016.10.024.

[21]

Y. LiL. She and R. Wang, Asymptotically autonomous dynamics for parabolic equation, J. Math. Anal. Appl., 459 (2018), 1106-1123.  doi: 10.1016/j.jmaa.2017.11.033.

[22]

Y. Li and R. Wang, Random attractors for 3D Benjamin-Bona-Mahony equations derived by a Laplace-multiplier noise, Stoch. Dyn., 18 (2018), 1850004, 26 pp. doi: 10.1142/S0219493718500041.

[23]

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

[24]

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.

[25]

J. Y. Park and S. H. Park, Pullback attractors for the non-autonomous Benjamin-Bona-Mahony equation in unbounded domains, Sci. China Math., 54 (2011), 741-752.  doi: 10.1007/s11425-011-4190-0.

[26]

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

[27]

B. Wang, Random attractors for a stochastic Benjamin-Bona-Mahony equation on unbounded domains, J. Differential Equations, 246 (2009), 2506-2537.  doi: 10.1016/j.jde.2008.10.012.

[28]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[29]

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

[30]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009, 31 pp. doi: 10.1142/S0219493714500099.

[31]

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.

[32]

M. Wang, Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity spaces, Nonlinear Anal., 105 (2014), 134-144.  doi: 10.1016/j.na.2014.04.013.

[33]

S. Wang and Y. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Phys. D, 382/383 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.

[34]

X. WangK. Lu and B. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047.  doi: 10.1137/140991819.

[35]

S. Yang and Y. Li, Asymptotic autonomous attractors for a stochastic lattice model with random viscosity, J. Difference Equ. Appl., 26 (2020), 540-560.  doi: 10.1080/10236198.2020.1755277.

[36]

F. Yin and X. Li, Fractal dimensions of random attractors for stochastic Benjamin-Bona-Mahony equation on unbounded domains, Comput. Math. Appl., 75 (2018), 1595-1615.  doi: 10.1016/j.camwa.2017.11.025.

[37]

Q. Zhang and Y. Li, Backward controller of a pullback attractor for delay Benjamin-Bona-Mahony equations, J. Dyn. Control Syst., 26 (2020), 423-441.  doi: 10.1007/s10883-019-09450-9.

[38]

Q. Zhang and Y. Li, Double stabilities of pullback random attractors for stochastic delayed p-Laplacian equations, Math. Meth. Appl. Sci., 43 (2020), 8406-8433.  doi: 10.1002/mma.6495.

[39]

M. ZhaoX.-G. YangX. Yan and X. Cui, Dynamics of a 3D Benjamin-Bona-Mahony equations with sublinear operator, Asymptot. Anal., 121 (2021), 75-100.  doi: 10.3233/ASY-201601.

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