Acta mathematica scientia,Series A ›› 2020, Vol. 40 ›› Issue (5): 1341-1353.
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Kaixuan Zhu1,*(),Yongqin Xie2(),Feng Zhou3(),Xijun Deng1()
Received:
2019-02-27
Online:
2020-10-26
Published:
2020-11-04
Contact:
Kaixuan Zhu
E-mail:zhukx12@163.com;xieyq@csust.edu.cn;zhoufeng13@upc.edu.cn;xijundeng@126.com
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CLC Number:
Kaixuan Zhu,Yongqin Xie,Feng Zhou,Xijun Deng. Pullback Attractors for the Complex Ginzburg-Landau Equations with Delays[J].Acta mathematica scientia,Series A, 2020, 40(5): 1341-1353.
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1 | Chepyzhov V V, Vishik M I. Attractors for Equations of Mathematical Physics. Providence: American Mathematical Society, 2002 |
2 | Carvalho A N , Langa J A , Robinson J C . Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems. New York: Springer, 2013 |
3 |
Caraballo T , Łukaszewicz G , Real J . Pullback attractors for asymptotically compact non-autonomous dynamical systems. Nonlinear Anal, 2006, 64, 484- 498
doi: 10.1016/j.na.2005.03.111 |
4 |
Caraballo T , Łukaszewica G , Real J . Pullback attractors for non-autonomous 2D-Navier-Stokes equations in some unbounded domains. C R Math Acad Sci Paris, 2006, 342, 263- 268
doi: 10.1016/j.crma.2005.12.015 |
5 |
Chueshov I , Lasiecka I . Long-time dynamics of von Karman semi-flows with non-linear boundary/interior damping. J Differential Equations, 2007, 233, 42- 86
doi: 10.1016/j.jde.2006.09.019 |
6 |
Caraballo T , Real J . Attractors for 2D-Navier-Stokes models with delays. J Differential Equations, 2004, 205, 271- 297
doi: 10.1016/j.jde.2004.04.012 |
7 |
Caraballo T , Marín-Rubio P , Valero J . Autonomous and non-autonomous attractors for differential equations with delays. J Differential Equations, 2005, 208, 9- 41
doi: 10.1016/j.jde.2003.09.008 |
8 |
Caraballo T , Marín-Rubio P , Valero J . Attractors for differential equations with unbounded delays. J Differential Equations, 2007, 239, 311- 342
doi: 10.1016/j.jde.2007.05.015 |
9 |
Caraballo T , Real J , Márquez A M . Three-dimensional system of globally modified Navier-Stokes equations with delay. Internat J Bifur Chaos Appl Sci Engrg, 2010, 20, 2869- 2883
doi: 10.1142/S0218127410027428 |
10 |
Caraballo T , Han X Y , Kloeden P E . Nonautonomous chemostats with variable delays. SIAM J Math Anal, 2015, 47, 2178- 2199
doi: 10.1137/14099930X |
11 |
Caraballo T , Kloeden P E , Marín-Rubio P . Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay. Discrete Contin Dyn Syst, 2007, 19, 177- 196
doi: 10.3934/dcds.2007.19.177 |
12 |
Cross M C , Hohenberg P C . Pattern formation outside of equilibrium. Rev Modern Phys, 1993, 65, 851- 1089
doi: 10.1103/RevModPhys.65.851 |
13 |
García-Luengo J , Marín-Rubio P . Reaction-diffusion equations with non-autonomous force in $H^{-1}$ and delays under measurability conditions on the driving delay term. J Math Anal Appl, 2014, 417, 80- 95
doi: 10.1016/j.jmaa.2014.03.026 |
14 | García-Luengo J , Marín-Rubio P , Real J . Pullback attractors for 2D Navier-Stokes equations with delays and their regularity. Adv Nonlinear Stud, 2013, 13, 331- 357 |
15 |
García-Luengo J , Marín-Rubio P , Real J . Some new regularity results of pullback attractors for 2D Navier-Stokes equations with delays. Commun Pure Appl Anal, 2015, 14, 1603- 1621
doi: 10.3934/cpaa.2015.14.1603 |
16 |
García-Luengo J , Marín-Rubio P , Planas G . Attractors for a double time-delayed 2D-Navier-Stokes model. Discrete Contin Dyn Syst, 2014, 34, 4085- 4105
doi: 10.3934/dcds.2014.34.4085 |
17 |
García-Luengo J , Marín-Rubio P , Real J . Regularity of pullback attractors and attraction in $H^{1}$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay. Discrete Contin Dyn Syst, 2014, 34, 181- 201
doi: 10.3934/dcds.2014.34.181 |
18 | Hale J K. Asymptotic Behavior of Dissipative Systems. Providence: American Mathematical Society, 1988 |
19 | Hale J K , Verduyn Lunel S M . Introduction to Functional Differential Equations. New York: Springer, 1993 |
20 |
Khanmamedov A Kh . Global attractors for von Karman equations with nonlinear interior dissipation. J Math Anal Appl, 2006, 318, 92- 101
doi: 10.1016/j.jmaa.2005.05.031 |
21 |
Kloeden P E . Upper semi continuity of attractors of delay differential equations in the delay. Bull Austral Math Soc, 2006, 73, 299- 306
doi: 10.1017/S0004972700038880 |
22 |
Kloeden P E , Marín-Rubio P . Equi-attraction and the continuous dependence of attractors on time delays. Discrete Contin Dyn Syst Ser B, 2008, 9, 581- 593
doi: 10.3934/dcdsb.2008.9.581 |
23 |
Kapustyan O V , Valero J . On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion. J Math Anal Appl, 2009, 357, 254- 272
doi: 10.1016/j.jmaa.2009.04.010 |
24 | Lions J L. Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Paris: Dunod, 1969 |
25 |
Li F , You B . Global attractors for the complex Ginzburg-Landau equation. J Math Anal Appl, 2014, 415, 14- 24
doi: 10.1016/j.jmaa.2014.01.059 |
26 | Marín-Rubio P , Márquez-Durán A M , Real J . Three dimensional system of globally modified Navier-Stokes equations with infinite delays. Discrete Contin Dyn Syst Ser B, 2010, 14, 655- 673 |
27 |
Marín-Rubio P , Márquez-Durán A M , Real J . Pullback attractors for globally modified Navier-Stokes equations with infinite delays. Discrete Contin Dyn Syst, 2011, 31, 779- 796
doi: 10.3934/dcds.2011.31.779 |
28 |
Moon H T , Huerre P , Redekopp L G . Transitions to chaos in the Ginzburg-Landau equation. Phys D, 1983, 7, 135- 150
doi: 10.1016/0167-2789(83)90124-0 |
29 |
Newell A C , Whitehead J A . Finite bandwidth, finite amplitude convection. J Fluid Mech, 1969, 38, 279- 304
doi: 10.1017/S0022112069000176 |
30 |
Okazawa N , Yokota T . Monotonicity method applied to the complex Ginzburg-Landau and related equations. J Math Anal Appl, 2002, 267, 247- 263
doi: 10.1006/jmaa.2001.7770 |
31 |
Sun C Y , Cao D M , Duan J Q . Non-autonomous dynamics of wave equations with nonlinear damping and critical nonlinearity. Nonlinearity, 2006, 19, 2645- 2665
doi: 10.1088/0951-7715/19/11/008 |
32 |
Wang Y J , Kloeden P E . The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete Contin Dyn Syst, 2014, 34, 4343- 4370
doi: 10.3934/dcds.2014.34.4343 |
33 | Wu F , Kloeden P E . Mean-square random attractors of stochastic delay differential equations with random delay. Discrete Contin Dyn Syst Ser B, 2013, 18, 1715- 1734 |
34 |
Xie Y Q , Li Q S , Zhu K X . Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity. Nonlinear Anal Real World Appl, 2016, 31, 23- 37
doi: 10.1016/j.nonrwa.2016.01.004 |
35 |
Zhou F , Sun C Y . Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains Ⅰ: the diffeomorphism case. Discrete Contin Dyn Syst Ser B, 2016, 21, 3767- 3792
doi: 10.3934/dcdsb.2016120 |
36 |
Zhou F , Sun C Y , Cheng J Q . Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains Ⅱ: The monotone case. J Math Phys, 2018, 59, 022703
doi: 10.1063/1.5024214 |
37 |
Zhu K X , Xie Y Q , Zhou F . Pullback attractors for a damped semilinear wave equation with delays. Acta Math Sin, 2018, 34, 1131- 1150
doi: 10.1007/s10114-018-7420-3 |
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