# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin Heidelberg New York, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag Berlin Heidelberg New York, 2001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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