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January  2014, 34(1): 269-300. doi: 10.3934/dcds.2014.34.269

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

 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.
Citation: Bixiang Wang. Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 269-300. doi: 10.3934/dcds.2014.34.269
##### References:
 [1] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 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), 841-866. doi: 10.1080/03605309208820866.  Google Scholar [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.  Google Scholar [4] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynamical Systems, 10 (2004), 31-52. 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), 1-21. 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), 845-869. 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), 415-443. 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), 525-539. 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, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513.  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), 439-455. 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), 3671-3684. 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), 153-201. 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), 622-653. 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), 2646-2668. 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), 469-512. 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), 127-144. doi: 10.1080/1468936042000207792.  Google Scholar [17] I. Chueshow, "Monotone Random Systems-Theory and Applications," Lecture Notes in Mathematics, 1779, Springer, Berlin, 2001. doi: 10.1007/b83277.  Google Scholar [18] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225.  Google Scholar [19] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. 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), 133-151.  Google Scholar [21] E. Feireisl, Attractors for semilinear damped wave equations on $\mathbbR^3$, Nonlinear Analysis TMA, 23 (1994), 187-195. 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), 431-447. 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), 1539-1555. 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), 21-45.  Google Scholar [25] M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681. 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), 369-388. 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), 2761-2782. doi: 10.1142/S0218127410027349.  Google Scholar [28] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, RI, 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), 89-123. 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), 197-214. 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), 855-882. 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), 702-719. 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), 163-181. 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), 1-23. 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), 49-82.  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), 1-20.  Google Scholar [37] M. Prizzi, Regularity of invariant sets in semilinear damped wave equations, J. Differential Equations, 247 (2009), 3315-3337. doi: 10.1016/j.jde.2009.08.011.  Google Scholar [38] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in "International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior," Dresden, (1992), 185-192. Google Scholar [39] R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer-Verlag, New York, 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), 1432-1457. 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), 2645-2665. 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), 427-443. doi: 10.1016/j.jde.2005.09.010.  Google Scholar [43] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997.  Google Scholar [44] B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. 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), 3639-3663. 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), 1544-1583. 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), 1-18.  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, Real World Applications, 12 (2011), 3468-3482. doi: 10.1016/j.nonrwa.2011.06.008.  Google Scholar

show all references

##### References:
 [1] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 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), 841-866. doi: 10.1080/03605309208820866.  Google Scholar [3] A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.  Google Scholar [4] J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Continuous Dynamical Systems, 10 (2004), 31-52. 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), 1-21. 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), 845-869. 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), 415-443. 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), 525-539. 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, Discrete and Impulsive Systems Series A: Mathematical Analysis, 10 (2003), 491-513.  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), 439-455. 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), 3671-3684. 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), 153-201. 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), 622-653. 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), 2646-2668. 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), 469-512. 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), 127-144. doi: 10.1080/1468936042000207792.  Google Scholar [17] I. Chueshow, "Monotone Random Systems-Theory and Applications," Lecture Notes in Mathematics, 1779, Springer, Berlin, 2001. doi: 10.1007/b83277.  Google Scholar [18] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225.  Google Scholar [19] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. 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), 133-151.  Google Scholar [21] E. Feireisl, Attractors for semilinear damped wave equations on $\mathbbR^3$, Nonlinear Analysis TMA, 23 (1994), 187-195. 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), 431-447. 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), 1539-1555. 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), 21-45.  Google Scholar [25] M. J. Garrido-Atienza and B. Schmalfuss, Ergodicity of the infinite dimensional fractional Brownian motion, J. Dynam. Differential Equations, 23 (2011), 671-681. 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), 369-388. 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), 2761-2782. doi: 10.1142/S0218127410027349.  Google Scholar [28] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, RI, 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), 89-123. 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), 197-214. 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), 855-882. 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), 702-719. 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), 163-181. 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), 1-23. 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), 49-82.  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), 1-20.  Google Scholar [37] M. Prizzi, Regularity of invariant sets in semilinear damped wave equations, J. Differential Equations, 247 (2009), 3315-3337. doi: 10.1016/j.jde.2009.08.011.  Google Scholar [38] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, in "International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior," Dresden, (1992), 185-192. Google Scholar [39] R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer-Verlag, New York, 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), 1432-1457. 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), 2645-2665. 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), 427-443. doi: 10.1016/j.jde.2005.09.010.  Google Scholar [43] R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997.  Google Scholar [44] B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52. 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), 3639-3663. 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), 1544-1583. 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), 1-18.  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, Real World Applications, 12 (2011), 3468-3482. doi: 10.1016/j.nonrwa.2011.06.008.  Google Scholar
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