# American Institute of Mathematical Sciences

December  2016, 21(10): 3669-3708. doi: 10.3934/dcdsb.2016116

## On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems

 1 School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000

Received  November 2015 Revised  April 2016 Published  November 2016

In this paper, we first present some conditions for the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems with small perturbations. Then we establish the upper semicontinuity of pullback attractors for nonautonomous and stochastic reaction-diffusion delay equations defined on $\mathbb{R}^n$ with small time delay perturbation for which uniqueness of solutions need not hold. Finally, we prove the existence and upper semicontinuity of pullback attractors for nonautonomous and stochastic nonclassical diffusion equations with polynomial growth nonlinearity of arbitrary order and without the uniqueness of solutions.
Citation: Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116
##### References:
 [1] C. T. Anh and N. D. Toan, Existence and upper semicontinuity of uniform attractors in $H^1(\mathbb{R}^N)$ for nonautonomous nonclassical diffusion equations, Ann. Polon. Math., 111 (2014), 271-295. doi: 10.4064/ap111-3-5. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [3] J. P. Aubin and H. Franskowska, Set-Valued Analysis, Birkhäuser, Bosten, Basel, Berlin, 1990. [4] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017. [5] T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385. [6] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513. [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 Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415. [8] 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., 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439. [9] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. [10] I. D. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002. doi: 10.1007/b83277. [11] M. Coti Zelati, On the theory of global attractors and Lyapunov functionals, Set-Valued Var. Anal., 21 (2013), 127-149. doi: 10.1007/s11228-012-0215-2. [12] M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561. doi: 10.1137/140978995. [13] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225. [14] K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992. doi: 10.1515/9783110874228. [15] J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380. [16] F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. [17] G. Hines, Upper semicontinuity of the attractor with respect to parameter dependent delays, J. Differ. Equ., 123 (1995), 56-92. doi: 10.1006/jdeq.1995.1157. [18] A. V. Kapustian and J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstr. Appl. Anal., 5 (2000), 33-46. doi: 10.1155/S1085337500000191. [19] P. E. Kloeden and B. Schmalfuß, Asymptotic behaviour of nonautonomous difference inclusions, Systems Control Lett., 33 (1998), 275-280. doi: 10.1016/S0167-6911(97)00107-2. [20] P. E. Kloeden, Upper semicontinuity of attractors of delay differential equations in the delay, Bull. Austral. Math. Soc., 73 (2006), 299-306. doi: 10.1017/S0004972700038880. [21] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proceedings of the Royal Society of London, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. [22] D. S. Li, Y. J. Wang and S. Y. Wang, On the dynamics of nonautonomous general dynamical systems and differential inclusions, Set-Valued Anal., 16 (2008), 651-671. doi: 10.1007/s11228-007-0054-8. [23] Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. [24] V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399. [25] V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403. doi: 10.1023/A:1026514727329. [26] B. Schmalfuß and K. R. Schneider, Invariant manifolds for random dynamical systems with slow and fast variables, J. Dynam. Differential Equations, 20 (2008), 133-164. doi: 10.1007/s10884-007-9089-7. [27] R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, 2nd Edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [28] B. X. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differ. Equ., 139 (2009), 1-18. [29] B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. [30] J. Y. Wang and Y. J. Wang, Pullback attractors for reaction-diffusion delay equations on unbounded domains with non-autonomous deterministic and stochastic forcing terms, J. Math. Phys., 54 (2013), 082703, 25 pp. doi: 10.1063/1.4817862. [31] X. H. Wang, K. N. Lu and B. X. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047. doi: 10.1137/140991819. [32] Y. H. Wang and Y. M. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equation, J. Math. Phys., 51 (2010), 022701, 12 pp. doi: 10.1063/1.3277152. [33] Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differ. Equ., 232 (2007), 573-622. doi: 10.1016/j.jde.2006.07.005. [34] Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes, Quarterly of Applied Mathematics, 71 (2013), 369-399. doi: 10.1090/S0033-569X-2013-01306-1. [35] Y. J. Wang and J. Y. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differ. Equ., 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026. [36] Y. C. You, Robustness of global attractors for reversible Gray-Scott systems, J. Dynam. Differential Equations, 24 (2012), 495-520. doi: 10.1007/s10884-012-9252-7. [37] Y. C. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Discrete Contin. Dyn. Syst., 34 (2014), 301-333. doi: 10.3934/dcds.2014.34.301. [38] C. K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equ., 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008. [39] W. Q. Zhao and Y. R. Li, $(L^2, L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal. TMA, 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050. [40] S. F. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises, Nonlinear Anal. TMA, 75 (2012), 2793-2805. doi: 10.1016/j.na.2011.11.022.

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##### References:
 [1] C. T. Anh and N. D. Toan, Existence and upper semicontinuity of uniform attractors in $H^1(\mathbb{R}^N)$ for nonautonomous nonclassical diffusion equations, Ann. Polon. Math., 111 (2014), 271-295. doi: 10.4064/ap111-3-5. [2] L. Arnold, Random Dynamical Systems, Springer-Verlag, New York, Berlin, 1998. doi: 10.1007/978-3-662-12878-7. [3] J. P. Aubin and H. Franskowska, Set-Valued Analysis, Birkhäuser, Bosten, Basel, Berlin, 1990. [4] P. W. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differ. Equ., 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017. [5] T. Caraballo, J. A. Langa, V. S. Melnik and J. Valero, Pullback attractors of nonautonomous and stochastic multivalued dynamical systems, Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385. [6] T. Caraballo and J. A. Langa, On the upper semicontinuity of cocycle attractors for non-autonomous and random dynamical systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513. [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 Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415. [8] 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., 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439. [9] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, RI, 2002. [10] I. D. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, New York, 2002. doi: 10.1007/b83277. [11] M. Coti Zelati, On the theory of global attractors and Lyapunov functionals, Set-Valued Var. Anal., 21 (2013), 127-149. doi: 10.1007/s11228-012-0215-2. [12] M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors, SIAM J. Math. Anal., 47 (2015), 1530-1561. doi: 10.1137/140978995. [13] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225. [14] K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992. doi: 10.1515/9783110874228. [15] J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380. [16] F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083. [17] G. Hines, Upper semicontinuity of the attractor with respect to parameter dependent delays, J. Differ. Equ., 123 (1995), 56-92. doi: 10.1006/jdeq.1995.1157. [18] A. V. Kapustian and J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations, Abstr. Appl. Anal., 5 (2000), 33-46. doi: 10.1155/S1085337500000191. [19] P. E. Kloeden and B. Schmalfuß, Asymptotic behaviour of nonautonomous difference inclusions, Systems Control Lett., 33 (1998), 275-280. doi: 10.1016/S0167-6911(97)00107-2. [20] P. E. Kloeden, Upper semicontinuity of attractors of delay differential equations in the delay, Bull. Austral. Math. Soc., 73 (2006), 299-306. doi: 10.1017/S0004972700038880. [21] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proceedings of the Royal Society of London, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. [22] D. S. Li, Y. J. Wang and S. Y. Wang, On the dynamics of nonautonomous general dynamical systems and differential inclusions, Set-Valued Anal., 16 (2008), 651-671. doi: 10.1007/s11228-007-0054-8. [23] Q. F. Ma, S. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. [24] V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399. [25] V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions, Set-Valued Anal., 8 (2000), 375-403. doi: 10.1023/A:1026514727329. [26] B. Schmalfuß and K. R. Schneider, Invariant manifolds for random dynamical systems with slow and fast variables, J. Dynam. Differential Equations, 20 (2008), 133-164. doi: 10.1007/s10884-007-9089-7. [27] R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics, 2nd Edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [28] B. X. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differ. Equ., 139 (2009), 1-18. [29] B. X. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differ. Equ., 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015. [30] J. Y. Wang and Y. J. Wang, Pullback attractors for reaction-diffusion delay equations on unbounded domains with non-autonomous deterministic and stochastic forcing terms, J. Math. Phys., 54 (2013), 082703, 25 pp. doi: 10.1063/1.4817862. [31] X. H. Wang, K. N. Lu and B. X. Wang, Random attractors for delay parabolic equations with additive noise and deterministic nonautonomous forcing, SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047. doi: 10.1137/140991819. [32] Y. H. Wang and Y. M. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equation, J. Math. Phys., 51 (2010), 022701, 12 pp. doi: 10.1063/1.3277152. [33] Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses, J. Differ. Equ., 232 (2007), 573-622. doi: 10.1016/j.jde.2006.07.005. [34] Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes, Quarterly of Applied Mathematics, 71 (2013), 369-399. doi: 10.1090/S0033-569X-2013-01306-1. [35] Y. J. Wang and J. Y. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differ. Equ., 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026. [36] Y. C. You, Robustness of global attractors for reversible Gray-Scott systems, J. Dynam. Differential Equations, 24 (2012), 495-520. doi: 10.1007/s10884-012-9252-7. [37] Y. C. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Discrete Contin. Dyn. Syst., 34 (2014), 301-333. doi: 10.3934/dcds.2014.34.301. [38] C. K. Zhong, M. H. Yang and C. Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equ., 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008. [39] W. Q. Zhao and Y. R. Li, $(L^2, L^p)$-random attractors for stochastic reaction-diffusion equation on unbounded domains, Nonlinear Anal. TMA, 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050. [40] S. F. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises, Nonlinear Anal. TMA, 75 (2012), 2793-2805. doi: 10.1016/j.na.2011.11.022.
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