Advanced Search
Article Contents
Article Contents

Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case

Abstract Related Papers Cited by
  • The purpose of this article is to analyze the dynamics of the following complex Ginzburg-Landau equation \begin{align*} \partial_{t}u-(\lambda+i\alpha)\Delta u+(\kappa+i\beta)|u|^{p-2}u-\gamma u=f(t) \end{align*} on non-cylindrical domains, which are obtained by diffeomorphic transformation of a bounded base domain, without any upper restriction on $p>2$, only with some restriction on $\beta/\kappa$. We establish the existence and uniqueness of strong and weak solutions as well as some energy inequalities for this equation on variable domains. Moreover the existence of a $\mathscr{D}$-pullback attractor is established for the process generated by the weak solutions under a slightly weaker condition that the measure of the spatial domains in the past is uniformly bounded above.
    Mathematics Subject Classification: Primary: 37L30, 37N20; Secondary: 35Q56.


    \begin{equation} \\ \end{equation}
  • [1]

    G. P. Agrawal, Optical pulse propagation in doped fiber amplifiers, Phys. Rev. A, 44 (1991), 7493-7501.doi: 10.1103/PhysRevA.44.7493.


    I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys., 74 (2002), 99-143.doi: 10.1103/RevModPhys.74.99.


    M. Bartuccelli, E. Constantin, C. R. Doering, J. D. Gibbon, M. Gisselfalt and G. P. Agrawal, On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), 421-444.doi: 10.1016/0167-2789(90)90156-J.


    H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Masson, Paris, 1983.


    P. Cannarsa, G. Da Prato and J. P. Zolesto, The damped wave equation in a moving domain, J. Differential Equations, 85 (1990), 1-16.doi: 10.1016/0022-0396(90)90086-5.


    D. M. Cao, C. Y. Sun and M. H. Yang, Dynamics for a stochastic reaction-diffusion equation with additive noise, J. Differential Equations, 259 (2015), 838-872.doi: 10.1016/j.jde.2015.02.020.


    T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.doi: 10.1016/j.na.2005.03.111.


    A. N. Carvalho and J. W. Cholewa, Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl., 337 (2008), 932-948.doi: 10.1016/j.jmaa.2007.04.051.


    A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013.doi: 10.1007/978-1-4614-4581-4.


    X. Chen and A. Friedman, A free boundary problem for an elliptic-hyperbolic system: An application to tumor growth, SIAM J. Math. Anal., 35 (2003), 974-986.doi: 10.1137/S0036141002418388.


    P. Clément, N. Okazawa, M. Sobajima and T. Yokota, A simple approach to the Cauchy problem for complex Ginzburg-Landau equations by compactness methods, J. Differential Equations, 253 (2012), 1250-1263.doi: 10.1016/j.jde.2012.05.002.


    H. Crauel, P. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains, Stoch. Dyn., 11 (2011), 301-314.doi: 10.1142/S0219493711003292.


    M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys., 65 (1993), 851-1089.doi: 10.1103/RevModPhys.65.851.


    D. R. da Costa, C. P. Dettmann and E. D. Leonel, Escape of particles in a time-dependent potential well, Phys. Rev. E, 83 (2011), 066211.doi: 10.1103/PhysRevE.83.066211.


    J. Q. Duan and V. J. Ervin, On nonlinear amplitude evolution under stochastic forcing, Appl. Math. Comput., 109 (2000), 59-65.doi: 10.1016/S0096-3003(99)00016-8.


    J. Q. Duan, H. V. Ly and E. S. Titi, The effect of nonlocal interactions on the dynamics of the Ginzburg-Landau equation, Z. Ang. Math. Phys., 47 (1996), 432-455.doi: 10.1007/BF00916648.


    M. Efendiev, A. Miranville and S. V. Zelik, Exponential attractors and finite-dimensional reduction of non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730.doi: 10.1017/S030821050000408X.


    C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations, 73 (1988), 309-353.doi: 10.1016/0022-0396(88)90110-6.


    B. L. Guo, X. K. Pu and F. H. Huang, Fractional Partial Differential Equations and Their Numberical Solution, Originally published by Science Press in 2011. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015.doi: 10.1142/9543.


    C. He and L. Hsiao, Two-dimensional Euler equations in a time dependent domain, J. Differential Equations, 163 (2000), 265-291.doi: 10.1006/jdeq.1999.3702.


    N. James, A. Ilyasse and D. Stevan, Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process, Internat. J. Control, 86 (2013), 1467-1478.doi: 10.1080/00207179.2013.786187.


    S. Jimbo and Y. Morita, Ginzburg-Landau equation with magnetic effect in a thin domain, Calc. Var. Partial Differential Equations, 15 (2002), 325-352.doi: 10.1007/s005260100130.


    P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems, J. Differential Equations, 253 (2012), 1422-1438.doi: 10.1016/j.jde.2012.05.016.


    P. E. Kloeden, P. Marín-Rubio and J. Real, Pullback attractors for a semilinear heat equation in a non-cylindrical domain, J. Differential Equations, 244 (2008), 2062-2090.doi: 10.1016/j.jde.2007.10.031.


    P. E. Kloeden, J. Real and C. Sun, Pullback attractors for a semilinear heat equation on time-varying domains, J. Differential Equations, 246 (2009), 4702-4730.doi: 10.1016/j.jde.2008.11.017.


    E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.doi: 10.1007/s10440-014-9993-x.


    O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Springer, 1985.doi: 10.1007/978-1-4757-4317-3.


    F. Li and B. You, Global attractors for the complex Ginzburg-Landau equation, J. Math. Anal. Appl., 415 (2014), 14-24.doi: 10.1016/j.jmaa.2014.01.059.


    N. Okazawa, Sectorialness of second order elliptic operators in divergence form, Amer. Math. Soc., 113 (1991), 701-706.doi: 10.1090/S0002-9939-1991-1072347-4.


    N. Okazawa and T. Yokota, Global existence and smoothing effect for the complex Ginzburg-Landau equation with p-Laplacian, J. Differential Equations, 182 (2002), 541-576.doi: 10.1006/jdeq.2001.4097.


    N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations, J. Math. Anal. Appl., 267 (2002), 247-263.doi: 10.1006/jmaa.2001.7770.


    X. K. Pu and B. L. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation, Appl. Anal., 92 (2011), 318-334.doi: 10.1080/00036811.2011.614601.


    G. Raugel, Dynamics of partial differential equations on thin domains, Dynamical systems (Montecatini terme, 1994), 208-315, Lecture Notes in Math. 1609, Springer, Berlin, 1995.doi: 10.1007/BFb0095241.


    C. Y. Sun, Asymptotic regularity for some dissipative equations, J. Differential Equations, 248 (2010), 342-362.doi: 10.1016/j.jde.2009.08.007.


    C. Y. Sun, D. M. Cao and J. Q. Duan, Non-autonomous wave dynamics with memory-Asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743-761.doi: 10.3934/dcdsb.2008.9.743.


    C. Y. Sun, Y. P. Xiao, Z. T. Tang and Y. Y. Liu, Continuity and pullback attractors for a semilinear heat equation on time-varying domains, submitted.


    C. Y. Sun and Y. B. Yuan, $L^p$-type pullback attractors for a semilinear heat equation on time-varying domains, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052.doi: 10.1017/S0308210515000177.


    R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1997.doi: 10.1007/978-1-4612-0645-3.


    B. You, Y. R. Hou, F. Li and J. P. Jiang, Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1801-1814.doi: 10.3934/dcdsb.2014.19.1801.


    F. Zhou and C. Y. Sun, Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains II: the monotonicity case, work in progress.

  • 加载中

Article Metrics

HTML views() PDF downloads(114) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint