# zbMATH — the first resource for mathematics

Pullback attractors for the non-autonomous complex Ginzburg-Landau type equation with $$p$$-Laplacian. (English) Zbl 1418.35041
Summary: In this paper, we are concerned with the long-time behavior of the non-autonomous complex Ginzburg-Landau type equation with $$p$$-Laplacian. We first prove the existence of pullback absorbing sets in $$L^2(\Omega)\cap W_0^{1,p}(\Omega)\cap L^q(\Omega)$$ for the process $$\{U (t,\tau)\}_{t>\tau}$$ corresponding to the non-autonomous complex Ginzburg-Landau type equation 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 $$W_0^{1,p}(\Omega)$$ for the process $$\{U(t,\tau)\}_{t>\tau}$$ by asymptotic a priori estimates.
##### MSC:
 35B41 Attractors 35Q56 Ginzburg-Landau equations
Full Text:
##### References:
 [1] A.R. Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents and singular initial data, J. Differ. Equations, 181:165-196, 2002. Nonlinear Anal. Model. Control, 20(2):233-248 246F. Li, B. You [2] C. Bu, On the Cauchy problem for the 1 + 2 complex Ginzburg-Landau equation, J. Aust. Math. Soc., Ser. B, 36:313-324, 1994. · Zbl 0829.35119 [3] T. Caraballo, J.A. Langa, J. Valero, The dimension of attractors of non-autonomous partial differential equations, ANZIAM J., 45:207-222, 2003. · Zbl 1047.35024 [4] T. Caraballo, G. Łukasiewicz, J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal., Theory Methods Appl., 64:484-498, 2006. · Zbl 1128.37019 [5] D.N. Cheban, P.E. Kloeden, B. Schmalfuß, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2:9-28, 2002. [6] G.X. Chen, C.K. Zhong, Uniform attractors for non-autonomous p-Laplacian equations, Nonlinear Anal., Theory Methods Appl., 68:3349-3363, 2008. · Zbl 1162.35326 [7] P. Clément, N. Okazawa, M. Sobajima, T. Yokota, A simple approach to the Cauchy problem for complex Ginzburg-Landau equations by compactness methods, J. Differ. Equations, 253:1250-1263, 2012. · Zbl 1248.35203 [8] V.V. Chepyzhov, M.I. Vishik, Attractors for Equations of Mathematical Physics, Colloq. Publ., Am. Math. Soc., Vol. 49. AMS, Providence, RI, 2002. · Zbl 0986.35001 [9] H. Crauel, A. Debussche, F. Flandoli, Random attractors, J. Dyn. Differ. Equations, 9:307-341, 1997. [10] H. Crauel, F. Flandoli, Attractors for random dynamical systems, Probab. Theory Relat. Fields, 100:365-393, 1994. · Zbl 0819.58023 [11] M.C. Cross, P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65:851-1089, 1993. · Zbl 1371.37001 [12] C.R. Doering, J.D. Gibbon, D. Holm, B. Nicolaenko, Low-dimensional behavior in the complex Ginzburg-Landau equation, Nonlinearity, 1:279-309, 1988. · Zbl 0655.58021 [13] C.R. Doering, J.D. Gibbon, C.D. Levermore, Weak and strong solutions of complex Ginzburg- Landau equation, Physica D, 71:285-318, 1994. · Zbl 0810.35119 [14] J.M. Ghidaglia, B. Héron, Dimension of the attractor associated to the Ginzburg-Landau equation, Physica D, 28:282-304, 1987. · Zbl 0623.58049 [15] J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods, Physica D, 95:191-228, 1996. · Zbl 0889.35045 [16] J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Compactness methods, Commun. Math. Phys., 187:45-79, 1997. · Zbl 0889.35046 [17] N.I. Karachalios, N.B. Zographopoulos, Global attractors and convergence to equilibrium for degenerate Ginzburg-Landau and parabolic equations, Nonlinear Anal., Theory Methods Appl., 63:1749-1768, 2005. · Zbl 1224.35041 [18] P.E. Kloeden, B. Schmalfuß, Non-autonomous systems, cocycle attractors and variable timestep discretization, Numer. Algorithms, 14:141-152, 1997. [19] P.E. Kloeden, D.J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations, Dyn. Contin. Discrete Impulsive Syst., 4:211-226, 1998. http://www.mii.lt/NA The non-autonomous complex Ginzburg-Landau type equation withp-Laplacian247 · Zbl 0905.34047 [20] Y. Li, S. Wang, H. Wu, Pullback attractors for non-autonomous reaction-diffusion equations in Lp, Appl. Math. Comput., 207:373-379, 2009. · Zbl 1177.35038 [21] Y. Li, C.K. Zhong, Pullback attractor for the norm-to-weak continuous process and application to the non-autonomous reaction-diffusion equations, Appl. Math. Comput., 190:1020-1029, 2007. · Zbl 1126.37049 [22] S.S. Lu, H.Q. Wu, C.K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external force, Discrete Contin. Dyn. Syst., 23:701-719, 2005. · Zbl 1083.35094 [23] G. Łukaszewicz, On pullback attractors in H01for nonautonomous reaction-diffusion equations, Int. J. Bifurcation Chaos Appl. Sci. Eng., 20:2637-2644, 2010. · Zbl 1202.35036 [24] G. Łukaszewicz, On pullback attractors in Lpfor nonautonomous reaction-diffusion equations, Nonlinear Anal., Theory Methods Appl., 73:350-357, 2010. [25] S. Lú, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation, Numer. Math., 22:1-9, 2000. [26] H.T. Moon, P. Huerre, L.G. Redekopp, Transitions to chaos in the Ginzburg-Landau equation, Physica D, 7:135-150, 1983. · Zbl 0558.58030 [27] A.C. Newell, J.A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38:279-304, 1969. · Zbl 0187.25102 [28] N. Okazawa, T. Yokota, Global existence and smoothing effect for the complex Ginzburg- Landau equation with p-Laplacian, J. Differ. Equations, 182:541-576, 2001. · Zbl 1005.35086 [29] N. Okazawa, T. Yokota, Monotonicity method for the complex Ginzburg-Landau equation, including smoothing effect, Nonlinear Anal., Theory Methods Appl., 47:79-88, 2001. · Zbl 1042.35615 [30] N. Okazawa, T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations, J. Differ. Equations, 267:247-263, 2002. · Zbl 0995.35029 [31] T. Ogawa, T. Yokota, Uniqueness and inviscid limits of solutions for the complex Ginzburg- Landau equation in a two-dimensional domain, Commun. Math. Phys., 245:105-121, 2004. · Zbl 1072.35171 [32] K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg- Landau partial differential equation, Physica D, 41:232-252, 1990. · Zbl 0696.35177 [33] B. Schmalfuß, Attractors for the non-autonomous dynamical systems, in B. Fiedler, K. Gröer, J. Sprekels (Eds.), Equadiff 99. Proceedings of the International Conference on Differential Equations, Berlin, 1-7 August 1999, World Scientific, Singapore, 2000, pp. 684-689. [34] H.T. Song, Pullback attractors of non-autonomous reaction-diffusion equations in H01, J. Differ. Equations, 249:2357-2376, 2010. · Zbl 1207.35072 [35] H.T. Song, H.Q. Wu, Pullback attractors of non-autonomous reaction-diffusion equations, J. Math. Anal. Appl., 325:1200-1215, 2007. · Zbl 1104.37013 [36] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, SpringerVerlag, New York, 1997. · Zbl 0871.35001 [37] A. Unai, Global C1solutions of time-dependent complex Ginzburg-Landau equations, Nonlinear Anal., Theory Methods Appl., 46:329-334, 2001. Nonlinear Anal. Model. Control, 20(2):233-248 248F. Li, B. You [38] Y. Wang, C.K. Zhong, Pullback D-attractors for nonautonomous sine-Gordon equations, Nonlinear Anal., Theory Methods Appl., 67:2137-2148, 2007. · Zbl 1156.37017 [39] M.H. Yang, C.Y. Sun, C.K. Zhong, Global attractors for p-Laplacian equation, J. Math. Anal. Appl., 327:1130-1142, 2007. · Zbl 1112.35031 [40] L. Yang, M.H. Yang, P.E. Kloeden, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition, Discrete Contin. Dyn. Syst., Ser. B, 17:2635-2651, 2012. · Zbl 1261.37033 [41] T. Yokota, Monotonicity method applied to complex Ginzburg-Landau type equations, J. Math. Anal. Appl., 380:455-466, 2011. · Zbl 1219.35296 [42] B. You, Y.R. Hou, F. Li, Global attractors for the quasi-linear complex Ginzburg-Landau equation with p-Laplacian, Asymptotic Anal. (submitted). [43] B. You, C.K. Zhong, Global attractors for p-Laplacian equations with dynamic flux boundary conditions, Adv. Nonlinear Stud., 13:391-410, 2013. · Zbl 1304.35130 [44] C.K. Zhong, M.H. Yang, C.Y. Sun, The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differ. Equations, 223:367-399, 2006. · Zbl 1101.35022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.