doi: 10.3934/eect.2020025

Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain

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

* Corresponding author: liyr@swu.edu.cn (Yangrong Li)

Received  February 2019 Revised  June 2019 Published  December 2019

Fund Project: This work was supported by the Natural Science Foundation of China Grant 11571283.

A pullback random attractor is called forward controllable if its time-component is semi-continuous to a compact set in the future, and the minimum among all such compact limit-sets is called a forward controller. The existence of a forward controller closely relates to the forward compactness of the attractor, which is further argued by the forward-pullback asymptotic compactness of the system. The abstract results are applied to the non-autonomous stochastic sine-Gordon equation on an unbounded domain. The existence of a forward compact attractor is proved, which leads to the existence of a forward controller. The measurability of the attractor is proved by considering two different universes.

Citation: 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, doi: 10.3934/eect.2020025
References:
[1]

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

[2]

B. Birnir and R. Grauer, An explicit description of the global attractor of the damped and driven sine-Gordon equation, Comm. Math. Phys., 162 (1994), 539-590.  doi: 10.1007/BF02101747.  Google Scholar

[3]

Z. BrzezniakB. Goldys and Q. T. Le Gia, Random Attractors for the Stochastic Navier-Stokes Equations on the 2D Unit Sphere, J. Math. Fluid Mech., 20 (2018), 227-253.  doi: 10.1007/s00021-017-0351-4.  Google Scholar

[4]

T. CaraballoM. 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

[5]

V. V. ChepyzhovM. I. Vishik and W. L. Wendland, On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 15 (2005), 27-38.  doi: 10.3934/dcds.2005.12.27.  Google Scholar

[6]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, 1779. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[7]

I. ChueshovP. E. Kloeden and M. Yang, Synchronization in coupled stochastic sine-Gordon wave model, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2969-2990.  doi: 10.3934/dcdsb.2016082.  Google Scholar

[8]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of Amer Math Soc, 195 (2008). doi: 10.1090/memo/0912.  Google Scholar

[9]

I. ChueshovI. Lasiecka and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Differential Equations, 21 (2009), 269-314.  doi: 10.1007/s10884-009-9132-y.  Google Scholar

[10]

I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrodinger-Boussinesq equations, Evol. Equ. Control Theory, 1 (2012), 57-80.  doi: 10.3934/eect.2012.1.57.  Google Scholar

[11]

H. CrauelP. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[12]

H. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186.  doi: 10.1016/j.jde.2018.07.028.  Google Scholar

[13]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.  Google Scholar

[14]

H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.  Google Scholar

[15]

H. CuiJ. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar

[16]

X. M. 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

[17]

X. M. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76.  doi: 10.2140/pjm.2004.216.63.  Google Scholar

[18]

A. H. Gu and P. E. Kloeden, Asymptotic behavior of a nonautonomous p-Laplacian lattice system, Internat. J. Bifur. Chaos, 26 (2016), 9 pp. doi: 10.1142/S0218127416501741.  Google Scholar

[19]

X. M. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493.  doi: 10.1016/j.jmaa.2010.11.032.  Google Scholar

[20]

D. S. JorgeA. Marcio and V. Narciso, Long-time Dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.  Google Scholar

[21]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[22]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[23]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[24]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Equ. Appl., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550.  Google Scholar

[25]

I. Lasiecka and R. Triggiani, Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers, Nonlinear Anal., 121 (2015), 424-446.  doi: 10.1016/j.na.2015.03.012.  Google Scholar

[26]

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

[27]

F. Z. LiY. R. Li and R. H. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.  Google Scholar

[28]

Y. R. LiA. H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[29]

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

[30]

Y. R. LiL. B. She and J. Y. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058.  Google Scholar

[31]

Y. R. LiR. H. Wang and L. B. She, Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations, Evol. Equ. Control Theory, 7 (2018), 617-637.  doi: 10.3934/eect.2018030.  Google Scholar

[32]

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

[33]

Y. R. Li and S. Yang, Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 1155-1175.  doi: 10.3934/cpaa.2019056.  Google Scholar

[34]

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

[35]

Z. W. ShenS. F. Zhou and W. X. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.  Google Scholar

[36]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[37]

B. X. 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

[38]

B. X. 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

[39]

B. X. 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

[40]

S. L. Wang and Y. R. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Physica D, 382 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.  Google Scholar

[41]

X. H. WangK. N. Lu and B. X. 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.  Google Scholar

[42]

Z. J. Wang and Y. N. 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

[43]

J. Y. YinY. R. Li and A. H. 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

[44]

J. Y. YinY. R. Li and H. Y. 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

[45]

S. F. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.  Google Scholar

show all references

References:
[1]

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

[2]

B. Birnir and R. Grauer, An explicit description of the global attractor of the damped and driven sine-Gordon equation, Comm. Math. Phys., 162 (1994), 539-590.  doi: 10.1007/BF02101747.  Google Scholar

[3]

Z. BrzezniakB. Goldys and Q. T. Le Gia, Random Attractors for the Stochastic Navier-Stokes Equations on the 2D Unit Sphere, J. Math. Fluid Mech., 20 (2018), 227-253.  doi: 10.1007/s00021-017-0351-4.  Google Scholar

[4]

T. CaraballoM. 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

[5]

V. V. ChepyzhovM. I. Vishik and W. L. Wendland, On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging, Discrete Contin. Dyn. Syst., 15 (2005), 27-38.  doi: 10.3934/dcds.2005.12.27.  Google Scholar

[6]

I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, 1779. Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar

[7]

I. ChueshovP. E. Kloeden and M. Yang, Synchronization in coupled stochastic sine-Gordon wave model, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2969-2990.  doi: 10.3934/dcdsb.2016082.  Google Scholar

[8]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of Amer Math Soc, 195 (2008). doi: 10.1090/memo/0912.  Google Scholar

[9]

I. ChueshovI. Lasiecka and D. Toundykov, Global attractor for a wave equation with nonlinear localized boundary damping and a source term of critical exponent, J. Dynam. Differential Equations, 21 (2009), 269-314.  doi: 10.1007/s10884-009-9132-y.  Google Scholar

[10]

I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrodinger-Boussinesq equations, Evol. Equ. Control Theory, 1 (2012), 57-80.  doi: 10.3934/eect.2012.1.57.  Google Scholar

[11]

H. CrauelP. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.  doi: 10.1142/S0219493711003292.  Google Scholar

[12]

H. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186.  doi: 10.1016/j.jde.2018.07.028.  Google Scholar

[13]

H. CuiP. E. Kloeden and F. Wu, Pathwise upper semi-continuity of random pullback attractors along the time axis, Phys. D, 374 (2018), 21-34.  doi: 10.1016/j.physd.2018.03.002.  Google Scholar

[14]

H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268.  doi: 10.1016/j.jde.2017.03.018.  Google Scholar

[15]

H. CuiJ. A. Langa and Y. Li, Measurability of random attractors for quasi strong-to-weak continuous random dynamical systems, J. Dynam. Differential Equations, 30 (2018), 1873-1898.  doi: 10.1007/s10884-017-9617-z.  Google Scholar

[16]

X. M. 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

[17]

X. M. Fan, Random attractor for a damped sine-Gordon equation with white noise, Pacific J. Math., 216 (2004), 63-76.  doi: 10.2140/pjm.2004.216.63.  Google Scholar

[18]

A. H. Gu and P. E. Kloeden, Asymptotic behavior of a nonautonomous p-Laplacian lattice system, Internat. J. Bifur. Chaos, 26 (2016), 9 pp. doi: 10.1142/S0218127416501741.  Google Scholar

[19]

X. M. Han, Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise, J. Math. Anal. Appl., 376 (2011), 481-493.  doi: 10.1016/j.jmaa.2010.11.032.  Google Scholar

[20]

D. S. JorgeA. Marcio and V. Narciso, Long-time Dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory, 6 (2017), 437-470.  doi: 10.3934/eect.2017023.  Google Scholar

[21]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[22]

P. E. Kloeden and J. Simsen, Attractors of asymptotically autonomous quasi-linear parabolic equation with spatially variable exponents, J. Math. Anal. Appl., 425 (2015), 911-918.  doi: 10.1016/j.jmaa.2014.12.069.  Google Scholar

[23]

P. E. KloedenJ. Simsen and M. S. Simsen, Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents, J. Math. Anal. Appl., 445 (2017), 513-531.  doi: 10.1016/j.jmaa.2016.08.004.  Google Scholar

[24]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Equ. Appl., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550.  Google Scholar

[25]

I. Lasiecka and R. Triggiani, Stabilization to an equilibrium of the Navier-Stokes equations with tangential action of feedback controllers, Nonlinear Anal., 121 (2015), 424-446.  doi: 10.1016/j.na.2015.03.012.  Google Scholar

[26]

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

[27]

F. Z. LiY. R. Li and R. H. Wang, Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise, Discrete Contin. Dyn. Syst., 38 (2018), 3663-3685.  doi: 10.3934/dcds.2018158.  Google Scholar

[28]

Y. R. LiA. H. Gu and J. Li, Existence and continuity of bi-spatial random attractors and application to stochastic semilinear Laplacian equations, J. Differential Equations, 258 (2015), 504-534.  doi: 10.1016/j.jde.2014.09.021.  Google Scholar

[29]

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

[30]

Y. R. LiL. B. She and J. Y. Yin, Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1535-1557.  doi: 10.3934/dcdsb.2018058.  Google Scholar

[31]

Y. R. LiR. H. Wang and L. B. She, Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations, Evol. Equ. Control Theory, 7 (2018), 617-637.  doi: 10.3934/eect.2018030.  Google Scholar

[32]

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

[33]

Y. R. Li and S. Yang, Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise, Commun. Pure Appl. Anal., 18 (2019), 1155-1175.  doi: 10.3934/cpaa.2019056.  Google Scholar

[34]

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

[35]

Z. W. ShenS. F. Zhou and W. X. Shen, One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations, 248 (2010), 1432-1457.  doi: 10.1016/j.jde.2009.10.007.  Google Scholar

[36]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[37]

B. X. 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

[38]

B. X. 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

[39]

B. X. 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

[40]

S. L. Wang and Y. R. Li, Longtime robustness of pullback random attractors for stochastic magneto-hydrodynamics equations, Physica D, 382 (2018), 46-57.  doi: 10.1016/j.physd.2018.07.003.  Google Scholar

[41]

X. H. WangK. N. Lu and B. X. 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.  Google Scholar

[42]

Z. J. Wang and Y. N. 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

[43]

J. Y. YinY. R. Li and A. H. 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

[44]

J. Y. YinY. R. Li and H. Y. 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

[45]

S. F. Zhou, Random exponential attractor for cocycle and application to non-autonomous stochastic lattice systems with multiplicative white noise, J. Differential Equations, 263 (2017), 2247-2279.  doi: 10.1016/j.jde.2017.03.044.  Google Scholar

[1]

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

[2]

S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593

[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]

Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651

[7]

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

[8]

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

[9]

Wenqiang Zhao. Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3395-3438. doi: 10.3934/dcdsb.2018326

[10]

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

[11]

Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781

[12]

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

[13]

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

[14]

Shulin Wang, Yangrong Li. Probabilistic continuity of a pullback random attractor in time-sample. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020028

[15]

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

[16]

Yuncheng You. Random attractor for stochastic reversible Schnackenberg equations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1347-1362. doi: 10.3934/dcdss.2014.7.1347

[17]

Yangrong Li, Lianbing She, Jinyan Yin. Longtime robustness and semi-uniform compactness of a pullback attractor via nonautonomous PDE. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1535-1557. doi: 10.3934/dcdsb.2018058

[18]

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

[19]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

[20]

Shengfan Zhou, Min Zhao. Fractal dimension of random attractor for stochastic non-autonomous damped wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2887-2914. doi: 10.3934/dcds.2016.36.2887

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (23)
  • HTML views (123)
  • Cited by (0)

Other articles
by authors

[Back to Top]