American Institute of Mathematical Sciences

Advanced
  • Previous Article
    The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations
  • CPAA Home
  • This Issue
  • Next Article
    Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions
May  2019, 18(3): 1155-1175. doi: 10.3934/cpaa.2019056

Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise

Yangrong Li , and  Shuang Yang

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

* Corresponding author

Received  March 2018 Revised  August 2018 Published  November 2018

Fund Project: This work is supported by National Natural Science Foundation of China grant 11571283.

A non-autonomous random attractor is called backward compact if its backward union is pre-compact. We show that such a backward compact random attractor exists if a non-autonomous random dynamical system is bounded dissipative and backward asymptotically compact. We also obtain both backward compact and periodic random attractor from a periodic and locally asymptotically compact system. The abstract results are applied to the sine-Gordon equation with multiplicative noise and a time-dependent force. If we assume that the density of noise is small and that the force is backward tempered and backward complement-small, then, we obtain a backward compact random attractor on the universe consisted of all backward tempered sets. Also, we obtain both backward compactness and periodicity of the attractor under the assumption of a periodic force.

Keywords: Stochastic sine-Gordon equation, random attractor, backward compactness, periodicity, backward tempered universe.
Mathematics Subject Classification: Primary: 35B41, 60H15; Secondary: 35R60.
Citation: Yangrong Li, Shuang Yang. Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1155-1175. doi: 10.3934/cpaa.2019056
References:
[1]

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

[2]

M. Anguiano and P. E. Kloeden, Asymptotic behaviour of the nonautonomous SIR equations with diffusion, Commun. Pure Appl. Anal., 13 (2014), 157-173.  doi: 10.3934/cpaa.2014.13.157.  Google Scholar

[3]

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

[4]

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

[5]

T. Caraballo, A. N. Carvalho, J. 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.  Google Scholar

[6]

T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure Appl. Anal., 16 (2017), 151-162.  doi: 10.3934/cpaa.2017007.  Google Scholar

[7]

T. Caraballo, M. J. Garrido-Atienza and J. Lopez-de-la-Cruz, Dynamics of some stochastic chemostat models with multiplicative noise, Commun. Pure Appl. Anal., 16 (2017), 1893-1914.  doi: 10.3934/cpaa.2017092.  Google Scholar

[8]

H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems J. Dyn. Differ. Equ., online (2017), DOI: 10.1007/s10884-017-9617-z. doi: 10.1007/s10884-017-9617-z.  Google Scholar

[9]

H. Cui, J. A. Langa and Y. 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.  Google Scholar

[10]

H. Cui and Y. Li, Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises, Appl. Math. Comput., 271 (2015), 777-789.  doi: 10.1016/j.amc.2015.09.031.  Google Scholar

[11]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860.  Google Scholar

[12]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.  doi: 10.3934/cpaa.2014.13.2543.  Google Scholar

[13]

A. Krause and B. 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.  Google Scholar

[14]

X. J. Li, X. L. Li and K. N. Lu, Random attractors for stochastic parabolic equations with additive noise in wighted spaces, Commun. Pure Appl. Anal., 17 (2018), 729-749.  doi: 10.3934/cpaa.2018038.  Google Scholar

[15]

J. Li, Y. Li and B. Wang, Random attractors of reaction-diffusion equations with multiplicative noise in Lp, Appl. Math. Comp., 215 (2010), 3399-3407.  doi: 10.1016/j.amc.2009.10.033.  Google Scholar

[16]

Y. Li, H. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44.  doi: 10.1016/j.na.2014.06.013.  Google Scholar

[17]

Y. Li, A. 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.  Google Scholar

[18]

Y. Li and B. 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.  Google Scholar

[19]

Y. Li, R. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092.  Google Scholar

[20]

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

[21]

Y. Li, L. She and J. Yin, Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equations, Acta Math. Sci., 38 (2018), 591-609.  doi: 10.1016/S0252-9602(18)30768-9.  Google Scholar

[22]

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

[23]

L. Liu and X. Fu, Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-Laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-473.  doi: 10.3934/cpaa.2017023.  Google Scholar

[24]

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

[25]

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

[26]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $R^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar

[27]

F-Y Wang, Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift, J. Differ. Equ., 260 (2016), 2792-2829.  doi: 10.1016/j.jde.2015.10.020.  Google Scholar

[28]

R. Wang, Y. Li and F. Li, Probabilitistic robustness for dispersive-dissipative wave equations driven by small Lapace-multiplier noise, Dyn. Syst. Appl., 27 (2018), 165-183.  doi: 10.12732/dsa.v27i1.9.  Google Scholar

[29]

Z. Wang and Y. Liu, Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped sine-Gordon equation on unbounded domains, Comput. Math. Appl., 73 (2017), 1445-1460.  doi: 10.1016/j.camwa.2017.01.015.  Google Scholar

[30]

Z. Wang and S. Zhou, Existence and upper semicontinuity of attractors for non-autonomous stochastic lattices systems with random coupled coefficients, Commun. Pure Appl. Anal., 15 (2016), 2221-2245.  doi: 10.3934/cpaa.2016035.  Google Scholar

[31]

J. Yin, A. Gu and Y. Li, Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dynamics of PDE, 14 (2017), 201-218.  doi: 10.4310/DPDE.2017.v14.n2.a4.  Google Scholar

[32]

J. Yin, Y. Li and A. Gu, Regularity of pullback attractors for non-autonomous stochastic coupled reaction-diffusion systems, J. Appl. Anal. Comput., 7 (2017), 884-898.   Google Scholar

[33]

J. Yin, Y. Li and H. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in Lq, Appl. Math. Comput., 225 (2013), 526-540.  doi: 10.1016/j.amc.2013.09.051.  Google Scholar

[34]

J. Yin and Y. Li, Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic p-Laplacian equations on R-n, Math. Methods Appl. Sci., 40 (2017), 4863-4879.  doi: 10.1002/mma.4353.  Google Scholar

[35]

J. Yin, Y. Li and H. Cui, Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207.  doi: 10.1016/j.jmaa.2017.01.064.  Google Scholar

[36]

J. Yin, Y. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.  doi: 10.1016/j.camwa.2017.05.015.  Google Scholar

[37]

W. Zhao, Random dynamics of stochastic p-Laplacian equations on RN with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.   Google Scholar

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. B, 18 (2013), 643-666.  doi: 10.3934/dcdsb.2013.18.643.  Google Scholar

[2]

M. Anguiano and P. E. Kloeden, Asymptotic behaviour of the nonautonomous SIR equations with diffusion, Commun. Pure Appl. Anal., 13 (2014), 157-173.  doi: 10.3934/cpaa.2014.13.157.  Google Scholar

[3]

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

[4]

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

[5]

T. Caraballo, A. N. Carvalho, J. 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.  Google Scholar

[6]

T. Caraballo and R. Colucci, A comparison between random and stochastic modeling for a SIR model, Commun. Pure Appl. Anal., 16 (2017), 151-162.  doi: 10.3934/cpaa.2017007.  Google Scholar

[7]

T. Caraballo, M. J. Garrido-Atienza and J. Lopez-de-la-Cruz, Dynamics of some stochastic chemostat models with multiplicative noise, Commun. Pure Appl. Anal., 16 (2017), 1893-1914.  doi: 10.3934/cpaa.2017092.  Google Scholar

[8]

H. Cui, J. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems J. Dyn. Differ. Equ., online (2017), DOI: 10.1007/s10884-017-9617-z. doi: 10.1007/s10884-017-9617-z.  Google Scholar

[9]

H. Cui, J. A. Langa and Y. 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.  Google Scholar

[10]

H. Cui and Y. Li, Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises, Appl. Math. Comput., 271 (2015), 777-789.  doi: 10.1016/j.amc.2015.09.031.  Google Scholar

[11]

X. Fan, Attractors for a damped stochastic wave equation of sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793.  doi: 10.1080/07362990600751860.  Google Scholar

[12]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Commun. Pure Appl. Anal., 13 (2014), 2543-2557.  doi: 10.3934/cpaa.2014.13.2543.  Google Scholar

[13]

A. Krause and B. 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.  Google Scholar

[14]

X. J. Li, X. L. Li and K. N. Lu, Random attractors for stochastic parabolic equations with additive noise in wighted spaces, Commun. Pure Appl. Anal., 17 (2018), 729-749.  doi: 10.3934/cpaa.2018038.  Google Scholar

[15]

J. Li, Y. Li and B. Wang, Random attractors of reaction-diffusion equations with multiplicative noise in Lp, Appl. Math. Comp., 215 (2010), 3399-3407.  doi: 10.1016/j.amc.2009.10.033.  Google Scholar

[16]

Y. Li, H. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44.  doi: 10.1016/j.na.2014.06.013.  Google Scholar

[17]

Y. Li, A. 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.  Google Scholar

[18]

Y. Li and B. 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.  Google Scholar

[19]

Y. Li, R. Wang and J. Yin, Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels, Discrete Contin. Dyn. Syst. B, 22 (2017), 2569-2586.  doi: 10.3934/dcdsb.2017092.  Google Scholar

[20]

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

[21]

Y. Li, L. She and J. Yin, Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equations, Acta Math. Sci., 38 (2018), 591-609.  doi: 10.1016/S0252-9602(18)30768-9.  Google Scholar

[22]

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

[23]

L. Liu and X. Fu, Existence and upper semicontinuity of (L2, Lq) pullback attractors for a stochastic p-Laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-473.  doi: 10.3934/cpaa.2017023.  Google Scholar

[24]

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

[25]

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

[26]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $R^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.  doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar

[27]

F-Y Wang, Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift, J. Differ. Equ., 260 (2016), 2792-2829.  doi: 10.1016/j.jde.2015.10.020.  Google Scholar

[28]

R. Wang, Y. Li and F. Li, Probabilitistic robustness for dispersive-dissipative wave equations driven by small Lapace-multiplier noise, Dyn. Syst. Appl., 27 (2018), 165-183.  doi: 10.12732/dsa.v27i1.9.  Google Scholar

[29]

Z. Wang and Y. Liu, Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped sine-Gordon equation on unbounded domains, Comput. Math. Appl., 73 (2017), 1445-1460.  doi: 10.1016/j.camwa.2017.01.015.  Google Scholar

[30]

Z. Wang and S. Zhou, Existence and upper semicontinuity of attractors for non-autonomous stochastic lattices systems with random coupled coefficients, Commun. Pure Appl. Anal., 15 (2016), 2221-2245.  doi: 10.3934/cpaa.2016035.  Google Scholar

[31]

J. Yin, A. Gu and Y. Li, Backwards compact attractors for non-autonomous damped 3D Navier-Stokes equations, Dynamics of PDE, 14 (2017), 201-218.  doi: 10.4310/DPDE.2017.v14.n2.a4.  Google Scholar

[32]

J. Yin, Y. Li and A. Gu, Regularity of pullback attractors for non-autonomous stochastic coupled reaction-diffusion systems, J. Appl. Anal. Comput., 7 (2017), 884-898.   Google Scholar

[33]

J. Yin, Y. Li and H. Zhao, Random attractors for stochastic semi-linear degenerate parabolic equations with additive noise in Lq, Appl. Math. Comput., 225 (2013), 526-540.  doi: 10.1016/j.amc.2013.09.051.  Google Scholar

[34]

J. Yin and Y. Li, Two types of upper semi-continuity of bi-spatial attractors for non-autonomous stochastic p-Laplacian equations on R-n, Math. Methods Appl. Sci., 40 (2017), 4863-4879.  doi: 10.1002/mma.4353.  Google Scholar

[35]

J. Yin, Y. Li and H. Cui, Box-counting dimensions and upper semicontinuities of bi-spatial attractors for stochastic degenerate parabolic equations on an unbounded domain, J. Math. Anal. Appl., 450 (2017), 1180-1207.  doi: 10.1016/j.jmaa.2017.01.064.  Google Scholar

[36]

J. Yin, Y. Li and A. Gu, Backwards compact attractors and periodic attractors for non-autonomous damped wave equations on an unbounded domain, Comput. Math. Appl., 74 (2017), 744-758.  doi: 10.1016/j.camwa.2017.05.015.  Google Scholar

[37]

W. Zhao, Random dynamics of stochastic p-Laplacian equations on RN with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.   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]

Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054
[3]

Goong Chen, Zhonghai Ding, Shujie Li. On positive solutions of the elliptic sine-Gordon equation. Communications on Pure & Applied Analysis, 2005, 4 (2) : 283-294. doi: 10.3934/cpaa.2005.4.283
[4]

Qin Sheng, David A. Voss, Q. M. Khaliq. An adaptive splitting algorithm for the sine-Gordon equation. Conference Publications, 2005, 2005 (Special) : 792-797. doi: 10.3934/proc.2005.2005.792
[5]

Igor Chueshov, Peter E. Kloeden, Meihua Yang. Synchronization in coupled stochastic sine-Gordon wave model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2969-2990. doi: 10.3934/dcdsb.2016082
[6]

Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925
[7]

Cornelia Schiebold. Noncommutative AKNS systems and multisoliton solutions to the matrix sine-gordon equation. Conference Publications, 2009, 2009 (Special) : 678-690. doi: 10.3934/proc.2009.2009.678
[8]

V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27
[9]

Fuzhi Li, Dongmei Xu, Jiali Yu. Regular measurable backward compact random attractor for $ g $-Navier-Stokes equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3137-3157. doi: 10.3934/cpaa.2020136
[10]

Carl-Friedrich Kreiner, Johannes Zimmer. Heteroclinic travelling waves for the lattice sine-Gordon equation with linear pair interaction. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 915-931. doi: 10.3934/dcds.2009.25.915
[11]

Sara Cuenda, Niurka R. Quintero, Angel Sánchez. Sine-Gordon wobbles through Bäcklund transformations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1047-1056. doi: 10.3934/dcdss.2011.4.1047
[12]

Ivan Christov, C. I. Christov. The coarse-grain description of interacting sine-Gordon solitons with varying widths. Conference Publications, 2009, 2009 (Special) : 171-180. doi: 10.3934/proc.2009.2009.171
[13]

Linda J. S. Allen, P. van den Driessche. Stochastic epidemic models with a backward bifurcation. Mathematical Biosciences & Engineering, 2006, 3 (3) : 445-458. doi: 10.3934/mbe.2006.3.445
[14]

Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems & Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002
[15]

Nguyen Huy Tuan, Tran Ngoc Thach, Yong Zhou. On a backward problem for two-dimensional time fractional wave equation with discrete random data. Evolution Equations & Control Theory, 2020, 9 (2) : 561-579. doi: 10.3934/eect.2020024
[16]

Boling Guo, Guoli Zhou. On the backward uniqueness of the stochastic primitive equations with additive noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3157-3174. doi: 10.3934/dcdsb.2018305
[17]

Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[18]

Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[19]

Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447
[20]

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 - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (101)
  • HTML views (227)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences