# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801
##### References:
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Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition,, Discrete Continuous Dynam. Systems-B, 17 (2012), 2635.  doi: 10.3934/dcdsb.2012.17.2635.  Google Scholar [44] T. Yokota, Monotonicity method applied to complex Ginzburg-Landau type equations,, Journal of Mathematical Analysis and Applications, 380 (2011), 455.  doi: 10.1016/j.jmaa.2011.04.001.  Google Scholar [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.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar

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##### 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.  doi: 10.1006/jdeq.2001.4072.  Google Scholar [2] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992).   Google Scholar [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.  doi: 10.1017/S0334270000010468.  Google Scholar [4] G. X. Chen and C. K. Zhong, Uniform attractors for non-autonomous $p$-Laplacian equations,, Nonlinear Analysis, 68 (2008), 3349.  doi: 10.1016/j.na.2007.03.025.  Google Scholar [5] H. Crauel and F. Flandoli, Attractors for random dynamical systems,, Probability Theory and Related Fields, 100 (1994), 365.  doi: 10.1007/BF01193705.  Google Scholar [6] H. Crauel, A. Debussche and F. Flandoli, Random attractors,, Journal of Dynamics and Differential Equations, 9 (1997), 307.  doi: 10.1007/BF02219225.  Google Scholar [7] M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium,, Reviews of Modern Physics, 65 (1993), 851.  doi: 10.1103/RevModPhys.65.851.  Google Scholar [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.  doi: 10.1016/j.jde.2012.05.002.  Google Scholar [9] T. Caraballo, G. Lukasiewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Analysis, 64 (2006), 484.  doi: 10.1016/j.na.2005.03.111.  Google Scholar [10] T. Caraballo, J. A. Langa and J. Valero, The dimension of attractors of non-autonomous partial differential equations,, ANZIAM Journal, 45 (2003), 207.  doi: 10.1017/S1446181100013274.  Google Scholar [11] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society, (2002).   Google Scholar [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.  doi: 10.1088/0951-7715/1/2/001.  Google Scholar [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.  doi: 10.1016/0167-2789(94)90150-3.  Google Scholar [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.   Google Scholar [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.  doi: 10.1016/0167-2789(96)00055-3.  Google Scholar [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.  doi: 10.1007/s002200050129.  Google Scholar [17] J. M. Ghidaglia and B. Héron, Dimension of the attractor associated to the Ginzburg-Landau equation,, Physica D, 28 (1987), 282.  doi: 10.1016/0167-2789(87)90020-0.  Google Scholar [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.  doi: 10.1016/j.na.2005.03.022.  Google Scholar [19] P. E. Kloeden and B. Schmalfuß, Non-autonomous systems, cocycle attractors and variable time-step discretization,, Numerical Algorithms, 14 (1997), 141.  doi: 10.1023/A:1019156812251.  Google Scholar [20] P. E. Kloeden and D. J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics of Continuous, 4 (1998), 211.   Google Scholar [21] G. Łukaszewicz, On pullback attractors in $H_0^1$ for nonautonomous reaction-diffusion equations,, International Journal of Bifurcation and Chaos, 20 (2010), 2637.  doi: 10.1142/S0218127410027258.  Google Scholar [22] G. Łukaszewicz, On pullback attractors in $L^p$ for nonautonomous reaction-diffusion equations,, Nonlinear Analysis, 73 (2010), 350.  doi: 10.1016/j.na.2010.03.023.  Google Scholar [23] S. Lú, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation,, Numerische Mathematik, 22 (2000), 1.   Google Scholar [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.  doi: 10.3934/dcds.2005.13.701.  Google Scholar [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.  doi: 10.1016/j.amc.2006.11.187.  Google Scholar [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.  doi: 10.1016/j.amc.2008.10.065.  Google Scholar [27] H. T. Moon, P. Huerre and L. G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation,, Physica D, 7 (1983), 135.  doi: 10.1016/0167-2789(83)90124-0.  Google Scholar [28] A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection,, Journal of Fluid Mechanics, 38 (1969), 279.  doi: 10.1017/S0022112069000176.  Google Scholar [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.  doi: 10.1006/jdeq.2001.4097.  Google Scholar [30] N. Okazawa and T. Yokota, Monotonicity method for the complex Ginzburg-Landau equation, including smoothing effect,, Nonlinear Analysis, 47 (2001), 79.  doi: 10.1016/S0362-546X(01)00158-4.  Google Scholar [31] N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations,, Journal of Differential Equations, 267 (2002), 247.  doi: 10.1006/jmaa.2001.7770.  Google Scholar [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.  doi: 10.1007/s00220-003-1004-4.  Google Scholar [33] K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation,, Physica D, 41 (1990), 232.  doi: 10.1016/0167-2789(90)90125-9.  Google Scholar [34] B. Schmalfuß, Attractors for non-autonomous dynamical systems,, in Proc. Equadiff 99 (eds. B. Fiedler, 1 (2000), 684.   Google Scholar [35] H. T. Song, Pullback attractors of non-autonomous reaction-diffusion equations in $H_0^1,$, Journal of Differential Equations, 249 (2010), 2357.  doi: 10.1016/j.jde.2010.07.034.  Google Scholar [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.  doi: 10.1016/j.jmaa.2006.02.041.  Google Scholar [37] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, New York, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar [38] A. Unai, Global $C^1$ solutions of time-dependent complex Ginzburg-Landau equations,, Nonlinear Analysis, 46 (2001), 329.  doi: 10.1016/S0362-546X(99)00435-6.  Google Scholar [39] Y. H. Wang and C. K. Zhong, Pullback $\mathcalD$-attractors for nonautonomous sine-Gordon equations,, Nonlinear Analysis, 67 (2007), 2137.  doi: 10.1016/j.na.2006.09.019.  Google Scholar [40] B. You and C. K. Zhong, Global attractors for $p$-Laplacian equations with dynamic flux boundary conditions,, Advanced Nonlinear Studies, 13 (2013), 391.   Google Scholar [41] B. You, Y. R. Hou and F. Li, Global attractors of the quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian,, submitted., ().   Google Scholar [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.  doi: 10.1016/j.jmaa.2006.04.085.  Google Scholar [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.  doi: 10.3934/dcdsb.2012.17.2635.  Google Scholar [44] T. Yokota, Monotonicity method applied to complex Ginzburg-Landau type equations,, Journal of Mathematical Analysis and Applications, 380 (2011), 455.  doi: 10.1016/j.jmaa.2011.04.001.  Google Scholar [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.  doi: 10.1016/j.jde.2005.06.008.  Google Scholar
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