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

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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

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

Abstract
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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 and 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.
[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.
[3]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	I. Chueshow, "Monotone Random Systems-Theory and Applications," Lecture Notes in Mathematics, 1779, Springer, Berlin, 2001. doi: 10.1007/b83277.
[18]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225.
[19]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[20]	J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.
[21]	E. Feireisl, Attractors for semilinear damped wave equations on $\mathbb{R}^3$, Nonlinear Analysis TMA, 23 (1994), 187-195. doi: 10.1016/0362-546X(94)90041-8.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, RI, 1988.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[39]	R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer-Verlag, New York, 2002.
[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.
[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.
[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.
[43]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997.
[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.
[45]	B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Transactions of American Mathematical Society, 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.
[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.
[47]	B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, arXiv:1205.4658v1.
[48]	B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations, arXiv:1304.4884v1.
[49]	B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 1-18.
[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.

show all references

References:

[1]	L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.
[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.
[3]	A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," North-Holland, Amsterdam, 1992.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	I. Chueshow, "Monotone Random Systems-Theory and Applications," Lecture Notes in Mathematics, 1779, Springer, Berlin, 2001. doi: 10.1007/b83277.
[18]	H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dyn. Diff. Eqns., 9 (1997), 307-341. doi: 10.1007/BF02219225.
[19]	H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Th. Re. Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[20]	J. Duan and B. Schmalfuss, The 3D quasigeostrophic fluid dynamics under random forcing on boundary, Comm. Math. Sci., 1 (2003), 133-151.
[21]	E. Feireisl, Attractors for semilinear damped wave equations on $\mathbb{R}^3$, Nonlinear Analysis TMA, 23 (1994), 187-195. doi: 10.1016/0362-546X(94)90041-8.
[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.
[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.
[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.
[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.
[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.
[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.
[28]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems," American Mathematical Society, Providence, RI, 1988.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[39]	R. Sell and Y. You, "Dynamics of Evolutionary Equations," Springer-Verlag, New York, 2002.
[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.
[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.
[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.
[43]	R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Springer-Verlag, New York, 1997.
[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.
[45]	B. Wang, Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^3$, Transactions of American Mathematical Society, 363 (2011), 3639-3663. doi: 10.1090/S0002-9947-2011-05247-5.
[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.
[47]	B. Wang, Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms, arXiv:1205.4658v1.
[48]	B. Wang, Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations, arXiv:1304.4884v1.
[49]	B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electronic Journal of Differential Equations, 2009 (2009), 1-18.
[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.

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