# American Institute of Mathematical Sciences

July  2017, 22(5): 2011-2051. doi: 10.3934/dcdsb.2017119

## Multivalued non-autonomous random dynamical systems for wave equations without uniqueness

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

Received  July 2015 Revised  April 2016 Published  March 2017

This paper deals with the multivalued non-autonomous random dynamical system generated by the non-autonomous stochastic wave equations on unbounded domains, which has a non-Lipschitz nonlinearity with critical exponent in the three dimensional case. We introduce the concept of weak upper semicontinuity of multivalued functions and use such continuity to prove the measurability of multivalued functions from a metric space to a separable Banach space. By this approach, we show the measurability of pullback attractors of the multivalued random dynamical system of the wave equations regardless of the completeness of the underlying probability space. The asymptotic compactness of solutions is proved by the method of energy equations, and the difficulty caused by the non-compactness of Sobolev embeddings on $\mathbb{R}^n$ is overcome by the uniform estimates on the tails of solutions.

Citation: Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119
##### References:
 [1] L. Arnold, Random Dynamical Systems Springer-Verlag, 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] T. Caraballo, J. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise, J. Math. Anal. Appl., 260 (2001), 602-622. doi: 10.1006/jmaa.2001.7497. Google Scholar [14] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, Journal of Difference Equations and Applications, 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010. 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, 2002.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. doi: 10.4310/CMS.2003.v1.n1.a9. Google Scholar [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. 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. doi: 10.1080/17442509608834083. 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. 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 [30] 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 [31] 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 [32] 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 [33] 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 [34] 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 [35] 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 [36] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185-192, Dresden, 1992.Google Scholar [37] R. Sell and Y. You, Dynamics of Evolutionary Equations Springer-Verlag, New York, 2002.Google Scholar [38] 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 [39] 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 [40] 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 [41] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997.Google Scholar [42] 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 [43] 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. Google Scholar [44] B. Wang, Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains, J. Differential Equations, 246 (2009), 2506-2537. doi: 10.1016/j.jde.2008.10.012. Google Scholar [45] 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 [46] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and Continuous Dynamical Systems, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar [47] Y. Wang and J. Wang, Pullback attractors for multivalued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026. Google Scholar [48] 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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##### References:
 [1] L. Arnold, Random Dynamical Systems Springer-Verlag, 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] T. Caraballo, J. A. Langa and J. Valero, Global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise, J. Math. Anal. Appl., 260 (2001), 602-622. doi: 10.1006/jmaa.2001.7497. Google Scholar [14] T. Caraballo, F. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, Journal of Difference Equations and Applications, 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010. 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, 2002.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. doi: 10.4310/CMS.2003.v1.n1.a9. Google Scholar [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. 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. doi: 10.1080/17442509608834083. 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. 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 [30] 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 [31] 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 [32] 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 [33] 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 [34] 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 [35] 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 [36] B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, 185-192, Dresden, 1992.Google Scholar [37] R. Sell and Y. You, Dynamics of Evolutionary Equations Springer-Verlag, New York, 2002.Google Scholar [38] 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 [39] 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 [40] 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 [41] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997.Google Scholar [42] 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 [43] 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. Google Scholar [44] B. Wang, Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains, J. Differential Equations, 246 (2009), 2506-2537. doi: 10.1016/j.jde.2008.10.012. Google Scholar [45] 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 [46] B. Wang, Random attractors for non-autonomous stochastic wave equations with multiplicative noise, Discrete and Continuous Dynamical Systems, 34 (2014), 269-300. doi: 10.3934/dcds.2014.34.269. Google Scholar [47] Y. Wang and J. Wang, Pullback attractors for multivalued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026. Google Scholar [48] 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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