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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 and Applied Analysis, 2019, 18 (3) : 1155-1175. doi: 10.3934/cpaa.2019056
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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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. 

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