Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (1): 86-102.
Previous Articles Next Articles
Uniform Attractors for the Sup-Cubic Weakly Damped Wave Equations with Delays
Kaixuan Zhu1,*(
),Yongqin Xie2(),Xinyu Mei3(),Xijun Deng1()
- 1 College of Mathematics and Physics Science, Hunan University of Arts and Science & Hunan Province Cooperative Innovation Center for the Construction and Development of Dongting Lake Ecological Economic Zone, Hunan Changde 415000
2 School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114
3 School of Mathematics and Statistics, Shenzhen University, Guangdong Shenzhen 518060
-
Received:2020-07-27Online:2022-02-26Published:2022-02-23 -
Contact:Kaixuan Zhu E-mail:zhukx12@163.com;xieyq@csust.edu.cn;meixy@szu.edu.cn;xijundeng@126.com -
Supported by:the NFS of Hunan Province(2018JJ2272);the Scientific Research Fund of Hunan Provincial Education Department(20C1263);the Hunan University of Arts and Science(STIT): Numerical Calculation and Stochastic Process with Their Applications
CLC Number:
- O193
Cite this article
Kaixuan Zhu,Yongqin Xie,Xinyu Mei,Xijun Deng. Uniform Attractors for the Sup-Cubic Weakly Damped Wave Equations with Delays[J].Acta mathematica scientia,Series A, 2022, 42(1): 86-102.
share this article
Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks
| 1 | Hale J K , Lunel S M . Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993 |
| 2 | Hale J K. Asymptotic Behavior of Dissipative Systems. Providence: American Mathematical Society, 1988 |
| 3 |
Caraballo T , Kloeden P E , Real J . Pullback and forward attractors for a damped wave equation with delays. Stoch Dyn, 2004, 4: 405- 423
doi: 10.1142/S0219493704001139 |
| 4 |
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 |
| 5 |
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 |
| 6 |
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 |
| 7 |
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 |
| 8 |
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 |
| 9 |
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 |
| 10 |
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
doi: 10.1515/ans-2013-0205 |
| 11 |
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 |
| 12 |
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 |
| 13 |
García-Luengo J , Marín-Rubio P , Real J . Regularity of pullback attractors and attraction in H1 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 |
| 14 |
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 |
| 15 |
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 |
| 16 | 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 |
| 17 |
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 |
| 18 |
Wang Y J . Pullback attractors for a damped wave equation with delays. Stoch Dyn, 2015, 15: 1550003
doi: 10.1142/S0219493715500033 |
| 19 |
Wang J T , Zhao C D , Caraballo T . Invariant measures for the 3D globally modified Navier-Stokes equations with unbounded variable delays. Commun Nonlinear Sci Numer Simul, 2020, 91: 105459
doi: 10.1016/j.cnsns.2020.105459 |
| 20 |
Zhao C D , Liu G W , An R . Global well-posedness and pullback attractors for an incompressible non-Newtonian fluid with infinite delays. Differ Equ Dyn Syst, 2017, 25: 39- 64
doi: 10.1007/s12591-014-0231-9 |
| 21 |
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 |
| 22 |
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 |
| 23 | Lions J L. Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Paris: Dunod, 1969 |
| 24 | Wang C Z , Zhang M S , Zhao C D . Existence of the uniform trajectory attractor for a 3D incompressible non-Newtonian fluid flow. Acta Math Sci, 2018, 38B: 187- 202 |
| 25 |
Zhao C D , Zhou S F , Liao X Y . Uniform attractors for nonautonomous incompressible non-Newtonian fluid with locally uniform integrable external forces. J Math Phys, 2006, 47: 052701
doi: 10.1063/1.2200145 |
| 26 | Zhao C D , Jia X L , Yang X B . Uniform attractor for nonautonomous incompressible non-Newtonian fluid with a new class of external forces. Acta Math Sci, 2011, 31B: 1803- 1812 |
| 27 | Temam R . Infinite-Dimensional Dynamical Systems in Mechanics and Physics. New York: Springer-Verlag, 1997 |
| 28 |
Thanh D T P . Asymptotic behavior of solutions to semilinear parabolic equations with infinite delay. Acta Math Vietnam, 2019, 44: 875- 892
doi: 10.1007/s40306-018-0289-5 |
| 29 |
Arrieta J , Carvalho A N , Hale J K . A damped hyperbolic equation with critical exponent. Comm Partial Differential Equations, 1992, 17: 841- 866
doi: 10.1080/03605309208820866 |
| 30 | Babin A V, Vishik M I. Attractors of Evolution Equations. Amsterdam: North-Holland, 1992 |
| 31 | Ball J M . Global attractors for damped semilinear wave equations. Discrete Contin Dyn Syst, 2004, 10: 31- 52 |
| 32 |
Kalantarov V , Savostianov A , Zelik S . Attractors for damped quintic wave equations in bounded domains. Ann Henri Poincaré, 2016, 17: 2555- 2584
doi: 10.1007/s00023-016-0480-y |
| 33 |
Burq N , Lebeau G , Planchon F . Global existence for energy critical waves in 3-D domains. J Amer Math Soc, 2008, 21: 831- 845
doi: 10.1090/S0894-0347-08-00596-1 |
| 34 |
Blair M D , Smith H F , Sogge C D . Strichartz estimates for the wave equation on manifolds with boundary. Ann Inst H Poincaré Anal Non Linéaire, 2009, 26: 1817- 1829
doi: 10.1016/j.anihpc.2008.12.004 |
| 35 |
Burq N , Planchon F . Global existence for energy critical waves in 3-D domains: Neumann boundary conditions. Amer J Math, 2009, 131: 1715- 1742
doi: 10.1353/ajm.0.0084 |
| 36 | Zhu K X , Xie Y Q , Mei X Y . Pullback attractors for a sup-cubic weakly damped wave equation with delays. Discrete Contin Dyn Syst Ser B, 2021, 26: 4433- 4458 |
| 37 |
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 |
| 38 |
Khanmamedov A K . 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 |
| 39 | Kloeden P E , Lorenz T . Pullback incremental attraction. Nonauton Dyn Syst, 2014, (1): 53- 60 |
| 40 | Sun C Y , Cao D M , Duan J Q . Uniform attractors for nonautonomous wave equations with nonlinear damping. SIAM J Appl Dyn Syst, 2006, 6: 293- 318 |
| 41 |
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 |
| 42 | Zhou F , Sun C Y , Li X . Dynamics for the damped wave equations on time-dependent domains. Discrete Contin Dyn Syst Ser B, 2018, 23: 1645- 1674 |
| 43 | Chepyzhov V V, Vishik M I. Attractors for Equations of Mathematical Physis. Providence: American Mathematical Society, 2002 |
| 44 | Zelik S . Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces. Discrete Contin Dyn Syst B, 2015, 20: 781- 810 |
| 45 | Mei X Y , Sun C Y . Attractors for a sup-cubic weakly damped wave equation in $\mathbb{R} ^{3}$. Discrete Contin Dyn Syst Ser B, 2019, 24: 4117- 4143 |
| 46 |
Mei X Y , Sun C Y . Uniform attractors for a weakly damped wave equation with sup-cubic nonlinearity. Appl Math Lett, 2019, 95: 179- 185
doi: 10.1016/j.aml.2019.04.003 |
| [1] | 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. |
| [2] | Qinghua Zhou,Li Wan,Jie Liu. Global Attracting Set for Neutral Type Hopfield Neural Networks with Time-Varying Delays [J]. Acta mathematica scientia,Series A, 2019, 39(4): 823-831. |
| [3] | Yin Chun, Zhou Shiwei, Wu Shanshan, Cheng Yuhua, Wei Xiuling, Wang Wei. New Unequal Delay Partitioning Methods to Stability Analysis for Neural Networks with Discrete and Distributed Delays [J]. Acta mathematica scientia,Series A, 2017, 37(2): 374-389. |
| [4] | Zhang Shuwen. A Stochastic Predator-Prey System with Time Delays and Prey Dispersal [J]. Acta mathematica scientia,Series A, 2015, 35(3): 592-603. |
| [5] | Luo Ricai, Xu Honglei, Wang Wusheng. Globally Asymptotic Stability of A New Class of Neutral Neural Networks with Time-Varying Delays [J]. Acta mathematica scientia,Series A, 2015, 35(3): 634-640. |
| [6] | YANG Xin-Bo, ZHAO Cai-De, JIA Xiao-Lin. The Uniform Attractor and Entropy of the Autonomous Coupled Nonlinear SchrÖdinger Equations on Infinite Lattices [J]. Acta mathematica scientia,Series A, 2013, 33(4): 636-645. |
| [7] | ZhAO Yong-Xiang, XIAO Ai-Guo. Error Estimate of Parallel Two-step W-methods for Singular-perturbation Problems with Delays [J]. Acta mathematica scientia,Series A, 2011, 31(5): 1239-1252. |
| [8] | LI Jun-Min. Adaptive Iterative Learning Control for Nonlinearly Parameterized Systems with Time-Varying Delays [J]. Acta mathematica scientia,Series A, 2011, 31(3): 682-690. |
| [9] | ZHANG Ruo-Jun, WANG Lin-Shan. Almost Periodic Solutions for Cellular Neural Networks with Distributed Delays [J]. Acta mathematica scientia,Series A, 2011, 31(2): 422-429. |
| [10] | WANG Hong-Chu, HU Shi-Geng. Exponential Stability for Stochastic Neural Networks with Multiple Delays: an LMI Approach [J]. Acta mathematica scientia,Series A, 2010, 30(1): 42-53. |
| [11] |
Liu Bingwen;Hunang Lihong.
Existence and Exponential Stability of Almost Periodic Solutions for Cellular Neural Networks with Delays [J]. Acta mathematica scientia,Series A, 2007, 27(6): 1082-1088. |
| [12] |
Liu Bingwen;Huang Lihong.
eriodic Solutions for a Liénard-type Equation with Delays [J]. Acta mathematica scientia,Series A, 2006, 26(3): 329-336. |
| [13] | ZHANG Qiang, MA Run-Nian. Delay dependent Stability of Cellular Neural Networks with Delays [J]. Acta mathematica scientia,Series A, 2004, 4(6): 694-698. |
|