- Previous Article
- KRM Home
- This Issue
-
Next Article
A local existence result for a plasma physics model containing a fully coupled magnetic field
Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space
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 |
$u_t-$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.
[1] |
Khaled El Dika. Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267 |
[10] |
Marjan Uddin, Hazrat Ali. Space-time kernel based numerical method for generalized Black-Scholes equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2905-2915. doi: 10.3934/dcdss.2020221 |
[11] |
Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure and Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835 |
[12] |
Xenia Kerkhoff, Sandra May. Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021054 |
[13] |
Qiangheng Zhang. Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021293 |
[14] |
Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021026 |
[15] |
Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations and Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21 |
[16] |
Shuai Liu, Yuzhu Wang. Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021041 |
[17] |
Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262 |
[18] |
Yangrong Li, Renhai Wang, Jinyan Yin. Backward compact attractors for non-autonomous Benjamin-Bona-Mahony equations on unbounded channels. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2569-2586. doi: 10.3934/dcdsb.2017092 |
[19] |
Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216 |
[20] |
Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037 |
2020 Impact Factor: 1.432
Tools
Metrics
Other articles
by authors
[Back to Top]