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Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing
Uniform attractor of the non-autonomous discrete Selkov model
1. | Department of Mathematics and Information Science, Wenzhou University, Zhejiang Province, 325035, China, China |
2. | College of Teacher Education, Wenzhou University, Zhejiang Province, 325035, China |
References:
[1] |
Ahmed Y. Abdallah, Uniform exponential attractor for first order non-autonomous lattice dynamical systems, J. Differential Equations, 251 (2011), 1489-1504.
doi: 10.1016/j.jde.2011.05.030. |
[2] |
Ahmed Y. Abdallah, Exponential attractors for first-order lattice dynamical systems, J. Math. Anal. Appl., 339 (2008), 217-224.
doi: 10.1016/j.jmaa.2007.06.054. |
[3] |
P. W. Bates, X. Chen and A. Chmaj, Travelling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.
doi: 10.1137/S0036141000374002. |
[4] |
P. W. Bates, H. Lisei and K. Lu, Attrators for stochastic lattice dynamical systems, Stoch. Dyna., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[5] |
P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Inter. J. Bifur. Chaos, 11 (2001), 143-153.
doi: 10.1142/S0218127401002031. |
[6] |
W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Diff. Eqs., 15 (2003), 485-515.
doi: 10.1023/B:JODY.0000009745.41889.30. |
[7] |
H. Chate and M. Courbage, Lattice systems, Phys. D, 103 (1997), 1-612.
doi: 10.1016/S0167-2789(96)00249-7. |
[8] |
S.-N. Chow, Lattice dynamical systems, in "Dynamical Systems," Lecture Notes in Math., 1822, Springer, Berlin, (2003), 1-102.
doi: 10.1007/978-3-540-45204-1_1. |
[9] |
S.-N. Chow and J. M. Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751.
doi: 10.1109/81.473583. |
[10] |
T. L. Carrol and L. M. Pecora, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.
doi: 10.1103/PhysRevLett.64.821. |
[11] |
S.-N. Chow, J. M. Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.
doi: 10.1006/jdeq.1998.3478. |
[12] |
S.-N. Chow, J. M. Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comp. Dyna., 4 (1996), 109-178. |
[13] |
L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156.
doi: 10.1109/81.222795. |
[14] |
L. O. Chua and Y. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272.
doi: 10.1109/31.7600. |
[15] |
V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002. |
[16] |
T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Phys. D, 67 (1993), 237-244.
doi: 10.1016/0167-2789(93)90208-I. |
[17] |
L. Fabiny, P. Colet and R. Roy, Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 47 (1993), 4287-4296.
doi: 10.1103/PhysRevA.47.4287. |
[18] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Math. Surveys and Monographs, AMS, Providence, RI, 1988. |
[19] |
M. Hillert, A solid-solution model for inhomogeneous systems, Acta Metall., 9 (1961), 525-535.
doi: 10.1016/0001-6160(61)90155-9. |
[20] |
X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024, 20 pp.
doi: 10.1142/S0219493711500249. |
[21] |
X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.
doi: 10.1016/j.jde.2010.10.018. |
[22] |
X. Jia, C. Zhao and X. Yang, Global attractor and Kolmogorov entropy of three component reversible Gray-Scott model on infinite lattices, Appl. Math. Comp., 218 (2012), 9781-9789.
doi: 10.1016/j.amc.2012.03.036. |
[23] |
J. P. Keener, Propagation and its failure in coupled systems of discret excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.
doi: 10.1137/0147038. |
[24] |
R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.
doi: 10.1007/BF01192578. |
[25] |
N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differntial Equations, 217 (2005), 88-123.
doi: 10.1016/j.jde.2005.06.002. |
[26] |
O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[27] |
Y. Lv and J. H. Sun, Dynamical behavior for stochastic lattice systems, Chaos Soli. Fract., 27 (2006), 1080-1090.
doi: 10.1016/j.chaos.2005.04.089. |
[28] |
P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonl. Analysis, 71 (2009), 3956-3963.
doi: 10.1016/j.na.2009.02.065. |
[29] |
E. E. Selkov, Self-oscillations in glycolysis: A simple kinetic model, Euorpean J. Bio., 4 (1968), 79-86. |
[30] |
G. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. |
[31] |
R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin, 1997. |
[32] |
T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multipliative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.
doi: 10.1016/j.jde.2012.03.020. |
[33] |
T. Caraballo, F. Morillas and J. Valero, Random attractors for sotchastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184.
doi: 10.1080/10236198.2010.549010. |
[34] |
T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.
doi: 10.1007/s11464-008-0028-7. |
[35] |
E. V. Vlecka and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D, 212 (2005), 317-336.
doi: 10.1016/j.physd.2005.10.006. |
[36] |
B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[37] |
B. Wang, Asymptotic behavior of non-autonomous lattice system, J. Math. Anal. Appl., 331 (2007), 121-136.
doi: 10.1016/j.jmaa.2006.08.070. |
[38] |
B. Wang, Uniform attractors of non-autonomous discret reaction-diffusion systems in weighted spaces, Inter. J. Bifur. Chaos, 18 (2008), 695-716.
doi: 10.1142/S0218127408020598. |
[39] |
R. L. Winaow, A. L. Kimball and A. Varghese, Simulating cardiac sinus and atrial network dynamics on connection machine, Phys. D, 64 (1993), 281-298. |
[40] |
Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Syst. (S), 2 (2009), 193-219.
doi: 10.3934/dcdss.2009.2.193. |
[41] |
S. Zhou, Attractors for first order dissipative lattice dynamical systems, Phys. D, 178 (2003), 51-61.
doi: 10.1016/S0167-2789(02)00807-2. |
[42] |
S. Zhou, Attractors and approximations for lattice dynamincal systems, J. Differential Equations, 200 (2004), 342-368.
doi: 10.1016/j.jde.2004.02.005. |
[43] |
S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.
doi: 10.1006/jdeq.2001.4032. |
[44] |
S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204.
doi: 10.1016/j.jde.2005.06.024. |
[45] |
X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Disc. Cont. Dyn. Syst. (B), 9 (2008), 763-785.
doi: 10.3934/dcdsb.2008.9.763. |
[46] |
C. Zhao and S. Zhou, Compact kernel sections for nonautonomous Klein-Gordon-Schr\"odinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56.
doi: 10.1016/j.jmaa.2006.10.002. |
[47] |
C. Zhao and S. Zhou, Compact kernel sections of long-wave-short-wave resonance equations on infinite lattices, Nonl. Analysis, 68 (2008), 652-670.
doi: 10.1016/j.na.2006.11.027. |
[48] |
C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Disc. Cont. Dyn. Syst. (A), 21 (2008), 643-663.
doi: 10.3934/dcds.2008.21.643. |
[49] |
C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95.
doi: 10.1016/j.jmaa.2008.12.036. |
[50] |
S. Zhou, C. Zhao and X. Liao, Compact uniform attractors for dissipative non-autonomous lattice dynamical systems, Comm. Pure Appl. Anal., 6 (2007), 1087-1111.
doi: 10.3934/cpaa.2007.6.1087. |
[51] |
S. Zhou, C. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Disc. Cont. Dyn. Syst. (A), 21 (2008), 1259-1277.
doi: 10.3934/dcds.2008.21.1259. |
show all references
References:
[1] |
Ahmed Y. Abdallah, Uniform exponential attractor for first order non-autonomous lattice dynamical systems, J. Differential Equations, 251 (2011), 1489-1504.
doi: 10.1016/j.jde.2011.05.030. |
[2] |
Ahmed Y. Abdallah, Exponential attractors for first-order lattice dynamical systems, J. Math. Anal. Appl., 339 (2008), 217-224.
doi: 10.1016/j.jmaa.2007.06.054. |
[3] |
P. W. Bates, X. Chen and A. Chmaj, Travelling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.
doi: 10.1137/S0036141000374002. |
[4] |
P. W. Bates, H. Lisei and K. Lu, Attrators for stochastic lattice dynamical systems, Stoch. Dyna., 6 (2006), 1-21.
doi: 10.1142/S0219493706001621. |
[5] |
P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Inter. J. Bifur. Chaos, 11 (2001), 143-153.
doi: 10.1142/S0218127401002031. |
[6] |
W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Diff. Eqs., 15 (2003), 485-515.
doi: 10.1023/B:JODY.0000009745.41889.30. |
[7] |
H. Chate and M. Courbage, Lattice systems, Phys. D, 103 (1997), 1-612.
doi: 10.1016/S0167-2789(96)00249-7. |
[8] |
S.-N. Chow, Lattice dynamical systems, in "Dynamical Systems," Lecture Notes in Math., 1822, Springer, Berlin, (2003), 1-102.
doi: 10.1007/978-3-540-45204-1_1. |
[9] |
S.-N. Chow and J. M. Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751.
doi: 10.1109/81.473583. |
[10] |
T. L. Carrol and L. M. Pecora, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.
doi: 10.1103/PhysRevLett.64.821. |
[11] |
S.-N. Chow, J. M. Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.
doi: 10.1006/jdeq.1998.3478. |
[12] |
S.-N. Chow, J. M. Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comp. Dyna., 4 (1996), 109-178. |
[13] |
L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156.
doi: 10.1109/81.222795. |
[14] |
L. O. Chua and Y. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272.
doi: 10.1109/31.7600. |
[15] |
V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002. |
[16] |
T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Phys. D, 67 (1993), 237-244.
doi: 10.1016/0167-2789(93)90208-I. |
[17] |
L. Fabiny, P. Colet and R. Roy, Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 47 (1993), 4287-4296.
doi: 10.1103/PhysRevA.47.4287. |
[18] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Math. Surveys and Monographs, AMS, Providence, RI, 1988. |
[19] |
M. Hillert, A solid-solution model for inhomogeneous systems, Acta Metall., 9 (1961), 525-535.
doi: 10.1016/0001-6160(61)90155-9. |
[20] |
X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024, 20 pp.
doi: 10.1142/S0219493711500249. |
[21] |
X. Han, W. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.
doi: 10.1016/j.jde.2010.10.018. |
[22] |
X. Jia, C. Zhao and X. Yang, Global attractor and Kolmogorov entropy of three component reversible Gray-Scott model on infinite lattices, Appl. Math. Comp., 218 (2012), 9781-9789.
doi: 10.1016/j.amc.2012.03.036. |
[23] |
J. P. Keener, Propagation and its failure in coupled systems of discret excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.
doi: 10.1137/0147038. |
[24] |
R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.
doi: 10.1007/BF01192578. |
[25] |
N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differntial Equations, 217 (2005), 88-123.
doi: 10.1016/j.jde.2005.06.002. |
[26] |
O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Cambridge University Press, Cambridge, 1991.
doi: 10.1017/CBO9780511569418. |
[27] |
Y. Lv and J. H. Sun, Dynamical behavior for stochastic lattice systems, Chaos Soli. Fract., 27 (2006), 1080-1090.
doi: 10.1016/j.chaos.2005.04.089. |
[28] |
P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonl. Analysis, 71 (2009), 3956-3963.
doi: 10.1016/j.na.2009.02.065. |
[29] |
E. E. Selkov, Self-oscillations in glycolysis: A simple kinetic model, Euorpean J. Bio., 4 (1968), 79-86. |
[30] |
G. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. |
[31] |
R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin, 1997. |
[32] |
T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multipliative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.
doi: 10.1016/j.jde.2012.03.020. |
[33] |
T. Caraballo, F. Morillas and J. Valero, Random attractors for sotchastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184.
doi: 10.1080/10236198.2010.549010. |
[34] |
T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335.
doi: 10.1007/s11464-008-0028-7. |
[35] |
E. V. Vlecka and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D, 212 (2005), 317-336.
doi: 10.1016/j.physd.2005.10.006. |
[36] |
B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.
doi: 10.1016/j.jde.2005.01.003. |
[37] |
B. Wang, Asymptotic behavior of non-autonomous lattice system, J. Math. Anal. Appl., 331 (2007), 121-136.
doi: 10.1016/j.jmaa.2006.08.070. |
[38] |
B. Wang, Uniform attractors of non-autonomous discret reaction-diffusion systems in weighted spaces, Inter. J. Bifur. Chaos, 18 (2008), 695-716.
doi: 10.1142/S0218127408020598. |
[39] |
R. L. Winaow, A. L. Kimball and A. Varghese, Simulating cardiac sinus and atrial network dynamics on connection machine, Phys. D, 64 (1993), 281-298. |
[40] |
Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Syst. (S), 2 (2009), 193-219.
doi: 10.3934/dcdss.2009.2.193. |
[41] |
S. Zhou, Attractors for first order dissipative lattice dynamical systems, Phys. D, 178 (2003), 51-61.
doi: 10.1016/S0167-2789(02)00807-2. |
[42] |
S. Zhou, Attractors and approximations for lattice dynamincal systems, J. Differential Equations, 200 (2004), 342-368.
doi: 10.1016/j.jde.2004.02.005. |
[43] |
S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.
doi: 10.1006/jdeq.2001.4032. |
[44] |
S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204.
doi: 10.1016/j.jde.2005.06.024. |
[45] |
X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Disc. Cont. Dyn. Syst. (B), 9 (2008), 763-785.
doi: 10.3934/dcdsb.2008.9.763. |
[46] |
C. Zhao and S. Zhou, Compact kernel sections for nonautonomous Klein-Gordon-Schr\"odinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56.
doi: 10.1016/j.jmaa.2006.10.002. |
[47] |
C. Zhao and S. Zhou, Compact kernel sections of long-wave-short-wave resonance equations on infinite lattices, Nonl. Analysis, 68 (2008), 652-670.
doi: 10.1016/j.na.2006.11.027. |
[48] |
C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Disc. Cont. Dyn. Syst. (A), 21 (2008), 643-663.
doi: 10.3934/dcds.2008.21.643. |
[49] |
C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95.
doi: 10.1016/j.jmaa.2008.12.036. |
[50] |
S. Zhou, C. Zhao and X. Liao, Compact uniform attractors for dissipative non-autonomous lattice dynamical systems, Comm. Pure Appl. Anal., 6 (2007), 1087-1111.
doi: 10.3934/cpaa.2007.6.1087. |
[51] |
S. Zhou, C. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Disc. Cont. Dyn. Syst. (A), 21 (2008), 1259-1277.
doi: 10.3934/dcds.2008.21.1259. |
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