# American Institute of Mathematical Sciences

December  2016, 21(10): 3767-3792. doi: 10.3934/dcdsb.2016120

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

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received  January 2016 Revised  March 2016 Published  November 2016

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.
Citation: Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120
##### References:
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B, 9 (2008), 743. doi: 10.3934/dcdsb.2008.9.743. Google Scholar [36] 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., (). Google Scholar [37] 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. doi: 10.1017/S0308210515000177. Google Scholar [38] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar [39] 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. doi: 10.3934/dcdsb.2014.19.1801. Google Scholar [40] F. Zhou and C. Y. Sun, Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains II: the monotonicity case,, work in progress., (). Google Scholar

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##### References:
 [1] G. P. Agrawal, Optical pulse propagation in doped fiber amplifiers,, Phys. Rev. A, 44 (1991), 7493. doi: 10.1103/PhysRevA.44.7493. Google Scholar [2] I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation,, Rev. Modern Phys., 74 (2002), 99. doi: 10.1103/RevModPhys.74.99. Google Scholar [3] 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. doi: 10.1016/0167-2789(90)90156-J. Google Scholar [4] H. Brezis, Analyse Fonctionnelle. Théorie et Applications,, Masson, (1983). Google Scholar [5] P. Cannarsa, G. Da Prato and J. P. Zolesto, The damped wave equation in a moving domain,, J. Differential Equations, 85 (1990), 1. doi: 10.1016/0022-0396(90)90086-5. Google Scholar [6] 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. doi: 10.1016/j.jde.2015.02.020. Google Scholar [7] T. Caraballo, G. Łukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems,, Nonlinear Anal., 64 (2006), 484. doi: 10.1016/j.na.2005.03.111. Google Scholar [8] 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. doi: 10.1016/j.jmaa.2007.04.051. Google Scholar [9] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-dimensional Non-autonomous Dynamical Systems,, Applied Mathematical Sciences, (2013). doi: 10.1007/978-1-4614-4581-4. Google Scholar [10] 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. doi: 10.1137/S0036141002418388. Google Scholar [11] 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. doi: 10.1016/j.jde.2012.05.002. Google Scholar [12] H. Crauel, P. E. Kloeden and M. Yang, Random attractors of stochastic reaction-diffusion equations on variable domains,, Stoch. Dyn., 11 (2011), 301. doi: 10.1142/S0219493711003292. Google Scholar [13] M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium,, Rev. Mod. Phys., 65 (1993), 851. doi: 10.1103/RevModPhys.65.851. Google Scholar [14] 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). doi: 10.1103/PhysRevE.83.066211. Google Scholar [15] J. Q. Duan and V. J. Ervin, On nonlinear amplitude evolution under stochastic forcing,, Appl. Math. Comput., 109 (2000), 59. doi: 10.1016/S0096-3003(99)00016-8. Google Scholar [16] 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. doi: 10.1007/BF00916648. Google Scholar [17] 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. doi: 10.1017/S030821050000408X. Google Scholar [18] C. Foias, G. R. Sell and R. Temam, Inertial manifolds for nonlinear evolutionary equations,, J. Differential Equations, 73 (1988), 309. doi: 10.1016/0022-0396(88)90110-6. Google Scholar [19] 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., (2011). doi: 10.1142/9543. Google Scholar [20] C. He and L. Hsiao, Two-dimensional Euler equations in a time dependent domain,, J. Differential Equations, 163 (2000), 265. doi: 10.1006/jdeq.1999.3702. Google Scholar [21] 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. doi: 10.1080/00207179.2013.786187. Google Scholar [22] S. Jimbo and Y. Morita, Ginzburg-Landau equation with magnetic effect in a thin domain,, Calc. Var. Partial Differential Equations, 15 (2002), 325. doi: 10.1007/s005260100130. Google Scholar [23] P. E. Kloeden and T. Lorenz, Mean-square random dynamical systems,, J. Differential Equations, 253 (2012), 1422. doi: 10.1016/j.jde.2012.05.016. Google Scholar [24] 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. doi: 10.1016/j.jde.2007.10.031. Google Scholar [25] 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. doi: 10.1016/j.jde.2008.11.017. Google Scholar [26] E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges,, Acta Appl. Math., 137 (2015), 123. doi: 10.1007/s10440-014-9993-x. Google Scholar [27] O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics,, Springer, (1985). doi: 10.1007/978-1-4757-4317-3. Google Scholar [28] F. Li and B. You, Global attractors for the complex Ginzburg-Landau equation,, J. Math. Anal. Appl., 415 (2014), 14. doi: 10.1016/j.jmaa.2014.01.059. Google Scholar [29] N. Okazawa, Sectorialness of second order elliptic operators in divergence form,, Amer. Math. Soc., 113 (1991), 701. doi: 10.1090/S0002-9939-1991-1072347-4. Google Scholar [30] 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. doi: 10.1006/jdeq.2001.4097. Google Scholar [31] N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations,, J. Math. Anal. Appl., 267 (2002), 247. doi: 10.1006/jmaa.2001.7770. Google Scholar [32] X. K. Pu and B. L. Guo, Well-posedness and dynamics for the fractional Ginzburg-Landau equation,, Appl. Anal., 92 (2011), 318. doi: 10.1080/00036811.2011.614601. Google Scholar [33] G. Raugel, Dynamics of partial differential equations on thin domains,, Dynamical systems (Montecatini terme, (1994), 208. doi: 10.1007/BFb0095241. Google Scholar [34] C. Y. Sun, Asymptotic regularity for some dissipative equations,, J. Differential Equations, 248 (2010), 342. doi: 10.1016/j.jde.2009.08.007. Google Scholar [35] 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. doi: 10.3934/dcdsb.2008.9.743. Google Scholar [36] 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., (). Google Scholar [37] 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. doi: 10.1017/S0308210515000177. Google Scholar [38] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar [39] 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. doi: 10.3934/dcdsb.2014.19.1801. Google Scholar [40] F. Zhou and C. Y. Sun, Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains II: the monotonicity case,, work in progress., (). Google Scholar
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