American Institute of Mathematical Sciences

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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

Yejuan Wang 1,

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.
Keywords: reaction-diffusion equation with delay, upper semicontinuity., Multi-valued noncompact random dynamical system, pullback attractor, nonclassical diffusion equation.
Mathematics Subject Classification: Primary: 34K26, 35K57, 37L5.
Citation: Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete & 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 $H^1(\mathbbR^N)$ for nonautonomous nonclassical diffusion equations,, Ann. Polon. Math., 111 (2014), 271. doi: 10.4064/ap111-3-5. Google Scholar

[2]

L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar

[3]

J. P. Aubin and H. Franskowska, Set-Valued Analysis,, Birkhäuser, (1990). Google Scholar

[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. doi: 10.1016/j.jde.2008.05.017. Google Scholar

[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. doi: 10.1023/A:1022902802385. Google Scholar

[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. 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 Contin. Dyn. Syst., 21 (2008), 415. doi: 10.3934/dcds.2008.21.415. Google Scholar

[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. doi: 10.3934/dcdsb.2010.14.439. Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002). Google Scholar

[10]

I. D. Chueshov, Monotone Random Systems Theory and Applications,, Springer-Verlag, (2002). doi: 10.1007/b83277. Google Scholar

[11]

M. Coti Zelati, On the theory of global attractors and Lyapunov functionals,, Set-Valued Var. Anal., 21 (2013), 127. doi: 10.1007/s11228-012-0215-2. Google Scholar

[12]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors,, SIAM J. Math. Anal., 47 (2015), 1530. doi: 10.1137/140978995. Google Scholar

[13]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differ. Equ., 9 (1997), 307. doi: 10.1007/BF02219225. Google Scholar

[14]

K. Deimling, Multivalued Differential Equations,, De Gruyter, (1992). doi: 10.1515/9783110874228. Google Scholar

[15]

J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations,, Ann. Probab., 31 (2003), 2109. doi: 10.1214/aop/1068646380. Google Scholar

[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. doi: 10.1080/17442509608834083. Google Scholar

[17]

G. Hines, Upper semicontinuity of the attractor with respect to parameter dependent delays,, J. Differ. Equ., 123 (1995), 56. doi: 10.1006/jdeq.1995.1157. Google Scholar

[18]

A. V. Kapustian and J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations,, Abstr. Appl. Anal., 5 (2000), 33. doi: 10.1155/S1085337500000191. Google Scholar

[19]

P. E. Kloeden and B. Schmalfuß, Asymptotic behaviour of nonautonomous difference inclusions,, Systems Control Lett., 33 (1998), 275. doi: 10.1016/S0167-6911(97)00107-2. Google Scholar

[20]

P. E. Kloeden, Upper semicontinuity of attractors of delay differential equations in the delay,, Bull. Austral. Math. Soc., 73 (2006), 299. doi: 10.1017/S0004972700038880. Google Scholar

[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. doi: 10.1098/rspa.2006.1753. Google Scholar

[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. doi: 10.1007/s11228-007-0054-8. Google Scholar

[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. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[24]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83. doi: 10.1023/A:1008608431399. Google Scholar

[25]

V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions,, Set-Valued Anal., 8 (2000), 375. doi: 10.1023/A:1026514727329. Google Scholar

[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. doi: 10.1007/s10884-007-9089-7. Google Scholar

[27]

R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics,, 2nd Edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[28]

B. X. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems,, Electron. J. Differ. Equ., 139 (2009), 1. Google Scholar

[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. doi: 10.1016/j.jde.2012.05.015. Google Scholar

[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). doi: 10.1063/1.4817862. Google Scholar

[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. doi: 10.1137/140991819. Google Scholar

[32]

Y. H. Wang and Y. M. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equation,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3277152. Google Scholar

[33]

Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses,, J. Differ. Equ., 232 (2007), 573. doi: 10.1016/j.jde.2006.07.005. Google Scholar

[34]

Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes,, Quarterly of Applied Mathematics, 71 (2013), 369. doi: 10.1090/S0033-569X-2013-01306-1. Google Scholar

[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. doi: 10.1016/j.jde.2015.02.026. Google Scholar

[36]

Y. C. You, Robustness of global attractors for reversible Gray-Scott systems,, J. Dynam. Differential Equations, 24 (2012), 495. doi: 10.1007/s10884-012-9252-7. Google Scholar

[37]

Y. C. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems,, Discrete Contin. Dyn. Syst., 34 (2014), 301. doi: 10.3934/dcds.2014.34.301. Google Scholar

[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. doi: 10.1016/j.jde.2005.06.008. Google Scholar

[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. doi: 10.1016/j.na.2011.08.050. Google Scholar

[40]

S. F. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises,, Nonlinear Anal. TMA, 75 (2012), 2793. doi: 10.1016/j.na.2011.11.022. Google Scholar

show all references

References:
[1]

C. T. Anh and N. D. Toan, Existence and upper semicontinuity of uniform attractors in $H^1(\mathbbR^N)$ for nonautonomous nonclassical diffusion equations,, Ann. Polon. Math., 111 (2014), 271. doi: 10.4064/ap111-3-5. Google Scholar

[2]

L. Arnold, Random Dynamical Systems,, Springer-Verlag, (1998). doi: 10.1007/978-3-662-12878-7. Google Scholar

[3]

J. P. Aubin and H. Franskowska, Set-Valued Analysis,, Birkhäuser, (1990). Google Scholar

[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. doi: 10.1016/j.jde.2008.05.017. Google Scholar

[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. doi: 10.1023/A:1022902802385. Google Scholar

[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. 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 Contin. Dyn. Syst., 21 (2008), 415. doi: 10.3934/dcds.2008.21.415. Google Scholar

[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. doi: 10.3934/dcdsb.2010.14.439. Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002). Google Scholar

[10]

I. D. Chueshov, Monotone Random Systems Theory and Applications,, Springer-Verlag, (2002). doi: 10.1007/b83277. Google Scholar

[11]

M. Coti Zelati, On the theory of global attractors and Lyapunov functionals,, Set-Valued Var. Anal., 21 (2013), 127. doi: 10.1007/s11228-012-0215-2. Google Scholar

[12]

M. Coti Zelati and P. Kalita, Minimality properties of set-valued processes and their pullback attractors,, SIAM J. Math. Anal., 47 (2015), 1530. doi: 10.1137/140978995. Google Scholar

[13]

H. Crauel, A. Debussche and F. Flandoli, Random attractors,, J. Dynam. Differ. Equ., 9 (1997), 307. doi: 10.1007/BF02219225. Google Scholar

[14]

K. Deimling, Multivalued Differential Equations,, De Gruyter, (1992). doi: 10.1515/9783110874228. Google Scholar

[15]

J. Duan, K. Lu and B. Schmalfuß, Invariant manifolds for stochastic partial differential equations,, Ann. Probab., 31 (2003), 2109. doi: 10.1214/aop/1068646380. Google Scholar

[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. doi: 10.1080/17442509608834083. Google Scholar

[17]

G. Hines, Upper semicontinuity of the attractor with respect to parameter dependent delays,, J. Differ. Equ., 123 (1995), 56. doi: 10.1006/jdeq.1995.1157. Google Scholar

[18]

A. V. Kapustian and J. Valero, Attractors of multivalued semiflows generated by differential inclusions and their approximations,, Abstr. Appl. Anal., 5 (2000), 33. doi: 10.1155/S1085337500000191. Google Scholar

[19]

P. E. Kloeden and B. Schmalfuß, Asymptotic behaviour of nonautonomous difference inclusions,, Systems Control Lett., 33 (1998), 275. doi: 10.1016/S0167-6911(97)00107-2. Google Scholar

[20]

P. E. Kloeden, Upper semicontinuity of attractors of delay differential equations in the delay,, Bull. Austral. Math. Soc., 73 (2006), 299. doi: 10.1017/S0004972700038880. Google Scholar

[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. doi: 10.1098/rspa.2006.1753. Google Scholar

[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. doi: 10.1007/s11228-007-0054-8. Google Scholar

[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. doi: 10.1512/iumj.2002.51.2255. Google Scholar

[24]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions,, Set-Valued Anal., 6 (1998), 83. doi: 10.1023/A:1008608431399. Google Scholar

[25]

V. S. Melnik and J. Valero, On global attractors of multivalued semiprocesses and nonautonomous evolution inclusions,, Set-Valued Anal., 8 (2000), 375. doi: 10.1023/A:1026514727329. Google Scholar

[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. doi: 10.1007/s10884-007-9089-7. Google Scholar

[27]

R. Temam, Infinite Dimensional Dynamical System in Mechanics and Physics,, 2nd Edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar

[28]

B. X. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems,, Electron. J. Differ. Equ., 139 (2009), 1. Google Scholar

[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. doi: 10.1016/j.jde.2012.05.015. Google Scholar

[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). doi: 10.1063/1.4817862. Google Scholar

[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. doi: 10.1137/140991819. Google Scholar

[32]

Y. H. Wang and Y. M. Qin, Upper semicontinuity of pullback attractors for nonclassical diffusion equation,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3277152. Google Scholar

[33]

Y. J. Wang and S. F. Zhou, Kernel sections and uniform attractors of multi-valued semiprocesses,, J. Differ. Equ., 232 (2007), 573. doi: 10.1016/j.jde.2006.07.005. Google Scholar

[34]

Y. J. Wang, On the upper semicontinuity of pullback attractors for multi-valued processes,, Quarterly of Applied Mathematics, 71 (2013), 369. doi: 10.1090/S0033-569X-2013-01306-1. Google Scholar

[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. doi: 10.1016/j.jde.2015.02.026. Google Scholar

[36]

Y. C. You, Robustness of global attractors for reversible Gray-Scott systems,, J. Dynam. Differential Equations, 24 (2012), 495. doi: 10.1007/s10884-012-9252-7. Google Scholar

[37]

Y. C. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems,, Discrete Contin. Dyn. Syst., 34 (2014), 301. doi: 10.3934/dcds.2014.34.301. Google Scholar

[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. doi: 10.1016/j.jde.2005.06.008. Google Scholar

[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. doi: 10.1016/j.na.2011.08.050. Google Scholar

[40]

S. F. Zhou, Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises,, Nonlinear Anal. TMA, 75 (2012), 2793. doi: 10.1016/j.na.2011.11.022. Google Scholar

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