• Previous Article
    Stage-structured models for interactive wild and periodically and impulsively released sterile mosquitoes
  • DCDS-B Home
  • This Issue
  • Next Article
    Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems
doi: 10.3934/dcdsb.2021293
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations & Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025

[2]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[3]

Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021107

[4]

Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559

[5]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210

[6]

Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326

[7]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[8]

Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1299-1316. doi: 10.3934/dcdsb.2019221

[9]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

[10]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

[11]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Random exponential attractor for stochastic non-autonomous suspension bridge equation with additive white noise. Discrete & Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2021318

[12]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23

[13]

Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275

[14]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345

[15]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715

[16]

Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095

[17]

Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441

[18]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343

[19]

Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663

[20]

István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134

2020 Impact Factor: 1.327

Article outline

[Back to Top]