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Permanence and extinction of a non-autonomous HIV-1 model with time delays
Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian
1. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an,710049, China |
2. | Department of Mathematics, Nanjing University, Nanjing, 210093 |
3. | School of Mathematics and Computer Sciences, Yan'an University, Yan'an, 716000, China |
References:
[1] |
A. R. Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents and singular initial data, Journal of Differential Equations, 181 (2002), 165-196.
doi: 10.1006/jdeq.2001.4072. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[3] |
C. Bu, On the Cauchy problem for the $1+2$ complex Ginzburg-Landau equation, Journal of the Australian Mathematical Society Series B-Applied Mathemati, 36 (1995), 313-324.
doi: 10.1017/S0334270000010468. |
[4] |
G. X. Chen and C. K. Zhong, Uniform attractors for non-autonomous $p$-Laplacian equations, Nonlinear Analysis, 68 (2008), 3349-3363.
doi: 10.1016/j.na.2007.03.025. |
[5] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[6] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, Journal of Dynamics and Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[7] |
M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Reviews of Modern Physics, 65 (1993), 851-1089.
doi: 10.1103/RevModPhys.65.851. |
[8] |
P. Clément, N. Okazawa, M. Sobajima and T. Yokota, A simple approach to the Cauchy problem for complex Ginzburg-Landau equations by compactness methods, Journal of Differential Equations, 253 (2012), 1250-1263.
doi: 10.1016/j.jde.2012.05.002. |
[9] |
T. Caraballo, G. Lukasiewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[10] |
T. Caraballo, J. A. Langa and J. Valero, The dimension of attractors of non-autonomous partial differential equations, ANZIAM Journal, 45 (2003), 207-222.
doi: 10.1017/S1446181100013274. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Rhode Isand, 2002. |
[12] |
C. R. Doering, J. D. Gibbon, D. Holm and B. Nicolaenko, Low-dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1988), 279-309.
doi: 10.1088/0951-7715/1/2/001. |
[13] |
C. R. Doering, J. D. Gibbon and C. D. Levermore, Weak and strong solutions of complex Ginzburg-Landau equation, Physica D, 71 (1994), 285-318.
doi: 10.1016/0167-2789(94)90150-3. |
[14] |
D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dynamics and Systems Theory, 2 (2002), 125-144. |
[15] |
J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Physica D, 95 (1996), 191-228.
doi: 10.1016/0167-2789(96)00055-3. |
[16] |
J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Compactness methods, Communications in Mathematical Physics, 187 (1997), 45-79.
doi: 10.1007/s002200050129. |
[17] |
J. M. Ghidaglia and B. Héron, Dimension of the attractor associated to the Ginzburg-Landau equation, Physica D, 28 (1987), 282-304.
doi: 10.1016/0167-2789(87)90020-0. |
[18] |
N. I. Karachalios and N. B. Zographopoulos, Global attractors and convergence to equilibrium for degenerate Ginzburg-Landau and parabolic equations, Nonlinear Analysis, 63 (2005), 1749-1768.
doi: 10.1016/j.na.2005.03.022. |
[19] |
P. E. Kloeden and B. Schmalfuß, Non-autonomous systems, cocycle attractors and variable time-step discretization, Numerical Algorithms, 14 (1997), 141-152.
doi: 10.1023/A:1019156812251. |
[20] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1998), 211-226. |
[21] |
G. Łukaszewicz, On pullback attractors in $H_0^1$ for nonautonomous reaction-diffusion equations, International Journal of Bifurcation and Chaos, 20 (2010), 2637-2644.
doi: 10.1142/S0218127410027258. |
[22] |
G. Łukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Analysis, 73 (2010), 350-357.
doi: 10.1016/j.na.2010.03.023. |
[23] |
S. Lú, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation, Numerische Mathematik, 22 (2000), 1-9. |
[24] |
S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external force, Discrete and Continuous Dynamical Systems-A, 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[25] |
Y. J. Li and C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Applied Mathematics and Computation, 190 (2007), 1020-1029.
doi: 10.1016/j.amc.2006.11.187. |
[26] |
Y. J. Li, S. Y. Wang and H. Q. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $L^p,$ Applied Mathematics and Computation, 207 (2009), 373-379.
doi: 10.1016/j.amc.2008.10.065. |
[27] |
H. T. Moon, P. Huerre and L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Physica D, 7 (1983), 135-150.
doi: 10.1016/0167-2789(83)90124-0. |
[28] |
A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, Journal of Fluid Mechanics, 38 (1969), 279-303.
doi: 10.1017/S0022112069000176. |
[29] |
N. Okazawa and T. Yokota, Global existence and smoothing effect for the complex Ginzburg-Landau equation with $p$-Laplacian, Journal of Differential Equations, 182 (2002), 541-576.
doi: 10.1006/jdeq.2001.4097. |
[30] |
N. Okazawa and T. Yokota, Monotonicity method for the complex Ginzburg-Landau equation, including smoothing effect, Nonlinear Analysis, 47 (2001), 79-88.
doi: 10.1016/S0362-546X(01)00158-4. |
[31] |
N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations, Journal of Differential Equations, 267 (2002), 247-263.
doi: 10.1006/jmaa.2001.7770. |
[32] |
T. Ogawa and T. Yokota, Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain, Communications in Mathematical Physics, 245 (2004), 105-121.
doi: 10.1007/s00220-003-1004-4. |
[33] |
K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation, Physica D, 41 (1990), 232-252.
doi: 10.1016/0167-2789(90)90125-9. |
[34] |
B. Schmalfuß, Attractors for non-autonomous dynamical systems, in Proc. Equadiff 99 (eds. B. Fiedler, K. Gröer and J. Sprekels), Berlin, World Scientific, Singapore, 1 (2000), 684-689. |
[35] |
H. T. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $H_0^1,$ Journal of Differential Equations, 249 (2010), 2357-2376.
doi: 10.1016/j.jde.2010.07.034. |
[36] |
H. T. Song and H. Q. Wu, Pullback attractors of non-autonomous reaction-diffusion equations, Journal of Mathematical Analysis and Applications, 325 (2007), 1200-1215.
doi: 10.1016/j.jmaa.2006.02.041. |
[37] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, New York, Springer-Verlag, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[38] |
A. Unai, Global $C^1$ solutions of time-dependent complex Ginzburg-Landau equations, Nonlinear Analysis, 46 (2001), 329-334.
doi: 10.1016/S0362-546X(99)00435-6. |
[39] |
Y. H. Wang and C. K. Zhong, Pullback $\mathcalD$-attractors for nonautonomous sine-Gordon equations, Nonlinear Analysis, 67 (2007), 2137-2148.
doi: 10.1016/j.na.2006.09.019. |
[40] |
B. You and C. K. Zhong, Global attractors for $p$-Laplacian equations with dynamic flux boundary conditions, Advanced Nonlinear Studies, 13 (2013), 391-410. |
[41] |
B. You, Y. R. Hou and F. Li, Global attractors of the quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian, submitted. |
[42] |
M. H. Yang, C. Y. Sun and C. K. Zhong, Global attractors for $p$-Laplacian equation, Journal of Mathematical Analysis and Applications, 327 (2007), 1130-1142.
doi: 10.1016/j.jmaa.2006.04.085. |
[43] |
L. Yang, M. H. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition, Discrete Continuous Dynam. Systems-B, 17 (2012), 2635-2651.
doi: 10.3934/dcdsb.2012.17.2635. |
[44] |
T. Yokota, Monotonicity method applied to complex Ginzburg-Landau type equations, Journal of Mathematical Analysis and Applications, 380 (2011), 455-466.
doi: 10.1016/j.jmaa.2011.04.001. |
[45] |
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, Journal of Differential Equations, 223 (2006), 367-399.
doi: 10.1016/j.jde.2005.06.008. |
show all references
References:
[1] |
A. R. Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents and singular initial data, Journal of Differential Equations, 181 (2002), 165-196.
doi: 10.1006/jdeq.2001.4072. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992. |
[3] |
C. Bu, On the Cauchy problem for the $1+2$ complex Ginzburg-Landau equation, Journal of the Australian Mathematical Society Series B-Applied Mathemati, 36 (1995), 313-324.
doi: 10.1017/S0334270000010468. |
[4] |
G. X. Chen and C. K. Zhong, Uniform attractors for non-autonomous $p$-Laplacian equations, Nonlinear Analysis, 68 (2008), 3349-3363.
doi: 10.1016/j.na.2007.03.025. |
[5] |
H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.
doi: 10.1007/BF01193705. |
[6] |
H. Crauel, A. Debussche and F. Flandoli, Random attractors, Journal of Dynamics and Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[7] |
M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Reviews of Modern Physics, 65 (1993), 851-1089.
doi: 10.1103/RevModPhys.65.851. |
[8] |
P. Clément, N. Okazawa, M. Sobajima and T. Yokota, A simple approach to the Cauchy problem for complex Ginzburg-Landau equations by compactness methods, Journal of Differential Equations, 253 (2012), 1250-1263.
doi: 10.1016/j.jde.2012.05.002. |
[9] |
T. Caraballo, G. Lukasiewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Analysis, 64 (2006), 484-498.
doi: 10.1016/j.na.2005.03.111. |
[10] |
T. Caraballo, J. A. Langa and J. Valero, The dimension of attractors of non-autonomous partial differential equations, ANZIAM Journal, 45 (2003), 207-222.
doi: 10.1017/S1446181100013274. |
[11] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society, Rhode Isand, 2002. |
[12] |
C. R. Doering, J. D. Gibbon, D. Holm and B. Nicolaenko, Low-dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1988), 279-309.
doi: 10.1088/0951-7715/1/2/001. |
[13] |
C. R. Doering, J. D. Gibbon and C. D. Levermore, Weak and strong solutions of complex Ginzburg-Landau equation, Physica D, 71 (1994), 285-318.
doi: 10.1016/0167-2789(94)90150-3. |
[14] |
D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dynamics and Systems Theory, 2 (2002), 125-144. |
[15] |
J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Physica D, 95 (1996), 191-228.
doi: 10.1016/0167-2789(96)00055-3. |
[16] |
J. Ginibre and G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Compactness methods, Communications in Mathematical Physics, 187 (1997), 45-79.
doi: 10.1007/s002200050129. |
[17] |
J. M. Ghidaglia and B. Héron, Dimension of the attractor associated to the Ginzburg-Landau equation, Physica D, 28 (1987), 282-304.
doi: 10.1016/0167-2789(87)90020-0. |
[18] |
N. I. Karachalios and N. B. Zographopoulos, Global attractors and convergence to equilibrium for degenerate Ginzburg-Landau and parabolic equations, Nonlinear Analysis, 63 (2005), 1749-1768.
doi: 10.1016/j.na.2005.03.022. |
[19] |
P. E. Kloeden and B. Schmalfuß, Non-autonomous systems, cocycle attractors and variable time-step discretization, Numerical Algorithms, 14 (1997), 141-152.
doi: 10.1023/A:1019156812251. |
[20] |
P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dynamics of Continuous, Discrete and Impulsive Systems, 4 (1998), 211-226. |
[21] |
G. Łukaszewicz, On pullback attractors in $H_0^1$ for nonautonomous reaction-diffusion equations, International Journal of Bifurcation and Chaos, 20 (2010), 2637-2644.
doi: 10.1142/S0218127410027258. |
[22] |
G. Łukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations, Nonlinear Analysis, 73 (2010), 350-357.
doi: 10.1016/j.na.2010.03.023. |
[23] |
S. Lú, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation, Numerische Mathematik, 22 (2000), 1-9. |
[24] |
S. S. Lu, H. Q. Wu and C. K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external force, Discrete and Continuous Dynamical Systems-A, 13 (2005), 701-719.
doi: 10.3934/dcds.2005.13.701. |
[25] |
Y. J. Li and C. K. Zhong, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations, Applied Mathematics and Computation, 190 (2007), 1020-1029.
doi: 10.1016/j.amc.2006.11.187. |
[26] |
Y. J. Li, S. Y. Wang and H. Q. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in $L^p,$ Applied Mathematics and Computation, 207 (2009), 373-379.
doi: 10.1016/j.amc.2008.10.065. |
[27] |
H. T. Moon, P. Huerre and L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Physica D, 7 (1983), 135-150.
doi: 10.1016/0167-2789(83)90124-0. |
[28] |
A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, Journal of Fluid Mechanics, 38 (1969), 279-303.
doi: 10.1017/S0022112069000176. |
[29] |
N. Okazawa and T. Yokota, Global existence and smoothing effect for the complex Ginzburg-Landau equation with $p$-Laplacian, Journal of Differential Equations, 182 (2002), 541-576.
doi: 10.1006/jdeq.2001.4097. |
[30] |
N. Okazawa and T. Yokota, Monotonicity method for the complex Ginzburg-Landau equation, including smoothing effect, Nonlinear Analysis, 47 (2001), 79-88.
doi: 10.1016/S0362-546X(01)00158-4. |
[31] |
N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations, Journal of Differential Equations, 267 (2002), 247-263.
doi: 10.1006/jmaa.2001.7770. |
[32] |
T. Ogawa and T. Yokota, Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain, Communications in Mathematical Physics, 245 (2004), 105-121.
doi: 10.1007/s00220-003-1004-4. |
[33] |
K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation, Physica D, 41 (1990), 232-252.
doi: 10.1016/0167-2789(90)90125-9. |
[34] |
B. Schmalfuß, Attractors for non-autonomous dynamical systems, in Proc. Equadiff 99 (eds. B. Fiedler, K. Gröer and J. Sprekels), Berlin, World Scientific, Singapore, 1 (2000), 684-689. |
[35] |
H. T. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $H_0^1,$ Journal of Differential Equations, 249 (2010), 2357-2376.
doi: 10.1016/j.jde.2010.07.034. |
[36] |
H. T. Song and H. Q. Wu, Pullback attractors of non-autonomous reaction-diffusion equations, Journal of Mathematical Analysis and Applications, 325 (2007), 1200-1215.
doi: 10.1016/j.jmaa.2006.02.041. |
[37] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, New York, Springer-Verlag, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[38] |
A. Unai, Global $C^1$ solutions of time-dependent complex Ginzburg-Landau equations, Nonlinear Analysis, 46 (2001), 329-334.
doi: 10.1016/S0362-546X(99)00435-6. |
[39] |
Y. H. Wang and C. K. Zhong, Pullback $\mathcalD$-attractors for nonautonomous sine-Gordon equations, Nonlinear Analysis, 67 (2007), 2137-2148.
doi: 10.1016/j.na.2006.09.019. |
[40] |
B. You and C. K. Zhong, Global attractors for $p$-Laplacian equations with dynamic flux boundary conditions, Advanced Nonlinear Studies, 13 (2013), 391-410. |
[41] |
B. You, Y. R. Hou and F. Li, Global attractors of the quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian, submitted. |
[42] |
M. H. Yang, C. Y. Sun and C. K. Zhong, Global attractors for $p$-Laplacian equation, Journal of Mathematical Analysis and Applications, 327 (2007), 1130-1142.
doi: 10.1016/j.jmaa.2006.04.085. |
[43] |
L. Yang, M. H. Yang and P. E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition, Discrete Continuous Dynam. Systems-B, 17 (2012), 2635-2651.
doi: 10.3934/dcdsb.2012.17.2635. |
[44] |
T. Yokota, Monotonicity method applied to complex Ginzburg-Landau type equations, Journal of Mathematical Analysis and Applications, 380 (2011), 455-466.
doi: 10.1016/j.jmaa.2011.04.001. |
[45] |
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, Journal of Differential Equations, 223 (2006), 367-399.
doi: 10.1016/j.jde.2005.06.008. |
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