August  2014, 19(6): 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

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

Received  May 2013 Revised  March 2014 Published  June 2014

In this paper, we are concerned with the long-time behavior of the following non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian \begin{align*} \frac{\partial u}{\partial t}-(\lambda+i\alpha)\Delta_p u+(\kappa+i\beta)|u|^{q-2}u-\gamma u=g(x,t) \end{align*} without any restriction on $q>2$ under additional assumptions. We first prove the existence of a pullback absorbing set in $L^2(\Omega) \cap W^{1,p}_0(\Omega)\cap L^q(\Omega)$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ corresponding to the non-autonomous quasi-linear complex Ginzburg-Landau equation (1)-(3) with $p$-Laplacian. Next, the existence of a pullback attractor in $L^2(\Omega)$ is established by the Sobolev compactness embedding theorem. Finally, we prove the existence of a pullback attractor in $W^{1,p}_0(\Omega)$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with the non-autonomous quasi-linear complex Ginzburg-Landau equation (1)-(3) with $p$-Laplacian by asymptotic a priori estimates.
Citation: Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801
References:
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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.

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A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

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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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