American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system
  • MCRF Home
  • This Issue
  • Next Article
    Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces
doi: 10.3934/mcrf.2020044

Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains

Ji Shu 1, , Linyan Li 2, , Xin Huang 3, and  Jian Zhang 4,,

1. 

School of Mathematical Sciences and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu, Sichuan 610068, China
2. 

School of Mathematical Sciences, Sichuan Normal University, Chengdu, Sichuan 610068, China
3. 

Department of Basic Courses, Sichuan Vocational College of Finance and Economics, Chengdu, Sichuan 610101, China
4. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

* Corresponding author: Jian Zhang

Received  May 2020 Revised  July 2020 Published  October 2020

We consider the dynamical behavior of fractional stochastic integro-differential equations with additive noise on unbounded domains. The existence and uniqueness of tempered random attractors for the equation in $ \mathbb{R}^{3} $ are proved. The upper semicontinuity of random attractors is also obtained when the intensity of noise approaches zero. The main difficulty is to show the pullback asymptotic compactness due to the lack of compactness on unbounded domains and the fact that the memory term includes the whole past history of the phenomenon. We establish such compactness by the tail-estimate method and the splitting method.

Keywords: Random attractor, stochastic integro-differential equation with memory, unbounded domains, asymptotic compactness, upper semicontinuity.
Mathematics Subject Classification: Primary: 37L55, 60H15; Secondary: 35Q56.
Citation: Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020044
References:
[1]

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

[2]

Q. Bai, J. Shu, L. Li and H. Li, Dynamical behavior of non-autonomous fractional stochastic reaction-diffusion equations, J. Math. Anal. Appl., 485 (2020), 123833. doi: 10.1016/j.jmaa.2019.123833.  Google Scholar

[3]

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

[4]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[5]

T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

[6]

T. Caraballo, I. D. Chueshov, P. Marin-Rubio and J. Real, Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst., 18 (2007), 253-270.  doi: 10.3934/dcds.2007.18.253.  Google Scholar

[7]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set–Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[8]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[9]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.  doi: 10.1512/iumj.2006.55.2661.  Google Scholar

[10]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.  Google Scholar

[11]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[12]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[13]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[14]

E. DiNezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[15]

J. Dong and M. Xu, Space-time fractional Schr$\ddot{o}$dinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.  doi: 10.1016/j.jmaa.2008.03.061.  Google Scholar

[16]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[17]

C. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.  Google Scholar

[18]

C. Giorgi, V. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049.  Google Scholar

[19]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Progress in Nonlinear Differential Equations and Their Applications, Vol. 50, Birkh$\ddot{a}$user, Basel, 2002,155–178.  Google Scholar

[20]

A. Gu, D. Li, B. 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

[21]

B. Guo, Y. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schr$\ddot{o}$dinger equation, Appl. Math. Comput., 204 (2008), 468-477.  doi: 10.1016/j.amc.2008.07.003.  Google Scholar

[22]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schr$\ddot{o}$dinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.  Google Scholar

[23]

B. Guo and M. Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138.  doi: 10.1016/j.jmaa.2009.09.009.  Google Scholar

[24]

B. Guo and G. Zhou, Ergodicity of the stochastic fractional reaction diffusion equation, Nonlinear Anal., 109 (2014), 1-22.  doi: 10.1016/j.na.2014.06.008.  Google Scholar

[25]

C. Guo, J. Shu and X. Wang, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations, Acta Math. Sin.(Engl. Ser.), 36 (2020), 318-336.  doi: 10.1007/s10114-020-8407-4.  Google Scholar

[26]

Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300.  doi: 10.1080/14689367.2018.1523368.  Google Scholar

[27]

Y. Lan and J. Shu, Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 2409-2431.  doi: 10.3934/cpaa.2019109.  Google Scholar

[28]

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

[29]

L. Li, J. Shu, Q. Bai and H. Li, Asymptotic behavior of fractional stochastic heat equations in materials with memory, Appl. Anal. (2019). doi: 10.1080/00036811.2019.1597057.  Google Scholar

[30]

J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[31]

L. Liu and T. Caraballo, Well-posedness and dynamics of a fractional stochastic integro-differential equation, Phys. D, 355 (2017), 45-57.  doi: 10.1016/j.physd.2017.05.006.  Google Scholar

[32]

H. Lu, P. W. Bates, J. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on $\mathbb{R}^{n}$, Nonlinear Anal., 128 (2015), 176-198.  doi: 10.1016/j.na.2015.06.033.  Google Scholar

[33]

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

[34]

G. Lv and J. Duan, Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals, J. Math. Anal. Appl., 449 (2017), 176-194.  doi: 10.1016/j.jmaa.2016.12.011.  Google Scholar

[35]

G. Lv, H. Gao, J. Wei and J. Wu, BMO and Morrey Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.  doi: 10.1016/j.jde.2018.08.042.  Google Scholar

[36]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^{n}$ with continuous nonlinearity, Asymptot. Anal., 44 (2005), 111-130.   Google Scholar

[37]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.   Google Scholar

[38]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[39]

X. Pu and B. Guo, Global weak solutions of the fractional Landau-Lifshitz-Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98.  doi: 10.1016/j.jmaa.2010.06.035.  Google Scholar

[40]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Laudau equation, Appl. Anal., 92 (2013), 318-334.  doi: 10.1080/00036811.2011.614601.  Google Scholar

[41]

B. Schmalfuss, Backward cocycle and atttractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universit$\ddot{a}$t, (1992), 185–192. Google Scholar

[42]

T. Shen and J. Huang, Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal., 110 (2014), 33-46.  doi: 10.1016/j.na.2014.06.018.  Google Scholar

[43]

J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1587-1599.  doi: 10.3934/dcdsb.2017077.  Google Scholar

[44]

J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702. doi: 10.1063/1.4934724.  Google Scholar

[45]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[46]

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.  Google Scholar

[47]

B. Wang, Upper semicontinuity of random attractors for non-compact random systems, Electron J. Differential Equations, 139 (2009), 1-18.   Google Scholar

[48]

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

[49]

B. Wang, Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009. 1–31. doi: 10.1142/S0219493714500099.  Google Scholar

[50]

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

[51]

X. Wang, S. 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.  Google Scholar

[52]

X. Wang, K. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[53]

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $\mathbb{R}^{3}$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.  Google Scholar

show all references

References:
[1]

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

[2]

Q. Bai, J. Shu, L. Li and H. Li, Dynamical behavior of non-autonomous fractional stochastic reaction-diffusion equations, J. Math. Anal. Appl., 485 (2020), 123833. doi: 10.1016/j.jmaa.2019.123833.  Google Scholar

[3]

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

[4]

P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[5]

T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 525-539.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

[6]

T. Caraballo, I. D. Chueshov, P. Marin-Rubio and J. Real, Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory, Discrete Contin. Dyn. Syst., 18 (2007), 253-270.  doi: 10.3934/dcds.2007.18.253.  Google Scholar

[7]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set–Valued Anal., 11 (2003), 153-201.  doi: 10.1023/A:1022902802385.  Google Scholar

[8]

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.  Google Scholar

[9]

M. Conti, V. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.  doi: 10.1512/iumj.2006.55.2661.  Google Scholar

[10]

H. Crauel and P. E. Kloeden, Nonautonomous and random attractors, Jahresber. Dtsch. Math.-Ver., 117 (2015), 173-206.  doi: 10.1365/s13291-015-0115-0.  Google Scholar

[11]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar

[12]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[13]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.  Google Scholar

[14]

E. DiNezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[15]

J. Dong and M. Xu, Space-time fractional Schr$\ddot{o}$dinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.  doi: 10.1016/j.jmaa.2008.03.061.  Google Scholar

[16]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastics Stochastics Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[17]

C. Gal and M. Warma, Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions, Discrete Contin. Dyn. Syst., 36 (2016), 1279-1319.  doi: 10.3934/dcds.2016.36.1279.  Google Scholar

[18]

C. Giorgi, V. Pata and A. Marzocchi, Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA Nonlinear Differential Equations Appl., 5 (1998), 333-354.  doi: 10.1007/s000300050049.  Google Scholar

[19]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, in Progress in Nonlinear Differential Equations and Their Applications, Vol. 50, Birkh$\ddot{a}$user, Basel, 2002,155–178.  Google Scholar

[20]

A. Gu, D. Li, B. 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

[21]

B. Guo, Y. Han and J. Xin, Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schr$\ddot{o}$dinger equation, Appl. Math. Comput., 204 (2008), 468-477.  doi: 10.1016/j.amc.2008.07.003.  Google Scholar

[22]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schr$\ddot{o}$dinger equation, Comm. Partial Differential Equations, 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.  Google Scholar

[23]

B. Guo and M. Zeng, Solutions for the fractional Landau-Lifshitz equation, J. Math. Anal. Appl., 361 (2010), 131-138.  doi: 10.1016/j.jmaa.2009.09.009.  Google Scholar

[24]

B. Guo and G. Zhou, Ergodicity of the stochastic fractional reaction diffusion equation, Nonlinear Anal., 109 (2014), 1-22.  doi: 10.1016/j.na.2014.06.008.  Google Scholar

[25]

C. Guo, J. Shu and X. Wang, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations, Acta Math. Sin.(Engl. Ser.), 36 (2020), 318-336.  doi: 10.1007/s10114-020-8407-4.  Google Scholar

[26]

Y. Lan and J. Shu, Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Dyn. Syst., 34 (2019), 274-300.  doi: 10.1080/14689367.2018.1523368.  Google Scholar

[27]

Y. Lan and J. Shu, Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 2409-2431.  doi: 10.3934/cpaa.2019109.  Google Scholar

[28]

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

[29]

L. Li, J. Shu, Q. Bai and H. Li, Asymptotic behavior of fractional stochastic heat equations in materials with memory, Appl. Anal. (2019). doi: 10.1080/00036811.2019.1597057.  Google Scholar

[30]

J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[31]

L. Liu and T. Caraballo, Well-posedness and dynamics of a fractional stochastic integro-differential equation, Phys. D, 355 (2017), 45-57.  doi: 10.1016/j.physd.2017.05.006.  Google Scholar

[32]

H. Lu, P. W. Bates, J. Xin and M. Zhang, Asymptotic behavior of stochastic fractional power dissipative equations on $\mathbb{R}^{n}$, Nonlinear Anal., 128 (2015), 176-198.  doi: 10.1016/j.na.2015.06.033.  Google Scholar

[33]

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

[34]

G. Lv and J. Duan, Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals, J. Math. Anal. Appl., 449 (2017), 176-194.  doi: 10.1016/j.jmaa.2016.12.011.  Google Scholar

[35]

G. Lv, H. Gao, J. Wei and J. Wu, BMO and Morrey Campanato estimates for stochastic convolutions and Schauder estimates for stochastic parabolic equations, J. Differential Equations, 266 (2019), 2666-2717.  doi: 10.1016/j.jde.2018.08.042.  Google Scholar

[36]

F. Morillas and J. Valero, Attractors for reaction-diffusion equations in $\mathbb{R}^{n}$ with continuous nonlinearity, Asymptot. Anal., 44 (2005), 111-130.   Google Scholar

[37]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 115-162.   Google Scholar

[38]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[39]

X. Pu and B. Guo, Global weak solutions of the fractional Landau-Lifshitz-Maxwell equation, J. Math. Anal. Appl., 372 (2010), 86-98.  doi: 10.1016/j.jmaa.2010.06.035.  Google Scholar

[40]

X. Pu and B. Guo, Well-posedness and dynamics for the fractional Ginzburg-Laudau equation, Appl. Anal., 92 (2013), 318-334.  doi: 10.1080/00036811.2011.614601.  Google Scholar

[41]

B. Schmalfuss, Backward cocycle and atttractors of stochastic differential equations, in International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, Technische Universit$\ddot{a}$t, (1992), 185–192. Google Scholar

[42]

T. Shen and J. Huang, Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials, Nonlinear Anal., 110 (2014), 33-46.  doi: 10.1016/j.na.2014.06.018.  Google Scholar

[43]

J. Shu, Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1587-1599.  doi: 10.3934/dcdsb.2017077.  Google Scholar

[44]

J. Shu, P. Li, J. Zhang and O. Liao, Random attractors for the stochastic coupled fractional Ginzburg-Landau equation with additive noise, J. Math. Phys., 56 (2015), 102702. doi: 10.1063/1.4934724.  Google Scholar

[45]

R. Temam, Infinite Dimension Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[46]

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.  Google Scholar

[47]

B. Wang, Upper semicontinuity of random attractors for non-compact random systems, Electron J. Differential Equations, 139 (2009), 1-18.   Google Scholar

[48]

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

[49]

B. Wang, Existence and upper-semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, Stoch. Dyn., 14 (2014), 1450009. 1–31. doi: 10.1142/S0219493714500099.  Google Scholar

[50]

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

[51]

X. Wang, S. 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.  Google Scholar

[52]

X. Wang, K. Lu and B. Wang, Wong-Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 264 (2018), 378-424.  doi: 10.1016/j.jde.2017.09.006.  Google Scholar

[53]

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equation with multiplicative noise in $\mathbb{R}^{3}$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.  Google Scholar

[1]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066
[2]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023
[3]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[4]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055
[5]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[6]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192
[7]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[8]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[9]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008
[10]

Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign. Networks & Heterogeneous Media, 2012, 7 (1) : 151-178. doi: 10.3934/nhm.2012.7.151
[11]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[12]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[13]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453
[14]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009
[15]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[16]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089
[17]

Seung-Yeal Ha, Shi Jin. Local sensitivity analysis for the Cucker-Smale model with random inputs. Kinetic & Related Models, 2018, 11 (4) : 859-889. doi: 10.3934/krm.2018034
[18]

Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051
[19]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205
[20]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021031

2019 Impact Factor: 0.857

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences