doi: 10.3934/dcdsb.2021271
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.

Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610065, China

*Corresponding author: Xiaohu Wang

Received  May 2021 Revised  September 2021 Early access November 2021

Fund Project: This work was supported by NSFC (11871049 and 12090013) and Young crop project of Sichuan University (2020SCUNL111)

In this paper, we study the asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise. The considered systems are driven by the fractional discrete Laplacian, which features the infinite-range interactions. We first prove the existence of pullback random attractor in $ \ell^2 $ for stochastic lattice systems. The upper semicontinuity of random attractors is also established when the intensity of noise approaches zero.

Citation: Yiju Chen, Xiaohu Wang. Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021271
References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. C. Bressloff, Waves in Neural Media: From Single Neurons to Neural Fields, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, New York, 2014. doi: 10.1007/978-1-4614-8866-8.

[3]

P. W. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[4]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[5]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[6]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[8]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits and Systems, 40 (1993), 147-156.  doi: 10.1109/81.222795.

[9]

L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Trans. Circuits and Systems, 35 (1998), 1257-1272.  doi: 10.1109/31.7600.

[10]

Ó. CiaurriT. A. GillespieL. RoncalJ. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6.

[11]

Ó. CiaurriC. LizamaL. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Lett., 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007.

[12]

Ó. Ciaurri and L. Roncal, Hardy's inequality for the fractional powers of a discrete Laplacian, J. Anal., 26 (2018), 211-225.  doi: 10.1007/s41478-018-0141-2.

[13]

Ó. CiaurriL. RoncalP. R. StingaJ. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math., 330 (2018), 688-738.  doi: 10.1016/j.aim.2018.03.023.

[14]

S. N. Chow, Lattice dynamical systems, Dynamical Systems, Lecture Notes in Math., 1822 (2003), 1-102.  doi: 10.1007/978-3-540-45204-1_1.

[15]

S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751.  doi: 10.1109/81.473583.

[16]

S. N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.

[17]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[18]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[19]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[20]

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.

[21]

C. GuoY. ChenJ. Shu and X. Yang, Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains, Front. Math. China, 16 (2021), 59-93.  doi: 10.1007/s11464-021-0896-7.

[22]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[23]

X. HanP. E. Kloeden and B. Usman, Long term behavior of a random Hopfield neural lattice model, Commun. Pure Appl. Anal., 18 (2019), 809-824.  doi: 10.3934/cpaa.2019039.

[24]

X. HanP. E. Kloeden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813.

[25]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[26]

Y. Hong and C. Yang, Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit, SIAM J. Math. Anal., 51 (2019), 1297-1320.  doi: 10.1137/18M120703X.

[27]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys., 317 (2013), 563-591.  doi: 10.1007/s00220-012-1621-x.

[28]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[29]

C. Lizama and L. Roncal, Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 1365-1403.  doi: 10.3934/dcds.2018056.

[30]

D. LiB. Wang and X. Wang, Random dynamics of fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$ without uniqueness, J. Math. Phys., 60 (2019), 072704.  doi: 10.1063/1.5063840.

[31]

D. LiX. Wang and J. Zhao, Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process, Commun. Pure Appl. Anal., 19 (2020), 2751-2776.  doi: 10.3934/cpaa.2020120.

[32]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of the 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.

[33]

C. Martínez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Math. Studies 187, Amsterdam, 2001.

[34]

M. SuiY. WangX. Han and P. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008.

[35]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[36]

W. M. Schouten and H. J. Hupkes, Nonlinear stability of pulse solutions for the discrete Fitzhugh-Nagumo equation with infinite-range interactions, Discrete Contin. Dyn. Syst., 39 (2019), 5017-5083.  doi: 10.3934/dcds.2019205.

[37]

B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006.

[38]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[39]

X. WangP. Kloeden and X. Han, Attractors of Hopfield-type lattice models with increasing neuronal input, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 799-813.  doi: 10.3934/dcdsb.2019268.

[40]

X. WangJ. ShenK. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026.

[41]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[42]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.

[43]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55.  doi: 10.1016/j.jmaa.2012.04.080.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. C. Bressloff, Waves in Neural Media: From Single Neurons to Neural Fields, Lecture Notes on Mathematical Modelling in the Life Sciences, Springer, New York, 2014. doi: 10.1007/978-1-4614-8866-8.

[3]

P. W. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[4]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.

[5]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[6]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[8]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits and Systems, 40 (1993), 147-156.  doi: 10.1109/81.222795.

[9]

L. O. Chua and L. Yang, Cellular neural networks: theory, IEEE Trans. Circuits and Systems, 35 (1998), 1257-1272.  doi: 10.1109/31.7600.

[10]

Ó. CiaurriT. A. GillespieL. RoncalJ. L. Torrea and J. L. Varona, Harmonic analysis associated with a discrete Laplacian, J. Anal. Math., 132 (2017), 109-131.  doi: 10.1007/s11854-017-0015-6.

[11]

Ó. CiaurriC. LizamaL. Roncal and J. L. Varona, On a connection between the discrete fractional Laplacian and superdiffusion, Appl. Math. Lett., 49 (2015), 119-125.  doi: 10.1016/j.aml.2015.05.007.

[12]

Ó. Ciaurri and L. Roncal, Hardy's inequality for the fractional powers of a discrete Laplacian, J. Anal., 26 (2018), 211-225.  doi: 10.1007/s41478-018-0141-2.

[13]

Ó. CiaurriL. RoncalP. R. StingaJ. L. Torrea and J. L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math., 330 (2018), 688-738.  doi: 10.1016/j.aim.2018.03.023.

[14]

S. N. Chow, Lattice dynamical systems, Dynamical Systems, Lecture Notes in Math., 1822 (2003), 1-102.  doi: 10.1007/978-3-540-45204-1_1.

[15]

S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746-751.  doi: 10.1109/81.473583.

[16]

S. N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.

[17]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.

[18]

T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.  doi: 10.1007/s11464-008-0028-7.

[19]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[20]

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.

[21]

C. GuoY. ChenJ. Shu and X. Yang, Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains, Front. Math. China, 16 (2021), 59-93.  doi: 10.1007/s11464-021-0896-7.

[22]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[23]

X. HanP. E. Kloeden and B. Usman, Long term behavior of a random Hopfield neural lattice model, Commun. Pure Appl. Anal., 18 (2019), 809-824.  doi: 10.3934/cpaa.2019039.

[24]

X. HanP. E. Kloeden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813.

[25]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[26]

Y. Hong and C. Yang, Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit, SIAM J. Math. Anal., 51 (2019), 1297-1320.  doi: 10.1137/18M120703X.

[27]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Commun. Math. Phys., 317 (2013), 563-591.  doi: 10.1007/s00220-012-1621-x.

[28]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[29]

C. Lizama and L. Roncal, Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 1365-1403.  doi: 10.3934/dcds.2018056.

[30]

D. LiB. Wang and X. Wang, Random dynamics of fractional stochastic reaction-diffusion equations on $\mathbb{R}^n$ without uniqueness, J. Math. Phys., 60 (2019), 072704.  doi: 10.1063/1.5063840.

[31]

D. LiX. Wang and J. Zhao, Limiting dynamical behavior of random fractional FitzHugh-Nagumo systems driven by a Wong-Zakai approximation process, Commun. Pure Appl. Anal., 19 (2020), 2751-2776.  doi: 10.3934/cpaa.2020120.

[32]

H. LuP. W. BatesS. Lu and M. Zhang, Dynamics of the 3-D fractional complex Ginzburg-Landau equation, J. Differential Equations, 259 (2015), 5276-5301.  doi: 10.1016/j.jde.2015.06.028.

[33]

C. Martínez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Math. Studies 187, Amsterdam, 2001.

[34]

M. SuiY. WangX. Han and P. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008.

[35]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.

[36]

W. M. Schouten and H. J. Hupkes, Nonlinear stability of pulse solutions for the discrete Fitzhugh-Nagumo equation with infinite-range interactions, Discrete Contin. Dyn. Syst., 39 (2019), 5017-5083.  doi: 10.3934/dcds.2019205.

[37]

B. Wang, Asymptotic behavior of non-autonomous fractional stochastic reaction-diffusion equations, Nonlinear Anal., 158 (2017), 60-82.  doi: 10.1016/j.na.2017.04.006.

[38]

R. WangY. Li and B. Wang, Random dynamics of fractional nonclassical diffusion equations driven by colored noise, Discrete Contin. Dyn. Syst., 39 (2019), 4091-4126.  doi: 10.3934/dcds.2019165.

[39]

X. WangP. Kloeden and X. Han, Attractors of Hopfield-type lattice models with increasing neuronal input, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 799-813.  doi: 10.3934/dcdsb.2019268.

[40]

X. WangJ. ShenK. Lu and B. Wang, Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems, J. Differential Equations, 280 (2021), 477-516.  doi: 10.1016/j.jde.2021.01.026.

[41]

X. WangK. Lu and B. Wang, Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise, J. Dynam. Differential Equations, 28 (2016), 1309-1335.  doi: 10.1007/s10884-015-9448-8.

[42]

X. WangS. Li and D. Xu, Random attractors for second-order stochastic lattice dynamical systems, Nonlinear Anal., 72 (2010), 483-494.  doi: 10.1016/j.na.2009.06.094.

[43]

S. Zhou and L. Wei, A random attractor for a stochastic second order lattice system with random coupled coefficients, J. Math. Anal. Appl., 395 (2012), 42-55.  doi: 10.1016/j.jmaa.2012.04.080.

[1]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

[2]

Linfang Liu, Xianlong Fu. Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-laplacian equation. Communications on Pure and Applied Analysis, 2017, 6 (2) : 443-474. doi: 10.3934/cpaa.2017023

[3]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035

[4]

Na Lei, Shengfan Zhou. Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 73-108. doi: 10.3934/dcds.2021108

[5]

Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116

[6]

Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1695-1724. doi: 10.3934/dcdsb.2021107

[7]

Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653

[8]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115

[9]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120

[10]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757

[11]

Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189

[12]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4899-4912. doi: 10.3934/dcdsb.2019036

[13]

Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252

[14]

María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations and Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001

[15]

Caibin Zeng, Xiaofang Lin, Jianhua Huang, Qigui Yang. Pathwise solution to rough stochastic lattice dynamical system driven by fractional noise. Communications on Pure and Applied Analysis, 2020, 19 (2) : 811-834. doi: 10.3934/cpaa.2020038

[16]

Wenlong Sun. The boundedness and upper semicontinuity of the pullback attractors for a 2D micropolar fluid flows with delay. Electronic Research Archive, 2020, 28 (3) : 1343-1356. doi: 10.3934/era.2020071

[17]

Shengfan Zhou, Caidi Zhao, Yejuan Wang. Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1259-1277. doi: 10.3934/dcds.2008.21.1259

[18]

Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055

[19]

Xiaoying Han, Peter E. Kloeden. Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021143

[20]

Hua Chen, Hong-Ge Chen. Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 301-317. doi: 10.3934/dcds.2021117

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (302)
  • HTML views (188)
  • Cited by (0)

Other articles
by authors

[Back to Top]