American Institute of Mathematical Sciences

Advanced
October  1998, 4(4): 783-793. doi: 10.3934/dcds.1998.4.783

Attractor for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities

Boling Guo 1, and  Zhengde Dai 2,

1. 

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088
2. 

Department of Mathematics, Yunnan University, Kunming 650091, China

Received  April 1997 Revised  May 1998 Published  July 1998

In this paper, the long time behavior of solution for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities is considered. First the global weak attractor for this equation in $E_1$ is constructed. And then by exact analysis of energy equations, it is showed that the global weak attractor is actually the global strong attractor in $E_1$.
Keywords: global attractor, dissipation., Hamiltonian amplitude equation.
Mathematics Subject Classification: 35K57, 35B4.
Citation: Boling Guo, Zhengde Dai. Attractor for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 783-793. doi: 10.3934/dcds.1998.4.783
[1]

Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043
[2]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345
[3]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015
[4]

Xumin Gu. Global wellposedness for a transport equation with super-critial dissipation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 653-665. doi: 10.3934/cpaa.2011.10.653
[5]

D. Blömker, S. Maier-Paape, G. Schneider. The stochastic Landau equation as an amplitude equation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 527-541. doi: 10.3934/dcdsb.2001.1.527
[6]

Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824
[7]

Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060
[8]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094
[9]

Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151
[10]

D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557
[11]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107
[12]

Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410
[13]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031
[14]

Deconinck Bernard, Olga Trichtchenko. High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1323-1358. doi: 10.3934/dcds.2017055
[15]

Guenbo Hwang, Byungsoo Moon. Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion. Electronic Research Archive, 2020, 28 (1) : 15-25. doi: 10.3934/era.2020002
[16]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078
[17]

George Avalos, Pelin G. Geredeli, Justin T. Webster. Finite dimensional smooth attractor for the Berger plate with dissipation acting on a portion of the boundary. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2301-2328. doi: 10.3934/cpaa.2016038
[18]

Juan-Ming Yuan, Jiahong Wu. The complex KdV equation with or without dissipation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 489-512. doi: 10.3934/dcdsb.2005.5.489
[19]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843
[20]

Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences