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 & Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116
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show all references

References:
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Ann. Polon. Math., 111 (2014), 271-295. doi: 10.4064/ap111-3-5.  Google Scholar

[2]

Springer-Verlag, New York, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar

[3]

Birkhäuser, Bosten, Basel, Berlin, 1990.  Google Scholar

[4]

J. Differ. Equ., 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.  Google Scholar

[5]

Set-Valued Anal., 11 (2003), 153-201. doi: 10.1023/A:1022902802385.  Google Scholar

[6]

Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 491-513.  Google Scholar

[7]

Discrete Contin. Dyn. Syst., 21 (2008), 415-443. doi: 10.3934/dcds.2008.21.415.  Google Scholar

[8]

Discrete Contin. Dyn. Syst., 14 (2010), 439-455. doi: 10.3934/dcdsb.2010.14.439.  Google Scholar

[9]

American Mathematical Society, Providence, RI, 2002.  Google Scholar

[10]

Springer-Verlag, New York, 2002. doi: 10.1007/b83277.  Google Scholar

[11]

Set-Valued Var. Anal., 21 (2013), 127-149. doi: 10.1007/s11228-012-0215-2.  Google Scholar

[12]

SIAM J. Math. Anal., 47 (2015), 1530-1561. doi: 10.1137/140978995.  Google Scholar

[13]

J. Dynam. Differ. Equ., 9 (1997), 307-341. doi: 10.1007/BF02219225.  Google Scholar

[14]

De Gruyter, Berlin, 1992. doi: 10.1515/9783110874228.  Google Scholar

[15]

Ann. Probab., 31 (2003), 2109-2135. doi: 10.1214/aop/1068646380.  Google Scholar

[16]

Stoch. Stoch. Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.  Google Scholar

[17]

J. Differ. Equ., 123 (1995), 56-92. doi: 10.1006/jdeq.1995.1157.  Google Scholar

[18]

Abstr. Appl. Anal., 5 (2000), 33-46. doi: 10.1155/S1085337500000191.  Google Scholar

[19]

Systems Control Lett., 33 (1998), 275-280. doi: 10.1016/S0167-6911(97)00107-2.  Google Scholar

[20]

Bull. Austral. Math. Soc., 73 (2006), 299-306. doi: 10.1017/S0004972700038880.  Google Scholar

[21]

Proceedings of the Royal Society of London, 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753.  Google Scholar

[22]

Set-Valued Anal., 16 (2008), 651-671. doi: 10.1007/s11228-007-0054-8.  Google Scholar

[23]

Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255.  Google Scholar

[24]

Set-Valued Anal., 6 (1998), 83-111. doi: 10.1023/A:1008608431399.  Google Scholar

[25]

Set-Valued Anal., 8 (2000), 375-403. doi: 10.1023/A:1026514727329.  Google Scholar

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2nd Edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[28]

Electron. J. Differ. Equ., 139 (2009), 1-18.  Google Scholar

[29]

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[31]

SIAM J. Appl. Dyn. Syst., 14 (2015), 1018-1047. doi: 10.1137/140991819.  Google Scholar

[32]

J. Math. Phys., 51 (2010), 022701, 12 pp. doi: 10.1063/1.3277152.  Google Scholar

[33]

J. Differ. Equ., 232 (2007), 573-622. doi: 10.1016/j.jde.2006.07.005.  Google Scholar

[34]

Quarterly of Applied Mathematics, 71 (2013), 369-399. doi: 10.1090/S0033-569X-2013-01306-1.  Google Scholar

[35]

J. Differ. Equ., 259 (2015), 728-776. doi: 10.1016/j.jde.2015.02.026.  Google Scholar

[36]

J. Dynam. Differential Equations, 24 (2012), 495-520. doi: 10.1007/s10884-012-9252-7.  Google Scholar

[37]

Discrete Contin. Dyn. Syst., 34 (2014), 301-333. doi: 10.3934/dcds.2014.34.301.  Google Scholar

[38]

J. Differ. Equ., 223 (2006), 367-399. doi: 10.1016/j.jde.2005.06.008.  Google Scholar

[39]

Nonlinear Anal. TMA, 75 (2012), 485-502. doi: 10.1016/j.na.2011.08.050.  Google Scholar

[40]

Nonlinear Anal. TMA, 75 (2012), 2793-2805. doi: 10.1016/j.na.2011.11.022.  Google Scholar

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