American Institute of Mathematical Sciences

Advanced
September  2009, 2(3): 521-550. doi: 10.3934/krm.2009.2.521

Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space

Hui Yin 1, and  Huijiang Zhao 2,

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received  June 2009 Revised  June 2009 Published  July 2009

This paper is concerned with the initial-boundary value problem of the generalized Benjamin-Bona-Mahony-Burgers equation in the half space $ R_+$

$u_t-$utxx-uxx$+f(u)_{x}=0,\ \ \ \ \ t>0,\ \ x\in R_+, $
$u(0,x)=u_0(x)\to u_+,\ \ \ as \ \ x\to +\infty,$
$u(t,0)=u_b$.

Here $u(t,x)$ is an unknown function of $t>0$ and $x\in R_+$, $u_+$≠$u_b$ are two given constant states and the nonlinear function $f(u)$ is a general smooth function.
    Asymptotic stability and convergence rates (both algebraic and exponential) of global solution $u(t,x)$ to the above initial-boundary value problem toward the boundary layer solution $\phi(x)$ are established in [9] for both the non-degenerate case $f'(u_+)<0$ and the degenerate case $f'(u_+)=0$. We note, however, that the analysis in [9] relies heavily on the assumption that $f(u)$ is strictly convex. Moreover, for the non-degenerate case, if the boundary layer solution $\phi(x)$ is monotonically decreasing, only the stability of weak boundary layer solution is obtained in [9]. This manuscript is concerned with the non-degenerate case and our main purpose is two-fold: Firstly, for general smooth nonlinear function $f(u)$, we study the global stability of weak boundary layer solutions to the above initial-boundary value problem. Secondly, when $f(u)$ is convex and the corresponding boundary layer solutions are monotonically decreasing, we discuss the local nonlinear stability of strong boundary layer solution. In both cases, the corresponding decay rates are also obtained.

Keywords: space-time weighted energy method., global stability, decay rate, Generalized Benjamin-Bona-Mahony-Burgers equation; boundary layer solution.
Mathematics Subject Classification: Primary: 35L70, 35B35; Secondary: 35B4.
Citation: Hui Yin, Huijiang Zhao. Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space. Kinetic & Related Models, 2009, 2 (3) : 521-550. doi: 10.3934/krm.2009.2.521
[1]

Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 583-622. doi: 10.3934/dcds.2005.13.583
[2]

Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824
[3]

Wenxia Chen, Ping Yang, Weiwei Gao, Lixin Tian. The approximate solution for Benjamin-Bona-Mahony equation under slowly varying medium. Communications on Pure & Applied Analysis, 2018, 17 (3) : 823-848. doi: 10.3934/cpaa.2018042
[4]

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
[5]

Vishal Vasan, Bernard Deconinck. Well-posedness of boundary-value problems for the linear Benjamin-Bona-Mahony equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3171-3188. doi: 10.3934/dcds.2013.33.3171
[6]

Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851
[7]

Anne-Sophie de Suzzoni. Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2905-2920. doi: 10.3934/dcds.2015.35.2905
[8]

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
[9]

Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267
[10]

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
[11]

Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations & Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21
[12]

Yangrong Li, Renhai Wang, Jinyan Yin. Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2569-2586. doi: 10.3934/dcdsb.2017092
[13]

Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262
[14]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[15]

Hassan Khodaiemehr, Dariush Kiani. High-rate space-time block codes from twisted Laurent series rings. Advances in Mathematics of Communications, 2015, 9 (3) : 255-275. doi: 10.3934/amc.2015.9.255
[16]

Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001
[17]

Norisuke Ioku. Some space-time integrability estimates of the solution for heat equations in two dimensions. Conference Publications, 2011, 2011 (Special) : 707-716. doi: 10.3934/proc.2011.2011.707
[18]

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
[19]

O. Guès, G. Métivier, M. Williams, K. Zumbrun. Boundary layer and long time stability for multi-D viscous shocks. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 131-160. doi: 10.3934/dcds.2004.11.131
[20]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences