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Kinetic and Related Models (KRM)
 

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

Pages: 521 - 550, Volume 2, Issue 3, September 2009      doi:10.3934/krm.2009.2.521

 
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Hui Yin - School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China (email)
Huijiang Zhao - School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China (email)

Abstract: 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:  Generalized Benjamin-Bona-Mahony-Burgers equation; boundary layer solution, global stability, decay rate, space-time weighted energy method.
Mathematics Subject Classification:  Primary: 35L70, 35B35; Secondary: 35B40.

Received: June 2009;      Revised: June 2009;      Available Online: July 2009.