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

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_+$

$u_t-$u_{txx}-u_{xx}$+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.