×

zbMATH — the first resource for mathematics

Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with \(p\)-Laplacian. (English) Zbl 1351.37261
Summary: 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 \[ \frac{\partial u}{\partial t}-(\lambda+i\alpha)\Delta_p u+(\kappa+i\beta)|u|^{q-2}u-\gamma u=g(x,t) \] 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.

MSC:
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
35B40 Asymptotic behavior of solutions to PDEs
35B41 Attractors
35B45 A priori estimates in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] A. R. Bernal, Attractors for parabolic equations with nonlinear boundary conditions, critical exponents and singular initial data,, Journal of Differential Equations, 181, 165, (2002) · Zbl 1042.35006
[2] A. V. Babin, <em>Attractors of Evolution Equations</em>,, North-Holland, (1992) · Zbl 0778.58002
[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, 313, (1995) · Zbl 0829.35119
[4] G. X. Chen, Uniform attractors for non-autonomous \(p\)-Laplacian equations,, Nonlinear Analysis, 68, 3349, (2008) · Zbl 1162.35326
[5] H. Crauel, Attractors for random dynamical systems,, Probability Theory and Related Fields, 100, 365, (1994) · Zbl 0819.58023
[6] H. Crauel, Random attractors,, Journal of Dynamics and Differential Equations, 9, 307, (1997) · Zbl 0884.58064
[7] M. C. Cross, Pattern formation outside of equilibrium,, Reviews of Modern Physics, 65, 851, (1993) · Zbl 1371.37001
[8] P. Clément, A simple approach to the Cauchy problem for complex Ginzburg-Landau equations by compactness methods,, Journal of Differential Equations, 253, 1250, (2012) · Zbl 1248.35203
[9] T. Caraballo, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Analysis, 64, 484, (2006) · Zbl 1128.37019
[10] T. Caraballo, The dimension of attractors of non-autonomous partial differential equations,, ANZIAM Journal, 45, 207, (2003) · Zbl 1047.35024
[11] V. V. Chepyzhov, <em>Attractors for Equations of Mathematical Physics</em>,, American Mathematical Society, (2002) · Zbl 0986.35001
[12] C. R. Doering, Low-dimensional behavior in the complex Ginzburg-Landau equation,, Nonlinearity, 1, 279, (1988) · Zbl 0655.58021
[13] C. R. Doering, Weak and strong solutions of complex Ginzburg-Landau equation,, Physica D, 71, 285, (1994) · Zbl 0810.35119
[14] D. N. Cheban, The relationship between pullback, forwards and global attractors of nonautonomous dynamical systems,, Nonlinear Dynamics and Systems Theory, 2, 125, (2002) · Zbl 1054.34087
[15] J. Ginibre, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. I. Compactness methods,, Physica D, 95, 191, (1996) · Zbl 0889.35045
[16] J. Ginibre, The Cauchy problem in local spaces for the complex Ginzburg-Landau equation. II. Compactness methods,, Communications in Mathematical Physics, 187, 45, (1997) · Zbl 0889.35046
[17] J. M. Ghidaglia, Dimension of the attractor associated to the Ginzburg-Landau equation,, Physica D, 28, 282, (1987) · Zbl 0623.58049
[18] N. I. Karachalios, Global attractors and convergence to equilibrium for degenerate Ginzburg-Landau and parabolic equations,, Nonlinear Analysis, 63, 1749, (2005) · Zbl 1224.35041
[19] P. E. Kloeden, Non-autonomous systems, cocycle attractors and variable time-step discretization,, Numerical Algorithms, 14, 141, (1997) · Zbl 0886.65077
[20] P. E. Kloeden, Cocycle attractors in nonautonomously perturbed differential equations,, Dynamics of Continuous, 4, 211, (1998) · Zbl 0905.34047
[21] G. Łukaszewicz, On pullback attractors in \(H_0^1\) for nonautonomous reaction-diffusion equations,, International Journal of Bifurcation and Chaos, 20, 2637, (2010) · Zbl 1202.35036
[22] G. Łukaszewicz, On pullback attractors in \(L^p\) for nonautonomous reaction-diffusion equations,, Nonlinear Analysis, 73, 350, (2010) · Zbl 1195.35061
[23] S. Lú, The dynamical behavior of the Ginzburg-Landau equation and its Fourier spectral approximation,, Numerische Mathematik, 22, 1, (2000) · Zbl 0962.65108
[24] S. S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with normal external force,, Discrete and Continuous Dynamical Systems-A, 13, 701, (2005) · Zbl 1083.35094
[25] Y. J. Li, Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction-diffusion equations,, Applied Mathematics and Computation, 190, 1020, (2007) · Zbl 1126.37049
[26] Y. J. Li, Pullback attractors for non-autonomous reaction-diffusion equations in \(L^p,\), Applied Mathematics and Computation, 207, 373, (2009) · Zbl 1177.35038
[27] H. T. Moon, Transitions to chaos in the Ginzburg-Landau equation,, Physica D, 7, 135, (1983) · Zbl 0558.58030
[28] A. C. Newell, Finite bandwidth, finite amplitude convection,, Journal of Fluid Mechanics, 38, 279, (1969) · Zbl 0187.25102
[29] N. Okazawa, Global existence and smoothing effect for the complex Ginzburg-Landau equation with \(p\)-Laplacian,, Journal of Differential Equations, 182, 541, (2002) · Zbl 1005.35086
[30] N. Okazawa, Monotonicity method for the complex Ginzburg-Landau equation, including smoothing effect,, Nonlinear Analysis, 47, 79, (2001) · Zbl 1042.35615
[31] N. Okazawa, Monotonicity method applied to the complex Ginzburg-Landau and related equations,, Journal of Differential Equations, 267, 247, (2002) · Zbl 0995.35029
[32] T. Ogawa, Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain,, Communications in Mathematical Physics, 245, 105, (2004) · Zbl 1072.35171
[33] K. Promislow, Induced trajectories and approximate inertial manifolds for the Ginzburg-Landau partial differential equation,, Physica D, 41, 232, (1990) · Zbl 0696.35177
[34] B. Schmalfuß, Attractors for non-autonomous dynamical systems,, in Proc. Equadiff 99 (eds. B. Fiedler, 1, 684, (2000) · Zbl 0971.37038
[35] H. T. Song, Pullback attractors of non-autonomous reaction-diffusion equations in \(H_0^1,\), Journal of Differential Equations, 249, 2357, (2010) · Zbl 1207.35072
[36] H. T. Song, Pullback attractors of non-autonomous reaction-diffusion equations,, Journal of Mathematical Analysis and Applications, 325, 1200, (2007) · Zbl 1104.37013
[37] R. Temam, <em>Infinite-dimensional Dynamical Systems in Mechanics and Physics</em>,, New York, (1997) · Zbl 0871.35001
[38] A. Unai, Global \(C^1\) solutions of time-dependent complex Ginzburg-Landau equations,, Nonlinear Analysis, 46, 329, (2001) · Zbl 0977.35064
[39] Y. H. Wang, Pullback \(\mathcalD\)-attractors for nonautonomous sine-Gordon equations,, Nonlinear Analysis, 67, 2137, (2007) · Zbl 1156.37017
[40] B. You, Global attractors for \(p\)-Laplacian equations with dynamic flux boundary conditions,, Advanced Nonlinear Studies, 13, 391, (2013) · Zbl 1304.35130
[41] B. You, Global attractors of the quasi-linear complex Ginzburg-Landau equation with \(p\)-Laplacian,, submitted.
[42] M. H. Yang, Global attractors for \(p\)-Laplacian equation,, Journal of Mathematical Analysis and Applications, 327, 1130, (2007) · Zbl 1112.35031
[43] L. Yang, Pullback attractors for non-autonomous quasi-linear parabolic equations with a dynamical boundary condition,, Discrete Continuous Dynam. Systems-B, 17, 2635, (2012) · Zbl 1261.37033
[44] T. Yokota, Monotonicity method applied to complex Ginzburg-Landau type equations,, Journal of Mathematical Analysis and Applications, 380, 455, (2011) · Zbl 1219.35296
[45] C. K. Zhong, 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, 367, (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.