American Institute of Mathematical Sciences

Advanced
August  2003, 3(3): 401-408. doi: 10.3934/dcdsb.2003.3.401

The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation

Shaoyong Lai 1, and  Yong Hong Wu 2,

1. 

Department of Applied Mathematics, Southwest Jiaotong University, 610066, Chengdu, China
2. 

Department of Mathematics and Statistics, Curtin University of Technology, GOP Box U1987, Perth, WA 6845, Australia

Received  April 2002 Revised  February 2003 Published  May 2003

In this paper, we consider the solution of an initial value problem for the generalized damped Boussinesq equation

$ u_{t t} - a u_{t t x x}- 2 b u_{t x x} = - c u_{x x x x}+ u_{x x} - p^2 u + \beta(u^2)_{x x}, $

where $x\in R^1,$ $t > 0,$ $a ,$ $b$ and $c $ are positive constants, $p \ne 0,$ and $\beta \in R^1$. For the case $a + c > b^2$ corresponding to damped oscillations with an infinite number of oscillation cycles, we establish the well-posedness theorem of the global solution to the problem and derive a large time asymptotic solution.

Keywords: well-posedness, Boussinesq equation., Asymptotic solution.
Mathematics Subject Classification: 34C25, 93C1.
Citation: Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401
[1]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[2]

Hongwei Wang, Amin Esfahani. Well-posedness and asymptotic behavior of the dissipative Ostrovsky equation. Evolution Equations and Control Theory, 2019, 8 (4) : 709-735. doi: 10.3934/eect.2019035
[3]

Seung-Yeal Ha, Jinyeong Park, Xiongtao Zhang. A global well-posedness and asymptotic dynamics of the kinetic Winfree equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1317-1344. doi: 10.3934/dcdsb.2019229
[4]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521
[5]

Qunyi Bie, Qiru Wang, Zheng-An Yao. On the well-posedness of the inviscid Boussinesq equations in the Besov-Morrey spaces. Kinetic and Related Models, 2015, 8 (3) : 395-411. doi: 10.3934/krm.2015.8.395
[6]

Alex M. Montes, Ricardo Córdoba. Local well-posedness for a class of 1D Boussinesq systems. Mathematical Control and Related Fields, 2022, 12 (2) : 447-473. doi: 10.3934/mcrf.2021030
[7]

Rainer Brunnhuber, Barbara Kaltenbacher. Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4515-4535. doi: 10.3934/dcds.2014.34.4515
[8]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[9]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036
[10]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241
[11]

A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469
[12]

Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527
[13]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control and Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[14]

Igor Chueshov, Alexey Shcherbina. Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations. Evolution Equations and Control Theory, 2012, 1 (1) : 57-80. doi: 10.3934/eect.2012.1.57
[15]

Renhui Wan. Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2709-2730. doi: 10.3934/dcds.2019113
[16]

Adalet Hanachi, Haroune Houamed, Mohamed Zerguine. On the global well-posedness of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6473-6506. doi: 10.3934/dcds.2020287
[17]

Saoussen Sokrani. On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1613-1650. doi: 10.3934/dcds.2019072
[18]

Xiaoqiang Dai, Shaohua Chen. Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4201-4211. doi: 10.3934/dcdss.2021114
[19]

Aiting Le, Chenyin Qian. Smoothing effect and well-posedness for 2D Boussinesq equations in critical Sobolev space. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022057
[20]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (112)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy