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The existence of nontrivial solutions to Chern-Simons-Schrödinger systems
Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise
1. | School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, China |
2. | Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China |
In this paper we study the asymptotic behavior of solutions of the non-autonomous stochastic strongly damped wave equation driven by multiplicative noise defined on unbounded domains. We first introduce a continuous cocycle for the equation. Then we consider the existence of a tempered pullback random attractor for the cocycle. Finally we establish the upper semicontinuity of random attractors as the coefficient of the white noise term tends to zero.
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L. Arnold, Random Dynamical Systems Springer-Verlag, Berlin, 1998.
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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. |
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P. Chow,
Stochastic wave equation with polynomial nonlinearity, Ann. Appl. Probab, 12 (2002), 361-381.
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I. Chueshov, Monotone Random Systems Theory and Applications Springer-Verlag, New York, 2002.
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H. Crauel, Random Probability Measure on Polish Spaces Taylor & Francis, London, 2002.
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H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.
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H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.
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[8] |
H. Crauel, P. E. Kloeden and M. Yang,
Random attractors of stochastic reaction-diffusion equations on variable domains, Stochastics and Dynamics, 11 (2011), 301-314.
doi: 10.1142/S0219493711003292. |
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J. Duan, K. Lu and B. Schmalfuss,
Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.
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[10] |
X. 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. |
[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] |
X. Fan and Y. Wang,
Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25 (2007), 381-396.
doi: 10.1080/07362990601139602. |
[13] |
X. Fan,
Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.
doi: 10.1142/S0129167X08004741. |
[14] |
R. Jones and B. Wang,
Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. Real World Appl., 14 (2013), 1308-1322.
doi: 10.1016/j.nonrwa.2012.09.019. |
[15] |
H. Li, Y. You and J. Tu,
Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.
doi: 10.1016/j.jde.2014.09.007. |
[16] |
K. Lu and B. Schmalfuß,
Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492.
doi: 10.1016/j.jde.2006.09.024. |
[17] |
Y. Lv and W. Wang,
Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.
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[18] |
A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983.
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[19] |
Z. Shen, S. Zhou and W. 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. |
[20] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997.
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B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discr. Contin. Dyn. Syst. Supplement, (2009), 800–809.
doi: 10.3934/proc.2009.2009.800. |
[22] |
B. Wang,
Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[23] |
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. |
[24] |
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. |
[25] |
B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms Stochastics and Dynamics 14 (2014), 1450009, 31 pp.
doi: 10.1142/S0219493714500099. |
[26] |
X. Wang,
An energy equation for the weakly damped driven nonlinear Schrodinger equations and its applications, Physica D, 88 (1995), 167-175.
doi: 10.1016/0167-2789(95)00196-B. |
[27] |
Z. Wang, S. Zhou and A. Gu,
Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal. Real World Appl., 12 (2011), 3468-3482.
doi: 10.1016/j.nonrwa.2011.06.008. |
[28] |
Z. Wang, S. Zhou and A. Gu,
Random attractor of the stochastic strongly damped wave equation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1649-1658.
doi: 10.1016/j.cnsns.2011.09.001. |
[29] |
Z. Wang and S. Zhou,
Random attractors for non-autonomous stochastic strongly damped wave equation on unbounded domains, J. Appl. Anal. Comput., 5 (2015), 363-387.
doi: 10.11948/2015031. |
[30] |
M. Yang, J. Duan and P. Kloeden,
Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.
doi: 10.1016/j.nonrwa.2010.06.032. |
[31] |
S. Zhou, F. Yin and Z. Ouyang,
Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903.
doi: 10.1137/050623097. |
[32] |
S. Zhou and M. Zhao,
Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonlinear. Anal., 133 (2016), 292-318.
doi: 10.1016/j.na.2015.12.013. |
[33] |
S. Zhou and M. Zhao,
Fractal dimension of random attractor for stochastic damped wave equation with multiplicative noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.
doi: 10.3934/dcds.2016.36.2887. |
show all references
References:
[1] |
L. Arnold, Random Dynamical Systems Springer-Verlag, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[2] |
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. |
[3] |
P. Chow,
Stochastic wave equation with polynomial nonlinearity, Ann. Appl. Probab, 12 (2002), 361-381.
doi: 10.1214/aoap/1015961168. |
[4] |
I. Chueshov, Monotone Random Systems Theory and Applications Springer-Verlag, New York, 2002.
doi: 10.1007/b83277. |
[5] |
H. Crauel, Random Probability Measure on Polish Spaces Taylor & Francis, London, 2002.
doi: 10.4324/9780203219119. |
[6] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[7] |
H. Crauel and F. Flandoli,
Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[8] |
H. Crauel, P. E. Kloeden and M. Yang,
Random attractors of stochastic reaction-diffusion equations on variable domains, Stochastics and Dynamics, 11 (2011), 301-314.
doi: 10.1142/S0219493711003292. |
[9] |
J. Duan, K. Lu and B. Schmalfuss,
Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.
doi: 10.1214/aop/1068646380. |
[10] |
X. 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. |
[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] |
X. Fan and Y. Wang,
Fractal dimension of attractors for a stochastic wave equation with nonlinear damping and white noise, Stoch. Anal. Appl., 25 (2007), 381-396.
doi: 10.1080/07362990601139602. |
[13] |
X. Fan,
Random attractors for damped stochastic wave equations with multiplicative noise, Internat. J. Math., 19 (2008), 421-437.
doi: 10.1142/S0129167X08004741. |
[14] |
R. Jones and B. Wang,
Asymptotic behavior of a class of stochastic nonlinear wave equations with dispersive and dissipative terms, Nonlinear Anal. Real World Appl., 14 (2013), 1308-1322.
doi: 10.1016/j.nonrwa.2012.09.019. |
[15] |
H. Li, Y. You and J. Tu,
Random attractors and averaging for non-autonomous stochastic wave equations with nonlinear damping, J. Differential Equations, 258 (2015), 148-190.
doi: 10.1016/j.jde.2014.09.007. |
[16] |
K. Lu and B. Schmalfuß,
Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492.
doi: 10.1016/j.jde.2006.09.024. |
[17] |
Y. Lv and W. Wang,
Limiting dynamics for stochastic wave equations, J. Differential Equations, 244 (2008), 1-23.
doi: 10.1016/j.jde.2007.10.009. |
[18] |
A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[19] |
Z. Shen, S. Zhou and W. 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. |
[20] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[21] |
B. Wang and X. Gao, Random attractors for wave equations on unbounded domains, Discr. Contin. Dyn. Syst. Supplement, (2009), 800–809.
doi: 10.3934/proc.2009.2009.800. |
[22] |
B. Wang,
Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Trans. Amer. Math. Soc., 363 (2011), 3639-3663.
doi: 10.1090/S0002-9947-2011-05247-5. |
[23] |
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. |
[24] |
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. |
[25] |
B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms Stochastics and Dynamics 14 (2014), 1450009, 31 pp.
doi: 10.1142/S0219493714500099. |
[26] |
X. Wang,
An energy equation for the weakly damped driven nonlinear Schrodinger equations and its applications, Physica D, 88 (1995), 167-175.
doi: 10.1016/0167-2789(95)00196-B. |
[27] |
Z. Wang, S. Zhou and A. Gu,
Random attractor for a stochastic damped wave equation with multiplicative noise on unbounded domains, Nonlinear Anal. Real World Appl., 12 (2011), 3468-3482.
doi: 10.1016/j.nonrwa.2011.06.008. |
[28] |
Z. Wang, S. Zhou and A. Gu,
Random attractor of the stochastic strongly damped wave equation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 1649-1658.
doi: 10.1016/j.cnsns.2011.09.001. |
[29] |
Z. Wang and S. Zhou,
Random attractors for non-autonomous stochastic strongly damped wave equation on unbounded domains, J. Appl. Anal. Comput., 5 (2015), 363-387.
doi: 10.11948/2015031. |
[30] |
M. Yang, J. Duan and P. Kloeden,
Asymptotic behavior of solutions for random wave equations with nonlinear damping and white noise, Nonlinear Anal. Real World Appl., 12 (2011), 464-478.
doi: 10.1016/j.nonrwa.2010.06.032. |
[31] |
S. Zhou, F. Yin and Z. Ouyang,
Random attractor for damped nonlinear wave equations with white noise, SIAM J. Appl. Dyn. Syst., 4 (2005), 883-903.
doi: 10.1137/050623097. |
[32] |
S. Zhou and M. Zhao,
Fractal dimension of random invariant sets for nonautonomous random dynamical systems and random attractor for stochastic damped wave equation, Nonlinear. Anal., 133 (2016), 292-318.
doi: 10.1016/j.na.2015.12.013. |
[33] |
S. Zhou and M. Zhao,
Fractal dimension of random attractor for stochastic damped wave equation with multiplicative noise, Discrete Contin. Dyn. Syst., 36 (2016), 2887-2914.
doi: 10.3934/dcds.2016.36.2887. |
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