May  2016, 21(3): 763-779. doi: 10.3934/dcdsb.2016.21.763

Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane

1. 

Department of Mathematics, National Central University, Chung-Li 32001, Taiwan

2. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 81148, Taiwan

3. 

Department of Mathematics, Oklahoma State University, Stillwater, OK 74078

4. 

Department of Financial and Computational Mathematics, Providence University, Taichung 43301

Received  April 2014 Revised  September 2015 Published  January 2016

This paper focuses on the two-dimensional Benjamin-Bona-Mahony and Benjamin-Bona-Mahony-Burgers equations with a general flux function. The aim is at the global (in time) well-posedness of the initial-and boundary-value problem for these equations defined in the upper half-plane. Under suitable growth conditions on the flux function, we are able to establish the global well-posedness in a Sobolev class. When the initial- and boundary-data become more regular, the corresponding solutions are shown to be classical. In addition, the continuous dependence on the data is also obtained.
Citation: C. H. Arthur Cheng, John M. Hong, Ying-Chieh Lin, Jiahong Wu, Juan-Ming Yuan. Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 763-779. doi: 10.3934/dcdsb.2016.21.763
References:
[1]

C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49. doi: 10.1016/0022-0396(89)90176-9.

[2]

J. Avrin and J. A. Goldstein, Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions, Nonlinear Anal., 9 (1985), 861-865. doi: 10.1016/0362-546X(85)90023-9.

[3]

J. L. Bona and P. J. Bryant, A mathematical model for long waves generated by wavemakers in non-linear dispersive systems, Proc. Camb. Phil. Soc., 73 (1973), 391-405. doi: 10.1017/S0305004100076945.

[4]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive system, Phil. Trans. R. Soc. London., Ser A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.

[5]

J. L. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations, Discrete Contin. Dyn. Syst., 23 (2009), 1253-1275. doi: 10.3934/dcds.2009.23.1253.

[6]

J. L. Bona, H. Chen, S. M. Sun and B.-Y. Zhang, Approximating initial-value problems with two-point boundary-value problems: BBM-equation, Bull. Iranian Math. Soc., 36 (2010), 1-25.

[7]

J. L. Bona, H. Chen, S. M. Sun and B.-Y. Zhang, Comparison of quarter-plane and two-point boundary value problem: The BBM-equation, Discrete Contin. Dyn. Syst., 13 (2005), 921-940. doi: 10.3934/dcds.2005.13.921.

[8]

J. L. Bona and V. A. Dougalis, An initial- and boundary-value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522. doi: 10.1016/0022-247X(80)90098-0.

[9]

J. L. Bona and L. Luo, More results on the decay of solutions to nonlinear, dispersive wave equations, Discrete Contin. Dyn. Syst., 1 (1995), 151-193. doi: 10.3934/dcds.1995.1.151.

[10]

J. L. Bona and L. Luo, Initial-boundary value problems for model equations for the propagation of long waves, Evolution equations (Baton Rouge, LA, 1992), 63, 65-94, Lecture Notes in Pure and Appl. Math., 168, Dekker, New York, 1995.

[11]

J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Phil. Trans. R. Soc. London., Ser A, 302 (1981), 457-510. doi: 10.1098/rsta.1981.0178.

[12]

J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252. doi: 10.3934/dcds.2009.23.1241.

[13]

J. L. Bona and J. Wu, Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane, Discrete Contin. Dyn. Syst., 23 (2009), 1141-1168. doi: 10.3934/dcds.2009.23.1141.

[14]

S. E. Buckley and M. C. Leverett, Mechanism of fluid displacements in sands, Trans. AIME, 146 (1942), 107-116. doi: 10.2118/942107-G.

[15]

C. Guo and S. Fang, Optimal decay rates of solutions for a multidimensional generalized Benjamin-Bona-Mahony equation, Nonlinear Anal., 75 (2012), 3385-3392. doi: 10.1016/j.na.2011.12.035.

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin Heidelberg New York, 2001.

[17]

J. A. Goldstein and B. J. Wichnoski, On the Benjamin-Bona-Mahony equation in higher dimensions, Nonlinear Anal., 4 (1980), 665-675. doi: 10.1016/0362-546X(80)90067-X.

[18]

J.-M. Hong, J. Wu and J.-M. Yuan, A new solution representation for the BBM equation in a quarter plane and the eventual periodicity, Nonlinearity, 22 (2009), 1927-1944. doi: 10.1088/0951-7715/22/8/009.

[19]

G. Karch, Asymptotic behaviour of solutions to some pseudoparabolic equations, Math. Methods Appl. Sci., 20 (1997), 271-289. doi: 10.1002/(SICI)1099-1476(199702)20:3<271::AID-MMA859>3.0.CO;2-F.

[20]

M. Mei, $L_q$-decay rates of solutions for Benjamin-Bona-Mahony-Burgers equations, J. Differential Equations, 158 (1999), 314-340. doi: 10.1006/jdeq.1999.3638.

[21]

Y. Martel, F. Merle and T. Mizumachi, Description of the inelastic collision of two solitary waves for the BBM equation, Arch. Rational Mech. Anal., 196 (2010), 517-574. doi: 10.1007/s00205-009-0244-7.

show all references

References:
[1]

C. J. Amick, J. L. Bona and M. E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations, 81 (1989), 1-49. doi: 10.1016/0022-0396(89)90176-9.

[2]

J. Avrin and J. A. Goldstein, Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions, Nonlinear Anal., 9 (1985), 861-865. doi: 10.1016/0362-546X(85)90023-9.

[3]

J. L. Bona and P. J. Bryant, A mathematical model for long waves generated by wavemakers in non-linear dispersive systems, Proc. Camb. Phil. Soc., 73 (1973), 391-405. doi: 10.1017/S0305004100076945.

[4]

T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive system, Phil. Trans. R. Soc. London., Ser A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.

[5]

J. L. Bona and H. Chen, Well-posedness for regularized nonlinear dispersive wave equations, Discrete Contin. Dyn. Syst., 23 (2009), 1253-1275. doi: 10.3934/dcds.2009.23.1253.

[6]

J. L. Bona, H. Chen, S. M. Sun and B.-Y. Zhang, Approximating initial-value problems with two-point boundary-value problems: BBM-equation, Bull. Iranian Math. Soc., 36 (2010), 1-25.

[7]

J. L. Bona, H. Chen, S. M. Sun and B.-Y. Zhang, Comparison of quarter-plane and two-point boundary value problem: The BBM-equation, Discrete Contin. Dyn. Syst., 13 (2005), 921-940. doi: 10.3934/dcds.2005.13.921.

[8]

J. L. Bona and V. A. Dougalis, An initial- and boundary-value problem for a model equation for propagation of long waves, J. Math. Anal. Appl., 75 (1980), 503-522. doi: 10.1016/0022-247X(80)90098-0.

[9]

J. L. Bona and L. Luo, More results on the decay of solutions to nonlinear, dispersive wave equations, Discrete Contin. Dyn. Syst., 1 (1995), 151-193. doi: 10.3934/dcds.1995.1.151.

[10]

J. L. Bona and L. Luo, Initial-boundary value problems for model equations for the propagation of long waves, Evolution equations (Baton Rouge, LA, 1992), 63, 65-94, Lecture Notes in Pure and Appl. Math., 168, Dekker, New York, 1995.

[11]

J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Phil. Trans. R. Soc. London., Ser A, 302 (1981), 457-510. doi: 10.1098/rsta.1981.0178.

[12]

J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252. doi: 10.3934/dcds.2009.23.1241.

[13]

J. L. Bona and J. Wu, Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane, Discrete Contin. Dyn. Syst., 23 (2009), 1141-1168. doi: 10.3934/dcds.2009.23.1141.

[14]

S. E. Buckley and M. C. Leverett, Mechanism of fluid displacements in sands, Trans. AIME, 146 (1942), 107-116. doi: 10.2118/942107-G.

[15]

C. Guo and S. Fang, Optimal decay rates of solutions for a multidimensional generalized Benjamin-Bona-Mahony equation, Nonlinear Anal., 75 (2012), 3385-3392. doi: 10.1016/j.na.2011.12.035.

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin Heidelberg New York, 2001.

[17]

J. A. Goldstein and B. J. Wichnoski, On the Benjamin-Bona-Mahony equation in higher dimensions, Nonlinear Anal., 4 (1980), 665-675. doi: 10.1016/0362-546X(80)90067-X.

[18]

J.-M. Hong, J. Wu and J.-M. Yuan, A new solution representation for the BBM equation in a quarter plane and the eventual periodicity, Nonlinearity, 22 (2009), 1927-1944. doi: 10.1088/0951-7715/22/8/009.

[19]

G. Karch, Asymptotic behaviour of solutions to some pseudoparabolic equations, Math. Methods Appl. Sci., 20 (1997), 271-289. doi: 10.1002/(SICI)1099-1476(199702)20:3<271::AID-MMA859>3.0.CO;2-F.

[20]

M. Mei, $L_q$-decay rates of solutions for Benjamin-Bona-Mahony-Burgers equations, J. Differential Equations, 158 (1999), 314-340. doi: 10.1006/jdeq.1999.3638.

[21]

Y. Martel, F. Merle and T. Mizumachi, Description of the inelastic collision of two solitary waves for the BBM equation, Arch. Rational Mech. Anal., 196 (2010), 517-574. doi: 10.1007/s00205-009-0244-7.

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