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

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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 & 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.
[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 L^p, 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, L^q) 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 L^q, 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 R^N 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 L^p, 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, L^q) 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 L^q, 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 R^N with an unbounded additive noise, J. Math. Anal. Appl., 455 (2017), 1178-1203.

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