American Institute of Mathematical Sciences

Advanced
October  2009, 23(4): 1141-1168. doi: 10.3934/dcds.2009.23.1141

Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane

Jerry Bona 1, and  Jiahong Wu 2,

1. 

Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago , 851 S. Morgan Street MC 249, Chicago, Illinois 60607-7045
2. 

Department of Mathematics, Oklahoma State University, United States

Received  January 2007 Revised  April 2008 Published  November 2008

Studied here is the large-time behavior and eventual periodicity of solutions of initial-boundary-value problems for the BBM equation and the KdV equation, with and without a Burgers-type dissipation appended. It is shown that the total energy of a solution of these problems grows at an algebraic rate which is in fact sharp for solutions of the associated linear equations. We also establish that solutions of the linear problems are eventually periodic if the boundary data are periodic.
Keywords: Large-time behavior, BBM-Burgers equation, KdV equation, BBM equation, KdV-Burgers equation, eventual periodicity.
Mathematics Subject Classification: Primary: 35Q20, 35B40.
Citation: Jerry Bona, Jiahong Wu. Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1141-1168. doi: 10.3934/dcds.2009.23.1141
[1]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15
[2]

Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036
[3]

Chi Hin Chan, Magdalena Czubak, Luis Silvestre. Eventual regularization of the slightly supercritical fractional Burgers equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 847-861. doi: 10.3934/dcds.2010.27.847
[4]

Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897
[5]

Zhaosheng Feng, Qingguo Meng. Exact solution for a two-dimensional KDV-Burgers-type equation with nonlinear terms of any order. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 285-291. doi: 10.3934/dcdsb.2007.7.285
[6]

Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47
[7]

Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121
[8]

Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure & Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835
[9]

Amin Esfahani. Remarks on a two dimensional BBM type equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1111-1127. doi: 10.3934/cpaa.2012.11.1111
[10]

Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253
[11]

Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565
[12]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241
[13]

Yvan Martel, Frank Merle. Inelastic interaction of nearly equal solitons for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 487-532. doi: 10.3934/dcds.2010.27.487
[14]

Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793
[15]

Juan-Ming Yuan, Jiahong Wu. The complex KdV equation with or without dissipation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 489-512. doi: 10.3934/dcdsb.2005.5.489
[16]

Karl Kunisch, Lijuan Wang. The bang-bang property of time optimal controls for the Burgers equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3611-3637. doi: 10.3934/dcds.2014.34.3611
[17]

Yanbin Tang, Ming Wang. A remark on exponential stability of time-delayed Burgers equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 219-225. doi: 10.3934/dcdsb.2009.12.219
[18]

Khaled El Dika. Smoothing effect of the generalized BBM equation for localized solutions moving to the right. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 973-982. doi: 10.3934/dcds.2005.12.973
[19]

Jerry L. Bona, Hongqiu Chen, Chun-Hsiung Hsia. Well-posedness for the BBM-equation in a quarter plane. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1149-1163. doi: 10.3934/dcdss.2014.7.1149
[20]

Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences