American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Validity and dynamics in the nonlinearly excited 6th-order phase equation
  • PROC Home
  • This Issue
  • Next Article
    Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents
2013, 2013(special): 709-717. doi: 10.3934/proc.2013.2013.709

Initial boundary value problem for the singularly perturbed Boussinesq-type equation

Changming Song 1, , Hong Li 1, and  Jina Li 1,

1. 

College of Science, Zhongyuan University of Technology, No.41, Zhongyuan Middle Road, Zhengzhou 450007, China, China, China

Received  September 2012 Revised  February 2013 Published  November 2013

We are concerned with the singularly perturbed Boussinesq-type equation including the singularly perturbed sixth-order Boussinesq equation, which describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to $1/3$. The existence and uniqueness of the global generalized solution and the global classical solution of the initial boundary value problem for the singularly perturbed Boussinesq-type equation are proved.
Keywords: Singularly perturbed Boussinesq-type equation, global classical solution, Boussinesq equation., global generalized solution, initial boundary value problem.
Mathematics Subject Classification: Primary: 35A01, 35L35; Secondary: 35G31, 35Q3.
Citation: Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709
References:
[1]

R. A. Admas, "Sobolev Space", Academic Press, New York, 1975. Google Scholar

[2]

P. Darapi and W. Hua, A numerical method for solving an ill-posed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput., 101 (1999), 159-207.  Google Scholar

[3]

P. Darapi and W. Hua, Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation, Math. Comput. Sim., 55 (2001), 393-405. Google Scholar

[4]

R. K. Dash and P. Darapi, Analytical and numerical studies of a singularly perturbed Boussinesq equation, Appl. Math. Comput., 126 (2002), 1-30.  Google Scholar

[5]

Z. S. Feng, Traveling solitary wave solutions to the generalized Boussinesq equation, Wave Motion, 37 (2003), 17-23.  Google Scholar

[6]

A. Friedman, "Partial Differential Equation of Parabolic Type", Prentice Hall, Eagliweed Cliffs, NJ, 1964.  Google Scholar

[7]

H. A. Levine and B. D. Sleeman, A note on the non-existence of global solutions of initial boundary value problems for the Boussinesq equation $u_{t t} = 3u_{x x x x} + u_{x x} - 12(u^2)_{x x}$, J. Math. Anal. Appl., 107 (1985), 206-210.  Google Scholar

[8]

Z.J. Yang, On local existence of solutions of initial boundary value problems for the "bad'' Boussinesq-type equation, Nonlinear Anal. TMA, 51 (2002), 1259-1271.  Google Scholar

[9]

Y. L. Zhou and H. Y. Fu, Nonlinear hyperbolic systems of higher order generalized Sine-Gordon type, Acta Math. Sinica, 26 (1983), 234-249.  Google Scholar

show all references

References:
[1]

R. A. Admas, "Sobolev Space", Academic Press, New York, 1975. Google Scholar

[2]

P. Darapi and W. Hua, A numerical method for solving an ill-posed Boussinesq equation arising in water waves and nonlinear lattices, Appl. Math. Comput., 101 (1999), 159-207.  Google Scholar

[3]

P. Darapi and W. Hua, Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation, Math. Comput. Sim., 55 (2001), 393-405. Google Scholar

[4]

R. K. Dash and P. Darapi, Analytical and numerical studies of a singularly perturbed Boussinesq equation, Appl. Math. Comput., 126 (2002), 1-30.  Google Scholar

[5]

Z. S. Feng, Traveling solitary wave solutions to the generalized Boussinesq equation, Wave Motion, 37 (2003), 17-23.  Google Scholar

[6]

A. Friedman, "Partial Differential Equation of Parabolic Type", Prentice Hall, Eagliweed Cliffs, NJ, 1964.  Google Scholar

[7]

H. A. Levine and B. D. Sleeman, A note on the non-existence of global solutions of initial boundary value problems for the Boussinesq equation $u_{t t} = 3u_{x x x x} + u_{x x} - 12(u^2)_{x x}$, J. Math. Anal. Appl., 107 (1985), 206-210.  Google Scholar

[8]

Z.J. Yang, On local existence of solutions of initial boundary value problems for the "bad'' Boussinesq-type equation, Nonlinear Anal. TMA, 51 (2002), 1259-1271.  Google Scholar

[9]

Y. L. Zhou and H. Y. Fu, Nonlinear hyperbolic systems of higher order generalized Sine-Gordon type, Acta Math. Sinica, 26 (1983), 234-249.  Google Scholar

[1]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[2]

Miao Liu, Weike Wang. Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1203-1222. doi: 10.3934/cpaa.2014.13.1203
[3]

Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431
[4]

Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401
[5]

Jing Li, Boling Guo, Lan Zeng, Yitong Pei. Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1345-1360. doi: 10.3934/dcdsb.2019230
[6]

Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917
[7]

Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015
[8]

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170
[9]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[10]

Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104
[11]

V. A. Dougalis, D. E. Mitsotakis, J.-C. Saut. On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1191-1204. doi: 10.3934/dcds.2009.23.1191
[12]

Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020040
[13]

Dongfen Bian. Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1591-1611. doi: 10.3934/dcdss.2016065
[14]

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 & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[15]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[16]

Zhijian Yang, Na Feng, Yanan Li. Robust attractors for a Kirchhoff-Boussinesq type equation. Evolution Equations & Control Theory, 2020, 9 (2) : 469-486. doi: 10.3934/eect.2020020
[17]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020
[18]

Xuecheng Wang. Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 5037-5048. doi: 10.3934/dcds.2017217
[19]

Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005
[20]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019

 Impact Factor: 

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences