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.

[2]

J. M. ArrietaA. 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. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.

[6]

P. W. BatesK. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoJ. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoM. 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. CaraballoJ. A. LangaV. 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]

T. CaraballoJ. 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.

[14]

T. CaraballoF. 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.

[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, 2002.

[18]

H. CrauelA. 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. doi: 10.4310/CMS.2003.v1.n1.a9.

[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. doi: 10.1080/17442509608834083.

[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-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.

[27]

M. J. Garrido-AtienzaB. 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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

R. Sell and Y. You, Dynamics of Evolutionary Equations Springer-Verlag, New York, 2002.

[38]

Z. ShenS. 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.

[39]

C. SunD. 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.

[40]

C. SunM. 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.

[41]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997.

[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.

[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.

[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.

[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.

[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.

[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.

[48]

Z. WangS. 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-Verlag, 1998.

[2]

J. M. ArrietaA. 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. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.

[6]

P. W. BatesK. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoJ. 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. CaraballoM. J. Garrido-AtienzaB. 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. CaraballoM. 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. CaraballoJ. A. LangaV. 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]

T. CaraballoJ. 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.

[14]

T. CaraballoF. 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.

[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, 2002.

[18]

H. CrauelA. 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. doi: 10.4310/CMS.2003.v1.n1.a9.

[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. doi: 10.1080/17442509608834083.

[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-AtienzaA. Ogrowsky and B. Schmalfuss, Random differential equations with random delays, Stoch. Dyn., 11 (2011), 369-388. doi: 10.1142/S0219493711003358.

[27]

M. J. Garrido-AtienzaB. 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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

R. Sell and Y. You, Dynamics of Evolutionary Equations Springer-Verlag, New York, 2002.

[38]

Z. ShenS. 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.

[39]

C. SunD. 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.

[40]

C. SunM. 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.

[41]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics Springer-Verlag, New York, 1997.

[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.

[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.

[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.

[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.

[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.

[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.

[48]

Z. WangS. 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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