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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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

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