American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Random attractors and robustness for stochastic reversible reaction-diffusion systems
  • DCDS Home
  • This Issue
  • Next Article
    Frequency domain conditions for finite-dimensional projectors and determining observations for the set of amenable solutions
January  2014, 34(1): 269-300. doi: 10.3934/dcds.2014.34.269

Random attractors for non-autonomous stochastic wave equations with multiplicative noise

Bixiang Wang 1,

1. 

Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801

Received  October 2012 Revised  February 2013 Published  June 2013

This paper is concerned with the asymptotic behavior of solutions of the damped non-autonomous stochastic wave equations driven by multiplicative white noise. We prove the existence of pullback random attractors in $H^1(\mathbb{R}^n) \times L^2(\mathbb{R}^n)$ when the intensity of noise is sufficiently small. We demonstrate that these random attractors are periodic in time if so are the deterministic non-autonomous external terms. We also establish the upper semicontinuity of random attractors when the intensity of noise approaches zero. In addition, we prove the measurability of random attractors even if the underlying probability space is not complete.
Keywords: periodic attractor, Random attractor, stochastic wave equation., upper semicontinuity, random complete solution.
Mathematics Subject Classification: Primary: 35B40; Secondary: 35B41, 37L3.
Citation: Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269
References:
[1]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar

[2]

J. M. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent,, Communications in Partial Differential Equations, 17 (1992), 841.  doi: 10.1080/03605309208820866.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).   Google Scholar

[4]

J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Continuous Dynamical Systems, 10 (2004), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[5]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, Stoch. Dyn., 6 (2006), 1.  doi: 10.1142/S0219493706001621.  Google Scholar

[6]

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

[7]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness,, Discrete Continuous Dynamical Systems, 21 (2008), 415.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[8]

T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory,, Discrete Continuous Dynamical Systems B, 9 (2008), 525.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

[9]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dynamics of Continuous, 10 (2003), 491.   Google Scholar

[10]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[11]

T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,, Nonlinear Anal., 74 (2011), 3671.  doi: 10.1016/j.na.2011.02.047.  Google Scholar

[12]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems,, Set-Valued Analysis, 11 (2003), 153.  doi: 10.1023/A:1022902802385.  Google Scholar

[13]

A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds,, J. Differential Equations, 233 (2007), 622.  doi: 10.1016/j.jde.2006.08.009.  Google Scholar

[14]

A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation,, J. Differential Equations, 246 (2009), 2646.  doi: 10.1016/j.jde.2009.01.007.  Google Scholar

[15]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469.  doi: 10.1007/s10884-004-4289-x.  Google Scholar

[16]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems,, Dynamical Systems, 19 (2004), 127.  doi: 10.1080/1468936042000207792.  Google Scholar

[17]

I. Chueshow, "Monotone Random Systems-Theory and Applications,", Lecture Notes in Mathematics, 1779 (2001).  doi: 10.1007/b83277.  Google Scholar

[18]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar

[19]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar

[20]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary,, Comm. Math. Sci., 1 (2003), 133.   Google Scholar

[21]

E. Feireisl, Attractors for semilinear damped wave equations on $\mathbbR^3$,, Nonlinear Analysis TMA, 23 (1994), 187.  doi: 10.1016/0362-546X(94)90041-8.  Google Scholar

[22]

E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent,, J. Differential Equations, 116 (1995), 431.  doi: 10.1006/jdeq.1995.1042.  Google Scholar

[23]

E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent,, Communications in Partial Differential Equations, 18 (1993), 1539.  doi: 10.1080/03605309308820985.  Google Scholar

[24]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Rep., 59 (1996), 21.   Google Scholar

[25]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion,, J. Dynam. Differential Equations, 23 (2011), 671.  doi: 10.1007/s10884-011-9222-5.  Google Scholar

[26]

M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays,, Stoch. Dyn., 11 (2011), 369.  doi: 10.1142/S0219493711003358.  Google Scholar

[27]

M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion,, International J. Bifurcation and Chaos, 20 (2010), 2761.  doi: 10.1142/S0218127410027349.  Google Scholar

[28]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", American Mathematical Society, (1988).   Google Scholar

[29]

J. K. Hale, X. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and PDE's,, Math. Comp., 50 (1988), 89.  doi: 10.1090/S0025-5718-1988-0917820-X.  Google Scholar

[30]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, J. Differential Equations, 73 (1988), 197.  doi: 10.1016/0022-0396(88)90104-0.  Google Scholar

[31]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains,, Discrete and Continuous Dynamical Systems, 24 (2009), 855.  doi: 10.3934/dcds.2009.24.855.  Google Scholar

[32]

A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents,, J. Differential Equations, 230 (2006), 702.  doi: 10.1016/j.jde.2006.06.001.  Google Scholar

[33]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Royal Soc. London Serie A., 463 (2007), 163.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[34]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations,, J. Differential Equations, 244 (2008), 1.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar

[35]

M. Prizzi and K. P. Rybakowski, Attractors for semilinear damped wave equations on arbitrary unbounded domains,, Topol. Methods Nonlinear Anal., 31 (2008), 49.   Google Scholar

[36]

M. Prizzi and K. P. Rybakowski, Attractors for singularly perturbed damped wave equations on unbounded domains,, Topol. Methods Nonlinear Anal., 32 (2008), 1.   Google Scholar

[37]

M. Prizzi, Regularity of invariant sets in semilinear damped wave equations,, J. Differential Equations, 247 (2009), 3315.  doi: 10.1016/j.jde.2009.08.011.  Google Scholar

[38]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, in, (1992), 185.   Google Scholar

[39]

R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Springer-Verlag, (2002).   Google Scholar

[40]

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.  doi: 10.1016/j.jde.2009.10.007.  Google Scholar

[41]

C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,, Nonlinearity, 19 (2006), 2645.  doi: 10.1088/0951-7715/19/11/008.  Google Scholar

[42]

C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping,, J. Differential Equation, 227 (2006), 427.  doi: 10.1016/j.jde.2005.09.010.  Google Scholar

[43]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer-Verlag, (1997).   Google Scholar

[44]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains,, Physica D, 128 (1999), 41.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[45]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbR^3$,, Transactions of American Mathematical Society, 363 (2011), 3639.  doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar

[46]

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

[47]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms,, arXiv:1205.4658v1., ().   Google Scholar

[48]

B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations,, arXiv:1304.4884v1., ().   Google Scholar

[49]

B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems,, Electronic Journal of Differential Equations, 2009 (2009), 1.   Google Scholar

[50]

Z. Wang, S. Zhou and A. Gu, Random attractors for a stochastic damped wave equation with multiplicative noise on unbounded domains,, Nonlinear Analysis, 12 (2011), 3468.  doi: 10.1016/j.nonrwa.2011.06.008.  Google Scholar

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar

[2]

J. M. Arrieta, A. N. Carvalho and J. K. Hale, A damped hyperbolic equation with critical exponent,, Communications in Partial Differential Equations, 17 (1992), 841.  doi: 10.1080/03605309208820866.  Google Scholar

[3]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations,", North-Holland, (1992).   Google Scholar

[4]

J. M. Ball, Global attractors for damped semilinear wave equations,, Discrete Continuous Dynamical Systems, 10 (2004), 31.  doi: 10.3934/dcds.2004.10.31.  Google Scholar

[5]

P. W. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems,, Stoch. Dyn., 6 (2006), 1.  doi: 10.1142/S0219493706001621.  Google Scholar

[6]

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

[7]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness,, Discrete Continuous Dynamical Systems, 21 (2008), 415.  doi: 10.3934/dcds.2008.21.415.  Google Scholar

[8]

T. Caraballo, J. Real and I. D. Chueshov, Pullback attractors for stochastic heat equations in materials with memory,, Discrete Continuous Dynamical Systems B, 9 (2008), 525.  doi: 10.3934/dcdsb.2008.9.525.  Google Scholar

[9]

T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems,, Dynamics of Continuous, 10 (2003), 491.   Google Scholar

[10]

T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 439.  doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[11]

T. Caraballo, M. J. Garrido-Atienza and T. Taniguchi, The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,, Nonlinear Anal., 74 (2011), 3671.  doi: 10.1016/j.na.2011.02.047.  Google Scholar

[12]

T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors for nonautonomous and stochastic multivalued dynamical systems,, Set-Valued Analysis, 11 (2003), 153.  doi: 10.1023/A:1022902802385.  Google Scholar

[13]

A. N. Carvalho and J. A. Langa, Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds,, J. Differential Equations, 233 (2007), 622.  doi: 10.1016/j.jde.2006.08.009.  Google Scholar

[14]

A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation,, J. Differential Equations, 246 (2009), 2646.  doi: 10.1016/j.jde.2009.01.007.  Google Scholar

[15]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469.  doi: 10.1007/s10884-004-4289-x.  Google Scholar

[16]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems,, Dynamical Systems, 19 (2004), 127.  doi: 10.1080/1468936042000207792.  Google Scholar

[17]

I. Chueshow, "Monotone Random Systems-Theory and Applications,", Lecture Notes in Mathematics, 1779 (2001).  doi: 10.1007/b83277.  Google Scholar

[18]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dyn. Diff. Eqns., 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar

[19]

H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probab. Th. Re. Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar

[20]

J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary,, Comm. Math. Sci., 1 (2003), 133.   Google Scholar

[21]

E. Feireisl, Attractors for semilinear damped wave equations on $\mathbbR^3$,, Nonlinear Analysis TMA, 23 (1994), 187.  doi: 10.1016/0362-546X(94)90041-8.  Google Scholar

[22]

E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent,, J. Differential Equations, 116 (1995), 431.  doi: 10.1006/jdeq.1995.1042.  Google Scholar

[23]

E. Feireisl and E. Zuazua, Global attractors for semilinear wave equations with locally distributed nonlinear damping and critical exponent,, Communications in Partial Differential Equations, 18 (1993), 1539.  doi: 10.1080/03605309308820985.  Google Scholar

[24]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise,, Stoch. Stoch. Rep., 59 (1996), 21.   Google Scholar

[25]

M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion,, J. Dynam. Differential Equations, 23 (2011), 671.  doi: 10.1007/s10884-011-9222-5.  Google Scholar

[26]

M. J. Garrido-Atienza, A. Ogrowsky and B. Schmalfuss, Random differential equations with random delays,, Stoch. Dyn., 11 (2011), 369.  doi: 10.1142/S0219493711003358.  Google Scholar

[27]

M. J. Garrido-Atienza, B. Maslowski and B. Schmalfuss, Random attractors for stochastic equations driven by a fractional Brownian motion,, International J. Bifurcation and Chaos, 20 (2010), 2761.  doi: 10.1142/S0218127410027349.  Google Scholar

[28]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", American Mathematical Society, (1988).   Google Scholar

[29]

J. K. Hale, X. Lin and G. Raugel, Upper semicontinuity of attractors for approximations of semigroups and PDE's,, Math. Comp., 50 (1988), 89.  doi: 10.1090/S0025-5718-1988-0917820-X.  Google Scholar

[30]

J. K. Hale and G. Raugel, Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation,, J. Differential Equations, 73 (1988), 197.  doi: 10.1016/0022-0396(88)90104-0.  Google Scholar

[31]

J. Huang and W. Shen, Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains,, Discrete and Continuous Dynamical Systems, 24 (2009), 855.  doi: 10.3934/dcds.2009.24.855.  Google Scholar

[32]

A. Kh. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents,, J. Differential Equations, 230 (2006), 702.  doi: 10.1016/j.jde.2006.06.001.  Google Scholar

[33]

P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors,, Proc. Royal Soc. London Serie A., 463 (2007), 163.  doi: 10.1098/rspa.2006.1753.  Google Scholar

[34]

Y. Lv and W. Wang, Limiting dynamics for stochastic wave equations,, J. Differential Equations, 244 (2008), 1.  doi: 10.1016/j.jde.2007.10.009.  Google Scholar

[35]

M. Prizzi and K. P. Rybakowski, Attractors for semilinear damped wave equations on arbitrary unbounded domains,, Topol. Methods Nonlinear Anal., 31 (2008), 49.   Google Scholar

[36]

M. Prizzi and K. P. Rybakowski, Attractors for singularly perturbed damped wave equations on unbounded domains,, Topol. Methods Nonlinear Anal., 32 (2008), 1.   Google Scholar

[37]

M. Prizzi, Regularity of invariant sets in semilinear damped wave equations,, J. Differential Equations, 247 (2009), 3315.  doi: 10.1016/j.jde.2009.08.011.  Google Scholar

[38]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations,, in, (1992), 185.   Google Scholar

[39]

R. Sell and Y. You, "Dynamics of Evolutionary Equations,", Springer-Verlag, (2002).   Google Scholar

[40]

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.  doi: 10.1016/j.jde.2009.10.007.  Google Scholar

[41]

C. Sun, D. Cao and J. Duan, Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity,, Nonlinearity, 19 (2006), 2645.  doi: 10.1088/0951-7715/19/11/008.  Google Scholar

[42]

C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping,, J. Differential Equation, 227 (2006), 427.  doi: 10.1016/j.jde.2005.09.010.  Google Scholar

[43]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Springer-Verlag, (1997).   Google Scholar

[44]

B. Wang, Attractors for reaction-diffusion equations in unbounded domains,, Physica D, 128 (1999), 41.  doi: 10.1016/S0167-2789(98)00304-2.  Google Scholar

[45]

B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbbR^3$,, Transactions of American Mathematical Society, 363 (2011), 3639.  doi: 10.1090/S0002-9947-2011-05247-5.  Google Scholar

[46]

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

[47]

B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms,, arXiv:1205.4658v1., ().   Google Scholar

[48]

B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations,, arXiv:1304.4884v1., ().   Google Scholar

[49]

B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems,, Electronic Journal of Differential Equations, 2009 (2009), 1.   Google Scholar

[50]

Z. Wang, S. Zhou and A. Gu, Random attractors for a stochastic damped wave equation with multiplicative noise on unbounded domains,, Nonlinear Analysis, 12 (2011), 3468.  doi: 10.1016/j.nonrwa.2011.06.008.  Google Scholar

[1]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of random attractors for non-autonomous stochastic strongly damped wave equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2787-2812. doi: 10.3934/dcds.2017120
[2]

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
[3]

Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055
[4]

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
[5]

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
[6]

Ling Xu, Jianhua Huang, Qiaozhen Ma. Upper semicontinuity of random attractors for the stochastic non-autonomous suspension bridge equation with memory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5959-5979. doi: 10.3934/dcdsb.2019115
[7]

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
[8]

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
[9]

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
[10]

Boling Guo, Yongqian Han, Guoli Zhou. Random attractor for the 2D stochastic nematic liquid crystals flows. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2349-2376. doi: 10.3934/cpaa.2019106
[11]

Chi Phan. Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3233-3256. doi: 10.3934/dcdsb.2020060
[12]

Zhaojuan Wang, Shengfan Zhou. Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice systems with random coupled coefficients. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2221-2245. doi: 10.3934/cpaa.2016035
[13]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899
[14]

María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations & Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001
[15]

Min Zhao, Shengfan Zhou. Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1683-1717. doi: 10.3934/dcdsb.2017081
[16]

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
[17]

Shulin Wang, Yangrong Li. Probabilistic continuity of a pullback random attractor in time-sample. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2699-2772. doi: 10.3934/dcdsb.2020028
[18]

Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116
[19]

Abdelghafour Atlas. Regularity of the attractor for symmetric regularized wave equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 695-704. doi: 10.3934/cpaa.2005.4.695
[20]

Cedric Galusinski, Serguei Zelik. Uniform Gevrey regularity for the attractor of a damped wave equation. Conference Publications, 2003, 2003 (Special) : 305-312. doi: 10.3934/proc.2003.2003.305

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (129)
  • HTML views (0)
  • Cited by (84)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences