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:
[1]

Phys. Rev. A, 44 (1991), 7493-7501. doi: 10.1103/PhysRevA.44.7493.  Google Scholar

[2]

Rev. Modern Phys., 74 (2002), 99-143. doi: 10.1103/RevModPhys.74.99.  Google Scholar

[3]

Phys. D, 44 (1990), 421-444. doi: 10.1016/0167-2789(90)90156-J.  Google Scholar

[4]

Masson, Paris, 1983.  Google Scholar

[5]

J. Differential Equations, 85 (1990), 1-16. doi: 10.1016/0022-0396(90)90086-5.  Google Scholar

[6]

J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020.  Google Scholar

[7]

Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.  Google Scholar

[8]

J. Math. Anal. Appl., 337 (2008), 932-948. doi: 10.1016/j.jmaa.2007.04.051.  Google Scholar

[9]

Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[10]

SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388.  Google Scholar

[11]

J. Differential Equations, 253 (2012), 1250-1263. doi: 10.1016/j.jde.2012.05.002.  Google Scholar

[12]

Stoch. Dyn., 11 (2011), 301-314. doi: 10.1142/S0219493711003292.  Google Scholar

[13]

Rev. Mod. Phys., 65 (1993), 851-1089. doi: 10.1103/RevModPhys.65.851.  Google Scholar

[14]

Phys. Rev. E, 83 (2011), 066211. doi: 10.1103/PhysRevE.83.066211.  Google Scholar

[15]

Appl. Math. Comput., 109 (2000), 59-65. doi: 10.1016/S0096-3003(99)00016-8.  Google Scholar

[16]

Z. Ang. Math. Phys., 47 (1996), 432-455. doi: 10.1007/BF00916648.  Google Scholar

[17]

Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X.  Google Scholar

[18]

J. Differential Equations, 73 (1988), 309-353. doi: 10.1016/0022-0396(88)90110-6.  Google Scholar

[19]

Originally published by Science Press in 2011. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9543.  Google Scholar

[20]

J. Differential Equations, 163 (2000), 265-291. doi: 10.1006/jdeq.1999.3702.  Google Scholar

[21]

Internat. J. Control, 86 (2013), 1467-1478. doi: 10.1080/00207179.2013.786187.  Google Scholar

[22]

Calc. Var. Partial Differential Equations, 15 (2002), 325-352. doi: 10.1007/s005260100130.  Google Scholar

[23]

J. Differential Equations, 253 (2012), 1422-1438. doi: 10.1016/j.jde.2012.05.016.  Google Scholar

[24]

J. Differential Equations, 244 (2008), 2062-2090. doi: 10.1016/j.jde.2007.10.031.  Google Scholar

[25]

J. Differential Equations, 246 (2009), 4702-4730. doi: 10.1016/j.jde.2008.11.017.  Google Scholar

[26]

Acta Appl. Math., 137 (2015), 123-157. doi: 10.1007/s10440-014-9993-x.  Google Scholar

[27]

Springer, 1985. doi: 10.1007/978-1-4757-4317-3.  Google Scholar

[28]

J. Math. Anal. Appl., 415 (2014), 14-24. doi: 10.1016/j.jmaa.2014.01.059.  Google Scholar

[29]

Amer. Math. Soc., 113 (1991), 701-706. doi: 10.1090/S0002-9939-1991-1072347-4.  Google Scholar

[30]

J. Differential Equations, 182 (2002), 541-576. doi: 10.1006/jdeq.2001.4097.  Google Scholar

[31]

J. Math. Anal. Appl., 267 (2002), 247-263. doi: 10.1006/jmaa.2001.7770.  Google Scholar

[32]

Appl. Anal., 92 (2011), 318-334. doi: 10.1080/00036811.2011.614601.  Google Scholar

[33]

Dynamical systems (Montecatini terme, 1994), 208-315, Lecture Notes in Math. 1609, Springer, Berlin, 1995. doi: 10.1007/BFb0095241.  Google Scholar

[34]

J. Differential Equations, 248 (2010), 342-362. doi: 10.1016/j.jde.2009.08.007.  Google Scholar

[35]

Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743-761. 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]

Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052. doi: 10.1017/S0308210515000177.  Google Scholar

[38]

$2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[39]

Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1801-1814. 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

show all references

References:
[1]

Phys. Rev. A, 44 (1991), 7493-7501. doi: 10.1103/PhysRevA.44.7493.  Google Scholar

[2]

Rev. Modern Phys., 74 (2002), 99-143. doi: 10.1103/RevModPhys.74.99.  Google Scholar

[3]

Phys. D, 44 (1990), 421-444. doi: 10.1016/0167-2789(90)90156-J.  Google Scholar

[4]

Masson, Paris, 1983.  Google Scholar

[5]

J. Differential Equations, 85 (1990), 1-16. doi: 10.1016/0022-0396(90)90086-5.  Google Scholar

[6]

J. Differential Equations, 259 (2015), 838-872. doi: 10.1016/j.jde.2015.02.020.  Google Scholar

[7]

Nonlinear Anal., 64 (2006), 484-498. doi: 10.1016/j.na.2005.03.111.  Google Scholar

[8]

J. Math. Anal. Appl., 337 (2008), 932-948. doi: 10.1016/j.jmaa.2007.04.051.  Google Scholar

[9]

Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[10]

SIAM J. Math. Anal., 35 (2003), 974-986. doi: 10.1137/S0036141002418388.  Google Scholar

[11]

J. Differential Equations, 253 (2012), 1250-1263. doi: 10.1016/j.jde.2012.05.002.  Google Scholar

[12]

Stoch. Dyn., 11 (2011), 301-314. doi: 10.1142/S0219493711003292.  Google Scholar

[13]

Rev. Mod. Phys., 65 (1993), 851-1089. doi: 10.1103/RevModPhys.65.851.  Google Scholar

[14]

Phys. Rev. E, 83 (2011), 066211. doi: 10.1103/PhysRevE.83.066211.  Google Scholar

[15]

Appl. Math. Comput., 109 (2000), 59-65. doi: 10.1016/S0096-3003(99)00016-8.  Google Scholar

[16]

Z. Ang. Math. Phys., 47 (1996), 432-455. doi: 10.1007/BF00916648.  Google Scholar

[17]

Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X.  Google Scholar

[18]

J. Differential Equations, 73 (1988), 309-353. doi: 10.1016/0022-0396(88)90110-6.  Google Scholar

[19]

Originally published by Science Press in 2011. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9543.  Google Scholar

[20]

J. Differential Equations, 163 (2000), 265-291. doi: 10.1006/jdeq.1999.3702.  Google Scholar

[21]

Internat. J. Control, 86 (2013), 1467-1478. doi: 10.1080/00207179.2013.786187.  Google Scholar

[22]

Calc. Var. Partial Differential Equations, 15 (2002), 325-352. doi: 10.1007/s005260100130.  Google Scholar

[23]

J. Differential Equations, 253 (2012), 1422-1438. doi: 10.1016/j.jde.2012.05.016.  Google Scholar

[24]

J. Differential Equations, 244 (2008), 2062-2090. doi: 10.1016/j.jde.2007.10.031.  Google Scholar

[25]

J. Differential Equations, 246 (2009), 4702-4730. doi: 10.1016/j.jde.2008.11.017.  Google Scholar

[26]

Acta Appl. Math., 137 (2015), 123-157. doi: 10.1007/s10440-014-9993-x.  Google Scholar

[27]

Springer, 1985. doi: 10.1007/978-1-4757-4317-3.  Google Scholar

[28]

J. Math. Anal. Appl., 415 (2014), 14-24. doi: 10.1016/j.jmaa.2014.01.059.  Google Scholar

[29]

Amer. Math. Soc., 113 (1991), 701-706. doi: 10.1090/S0002-9939-1991-1072347-4.  Google Scholar

[30]

J. Differential Equations, 182 (2002), 541-576. doi: 10.1006/jdeq.2001.4097.  Google Scholar

[31]

J. Math. Anal. Appl., 267 (2002), 247-263. doi: 10.1006/jmaa.2001.7770.  Google Scholar

[32]

Appl. Anal., 92 (2011), 318-334. doi: 10.1080/00036811.2011.614601.  Google Scholar

[33]

Dynamical systems (Montecatini terme, 1994), 208-315, Lecture Notes in Math. 1609, Springer, Berlin, 1995. doi: 10.1007/BFb0095241.  Google Scholar

[34]

J. Differential Equations, 248 (2010), 342-362. doi: 10.1016/j.jde.2009.08.007.  Google Scholar

[35]

Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743-761. 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]

Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 1029-1052. doi: 10.1017/S0308210515000177.  Google Scholar

[38]

$2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[39]

Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1801-1814. 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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