\documentstyle[twoside,epsf]{article}
\pagestyle{myheadings}
\markboth{\hfil Stability of strong detonation waves \hfil EJDE--1998/09}%
{EJDE--1998/09\hfil Tong Li \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~09, pp. 1--17. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
 Stability of strong detonation waves  \\ and rates of convergence
\thanks{ {\em 1991 Mathematics Subject Classifications:} 
35L65, 35B40, 35B50, 76L05, 76J10.
\hfil\break\indent
{\em Key words and phrases:} Strong detonation, shock wave, traveling wave, 
\hfil\break\indent asymptotic behavior, weighted energy estimate.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted October 14, 1997. Published March 18, 1998.} }
\date{}
\author{Tong Li}
\maketitle

\begin{abstract} 
In this article, we prove stability of  strong detonation waves and find
their rate of convergence for a combustion model. Our results read as follows:
I) There exists a global solution that converges exponentially in time to
a strong detonation wave, provided that the initial data is a small
perturbation of a strong detonation wave that decays exponentially in
$|x|$.
II) When the initial perturbation decays algebraically in $|x|$,  
the solution converges algebraically in time. 
That is, the perturbation decays in $t$ as `fast' as the initial 
perturbation decays in $|x|$.
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}

\section{Introduction} 
Physical experimentation has shown that in a sufficiently insensitive
mixture or in a typical condensed phase, explosive
detonation waves approach a steady state as time goes by.
The study of this steady state is a subject in explosive engineering and is
based on measurements of pressure, velocity and other observables of
detonation waves. To learn about the structure and the behavior of the
steady state, we formulate questions such as:
How does a detonation wave respond to a perturbation?
How quickly is the steady state is attained? And what are the details
of the flow as the steady solution is approached?
In particular, hydrodynamic stability of the steady detonation is very
interesting question, and  has received a lot of attention.
Fickett \cite{F} studied the decay of small planar perturbations
for strong steady detonation in a simple model.
His work uses the linearization technique of hydrodynamic stability theory
introduced by Erpenbeck \cite{ER1}.
Liu and Ying  \cite{LY} proved that the strong detonation is stable for a
combustion model, but did not show rates of convergence.
In the present paper, we show that a perturbation to strong detonation wave
in a combustion model decays in $t$ as `fast' as the initial perturbation
decays in $|x|$.
 
We study the dynamic combustion model
\begin{eqnarray}
&u_{t}+(f(u)-qz)_{x}=\epsilon u_{xx}& \label{1} \\
&z_{x}=k\varphi(u)z\,,&\label{2}
\end{eqnarray}
where $u=u(x,t)$ and $z=z(x,t)$ are scalar functions
representing the velocity or the temperature of the combustible gas,
and the concentration of the unburnt gas; and the constants
$q$, $\epsilon$, and $k>0$ represent the amount of heat released during
the chemical reaction, the viscous coefficient, and the reaction rate,
respectively.
The reaction rate function has the form
\begin{equation}
\varphi(u)=\left\{
\begin{array}{ll}
0 &u\leq u_{i}\\
\mbox{a smooth increasing function}&u_i<u\leq 2u_i\\
1&u> 2u_{i}\,,
\end{array}\right.\label{rr}
\end{equation}
where $u_i\geq 0$ is a constant related to the ignition temperature.
 
Motivated by the study of shock waves for gas dynamics, and by the
asymptotic analysis performed in \cite{M}, we require the that the
flux $f$ satisfy
\[
f(0)=0\,,  \quad f'(0)>0\,, \quad f''(u)>0\,.
\]
To make (\ref{1})-(\ref{2}) a well-posed problem, the data are assumed to
satisfy 
\begin{eqnarray}
&u(x,0)=u_0(x)\,,& \label{i1}\\
&z(+\infty,t)=1\,.&\label{i2}
\end{eqnarray}
This model was derived by Rosales and Majda \cite{M} under the assumptions
of weak nonlinearity, high activation energy, and
nearly sonic speed of the detonation wave.
It describes the one-dimensional flow of a reactive gas with a high Mach
number. It includes the two important physical mechanisms for this
type of problem: the nonlinear transport and the chemical reaction
through the energy release term.

Under appropriate conditions on the parameters $q$ and $k_0=\epsilon k$,
this model predicted the qualitative internal structure of the  strong
detonation assumed by Zeldovich-von Neumann-Doring \cite{M}.
 i.e., a detonation
wave traveling at speed $D$ has the internal structure of an ordinary
precursor fluid dynamic shock wave traveling at speed $D$, followed
by a reaction zone. The parameter $k_0$ measures the ratio of
the width of the analogue of the fluid dynamic shock layer
and the width of the reaction zone.
The detonation wave has the form
$$
(u(x,t),z(x,t))=(\psi(x-Dt),Z(x-Dt))=(\psi(\xi), Z(\xi))\,,
$$
where $\xi=x-Dt$ is the traveling wave variable, and the 
pair $(\psi,Z)(\xi)$ is a solution to the system 
\begin{eqnarray}
&-D\psi'+f'(\psi)\psi'=\epsilon \psi''+ qZ' &\label{t1}\\
&Z'= k\varphi(\psi)Z\,. &\label{t2}
\end{eqnarray}
When the boundary conditions are
\begin{eqnarray}
&\lim_{\xi\rightarrow-\infty}(\psi,Z)(\xi)=(u_l,0)& \label{t3}\\
&\lim_{\xi\rightarrow+\infty}(\psi,Z)(\xi)=(0,1)\,,&\label{t4}
\end{eqnarray}
then the propagation speed $D$ is determined by the boundary data,
$$
D={f(\psi(+\infty))-f(\psi(-\infty))-qZ(+\infty)\over\psi(+\infty)
-\psi(-\infty)}\,.
$$
We will consider only strong detonation in this paper, that is,
\begin{equation}
f'(\psi(+\infty))<D<f'(\psi(-\infty))\,.\label{strong}
\end{equation}

\begin{figure}
\begin{center}
  \leavevmode{
    \epsfxsize=12cm
    \epsffile{fig1.ps}
  }
\end{center}
\caption{Traveling wave profile with $O(\delta)=O(q)$}
\label{strong.fig}
\end{figure}
 
The traveling wave solution has a non-monotone spike
in the $u$-profile, which corresponds to a von Neumann spike.
There is a maximum point $\xi_0$ on the profile,
which separates the reaction zone and the viscous shock wave. See
Figure~\ref{strong.fig}.
The traveling wave solution decays exponentially to its end state,
as $|\xi|\rightarrow +\infty$.
 
We are concerned with the stability of the detonation waves described above.
Stability of nonlinear problems for viscous shock waves for systems of
nonlinear conservation laws have been studied by Goodman \cite{gd}, 
Kawashima and Matsumura \cite{KM},
Liu \cite{Liu}, and Liu and Xin \cite{LX}.
 
By restricting our attention to strong detonation waves, we avoid dealing
with the sonic point at the end of the reaction zone, which is present in
the Chapman-Jouguet detonation, and with the unsupported case.
Studies on the stability of a Chapman-Jouguet detonation are presented in
another article by the author, \cite{Li2}.

We establish the stability of strong detonation waves
modeled by (\ref{1})-(\ref{2}). Then, inspired by the
work of Kawashima and Matsumura \cite{KM}, we find rates of convergence.
Our results are stated as follows:
\begin{description}
\item{(I)} There exits a  global solution to (\ref{1})-(\ref{2}),
 (\ref{i1})-(\ref{i2}), which converges exponentially
in time to a traveling wave,  provided the initial data is a small perturbation
of the traveling wave that decays exponentially in $|x|$.
\item{(II)} If the rate of decay of the initial perturbation in the far
 fields is algebraic instead,
the solution converges algebraically in time.
\end{description}
The weighted-energy method developed in \cite{KM} is used to
obtain the rate of convergence.
The characteristic-energy method used in \cite{LX, LY}
is employed to deal with the difficulties arising from
the non-monotonicity of the profile.
Estimates involving $z$ will be obtained through detailed
$L^{2}$ and pointwise estimates.
 
In Section 2, we state our main results.
Section 3 is a detailed stability analysis for the exponential decay.
In Section 4, we study the algebraic decay.

\section{Assumptions and main results}
Due to the fact that (\ref{1}) is a conservation law, we choose to work
with the anti-derivative $v(x,t)$ of the perturbation $u(x,t)-\psi(x-Dt)$;
see \cite{gd, Liu}. The function $v(x,t)$ is defined as
\begin{equation}
v(x,t)=\int^x_{-\infty} (u(x,t)-\psi(x-Dt))\,dx\,.\label{anti} 
\end{equation}

We make the following assumptions on the data.
 
\paragraph{1.) Zero initial integral difference:}
\begin{equation}
\int^{+\infty}_{-\infty} (u_0(x)-\psi(x))dx=0\,. \label{2.1}
\end{equation}
Note that (\ref{anti}), (\ref{2.1}) imply 
\begin{equation}
v(\pm\infty, t)=0\,.\label{2.2}
\end{equation}
This because (\ref{1}) makes the 
integral $v(x,t)$ in (\ref{anti}) a conserved quantity, i.e.,
\[
{d\over dt }\int_{-\infty}^{+\infty}(u(x,t)-\psi(x-Dt))\,dx=0\,,
\]
and
\[
\int_{-\infty}^{+\infty}(u(x,t)-\psi(x-Dt))dx=\mbox
{constant}
\]
which equals zero due to our choice of data (\ref{2.1}).

\paragraph{2.) Small heat release:}
$0<q\ll \epsilon \ll 1$. So for $\xi\leq \xi_0$,
\begin{eqnarray}
&0<\int_{-\infty}^{\xi} f'(\psi(\xi))_\xi\,d\xi =\delta_1<C q\ll \epsilon 
\ll 1\,,&  \label{2.3} \\
 &0<f'(\psi(\xi))_\xi <\delta_2<Cq\ll\epsilon \ll  1\,.&\label{2.4}
\end{eqnarray}
These assumptions make the non-monotone spike of the strong detonation
profile small (see Figure~\ref{strong.fig}); so that the characteristic
 energy estimate can be obtained.

The smallness condition on the initial data and the stability analysis
to be performed imply  that there exist
$\xi_1$ and $\xi_2$, $m>0$ such that $\xi_0 <\xi_1 < \xi_2$ and
\begin{eqnarray}
&\varphi(\psi)=\varphi(u)=0\,,\quad \xi>\xi_2&\label{2.6} \\
&\varphi(\psi)=\varphi(u)=1 \,,\quad \xi<\xi_1\,.&\label{2.7}
\end{eqnarray}
Therefore,
\begin{eqnarray}
&-f'(\psi (\xi))_\xi > m > 0\,, \quad \xi_1 <\xi <\xi_2&\label{2.8} \\
&f'(\psi(\xi))-D>m>0\,, \quad \xi<\xi_1\,,& \label{2.9}
\end{eqnarray}
where (\ref{2.9}) holds  because the detonation under consideration
is strong (see (\ref{strong})).

Again because the detonation wave is  strong,
we can find a $\xi_*\in (\xi_1, \xi_2)$ such that
\begin{equation}
f'(\psi(\xi_*))=D\,.\label{velo}
\end{equation}
See Figure~\ref{strong.fig}.
\smallskip

 Now we introduce some notation. Let
\[
L^2=\{ v \mid \int_{-\infty}^{+\infty} v^2 dx < +\infty\}
\]
and
\[
H^2=\{v\mid v\in L^2, v_{x}\in L^2, v_{xx}\in L^2\}.
\]
Let $\omega(x)=\exp(\alpha \langle x-\xi_*\rangle)$ where
$\langle x\rangle =(1+x^2)^{1/2}$. Then we define the space
\[
H^2_{\omega}=\{v\mid ve^{{1 \over 2}\alpha \langle x-\xi_*\rangle}\in H^2\}\,,
\]
the associated norm  
\[
\| v \|_{H^2_{\omega}}=\left(\int_{-\infty}^{\infty}
\omega (v^2+v_x^2+v_{xx}^2)dx\right)^{1/2}.
\]
Our main result of exponential decay is:

\begin{theorem} \label{th2.1}
Suppose that $v_0\in H^2_{\omega}$,
$\| v_0\|_{H^2_{\omega}}\ll 1$, and
Assumptions 1 and 2 from the previous section hold.
Then there exists a global solution, $v(\cdot, t)\in H^2_{\omega}$,
to (\ref{1})-(\ref{2}), (\ref{i1})-(\ref{i2}) satisfying
\begin{equation}
\| v(\cdot, t)\|_{H^2_{\omega}}\leq \| v_0(\cdot)\|_{H^2_{\omega}}
e^{-\beta t}.  \label{2.10}
\end{equation}
Consequently, 
\begin{equation}
\sup_{-\infty <x<+\infty} |u(x,t)-\psi (x-Dt)|\leq C e^{-\beta t/2}
\,, \label{2.11}
\end{equation}
where $\beta$ is a positive constant that  depends on
$\alpha, k, f,  \epsilon$; and $C$ depends only on initial data.
\end{theorem}
 
To state the algebraic decay result, we introduce the following notation.
Let \[
L^2_{\alpha}=\{v\mid \langle x-\xi_*\rangle^{\alpha/2}v\in L^2\}\,,
\]
with the associated norm 
\[
 \| v \|_{\alpha}=\left(\int_{-\infty}^{+\infty}
 \langle x-\xi_*\rangle^{\alpha}|v|^2\, dx
\right)^{1/2}.
\] 

\begin{theorem} \label{th2.2}
Assume that $v_0\in L_{\alpha}^2$, $v_{0,x}\in H^1$,
$N_{\alpha}=\| v_{0,x} \|_1+\| v_0 \|_{\alpha}\ll 1$,
and Assumptions 1 and 2 from  the previous section hold. 
Then there exists a global solution $u(x,t)$
of problem (\ref{1})-(\ref{2}), (\ref{i1})-(\ref{i2}.)
 Moreover, this  solution tends to a traveling
wave solution at the rate $t^{-\alpha/2}$ as $t$ tends to infinity,
in the maximum norm.   i.e.,
\[
\sup_{-\infty<x<+\infty}|u(x, t)-\psi(x-Dt)|\leq C(1+t)^{-\alpha/2}
(\| u_0-\psi\|_1+\| v_0\|_{\alpha})\,,
\]
where $C$ is some positive constant depending on the initial data only.
\end{theorem}

\section{Proof of stability: Exponential decay}
In this section, we prove Theorem \ref{th2.1} by using weighted-energy
estimates and the characteristic-energy method.
 
Assume {\it a priori} that
\begin{equation}
0<\sup_{x,t} |v_{x}(x,t)|=\delta_3\ll 1\, .\label{22.5}
\end{equation}
This assumption will be guaranteed by the smallness of initial data and the
stability analysis to be performed.
 
Let 
\[
u(x,t)=\psi(x-Dt)+v_x(x,t)\,.
\]
Then the anti-derivative of the perturbation is
\[
v(x,t)=\int_{-\infty}^x (u(y,t)-\psi(y-Dt))\,dy\,.
\]

Subtract (\ref{t1}), which is satisfied by $\psi(x-D_{CJ}t)$,
from (\ref{1}), which is satisfied by $u(x,t)$. Then integrate  
from $-\infty$ to $x$, and write the result in
terms of the anti-derivative $v(x,t)$. In terms of the traveling-wave
variable $\xi$, we have
\[
v_t-Dv_{\xi}+f(\psi+v_{\xi})-f(\psi)-q(z-Z)=\epsilon v_{\xi \xi}.
\] 
This expression can be rewritten as
\begin{equation}
v_t(\xi,t)+(f'(\psi)-D)v_{\xi}= \epsilon v_{\xi\xi}+qw+F(v_{\xi}, \psi)\,,
 \label{3.1}
\end{equation}
where $w(\xi,t)=z(\xi,t)-Z(\xi)$, and for $|v_{\xi}|$ small,
\[
|F|=|-(f(\psi+v_{\xi})-f(\psi)-f'(\psi)v_{\xi})|\leq C|v_{\xi}|^2\,.
\]

Now, we prove a result that plays an important role
in deriving weighted energy estimates, as in [3].
 
\begin{lemma} \label{l1}
Let
\begin{eqnarray}
G_{\alpha}(\xi)=\left\{
\begin{array}{ll}
-{1\over 2}f'(\psi(\xi))_\xi-{\alpha\over 2}
{(\xi-\xi_*)\over \langle\xi-\xi_*\rangle}(f'(\psi(\xi))-D), & \xi_0<\xi<+\infty\\
-{\alpha\over 2}{(\xi-\xi_*)\over \langle\xi-\xi_*\rangle}(f'(\psi(\xi))-D), &
-\infty<\xi<\xi_0.
\end{array}\right.\label{er}
\end{eqnarray}
Then for some positive constant $\beta$,
\begin{equation}
G_{\alpha}(\xi)\geq \left\{
\begin{array}{ll}
\beta-{1\over 4}f'(\psi(\xi))_\xi, &\xi_0<\xi<+\infty\\
\beta, &-\infty<\xi<\xi_0\,.
\end{array}\right.\label{er1}
\end{equation}
\end{lemma}

\paragraph{Proof.} We consider two cases.
Recall that $\xi_*$ satisfies (\ref{velo}). 

Case i) When $\xi$ is close to $\xi_*$, we have that $G_{\alpha}(\xi_*)$
is close to $-{1\over 2}f'(\psi(\xi_*))_\xi>m>0$, 
see (\ref{2.8}) and Figure \ref{strong.fig}.
 Choose $\beta$ such that
\[
0< \beta< -{1\over 8}f'(\psi(\xi_*))_\xi\,.
\]
Then
\[
G_{\alpha}(\xi)\geq \beta-{1\over 8}f'(\psi(\xi_*))_\xi\geq  \beta-{1\over 4}
f'(\psi(\xi))_\xi\,.
\]

Case ii) When  $\xi$ is away from $\xi_*$, say, $|\xi-\xi_*|>\delta_0$,
then from (\ref{2.9}) it follows that
\[
G_{\alpha}(\xi)\geq \alpha c m>0
\]
for $-\infty<\xi<\xi_0$ and $c>0$.
For $\xi_0<\xi<+\infty$, the convexity of $f$ gives us
\[
G_{\alpha}(\xi)\geq \alpha c\delta_0-{1\over 4}f'(\psi(\xi))_\xi\,,
\]
where $c$ is some constant determined by the convexity of $f$.
The desired inequality (\ref{er1}) follows by choosing $\beta$ such that
\[
0<\beta< \min\{ -{1\over 8}f'(\psi(\xi_*))_\xi, \alpha c m, \alpha c\delta_0\}.
\]

\paragraph{Remark.} The condition  that the detonation is strong,
(\ref{strong}), is the key condition in this lemma.
For  Chapman-Jouguet waves  there is not such a result.\smallskip

Now establish our main estimates.
Multiplying (3.1) by $e^{\alpha \langle \xi-\xi_*\rangle}v$ and integrating,
we obtain 
\begin{eqnarray*}
\lefteqn{ {1\over 2}{d\over dt}\int^{+\infty}_{-\infty}
 e^{\alpha \langle\xi-\xi_*\rangle}v^2\, d\xi } && \hspace{9cm} \\
\lefteqn{ +\int^{+\infty}_{-\infty} e^{\alpha \langle\xi-\xi_*\rangle}
(f'(\psi(\xi))-D) vv_\xi\, d\xi 
- \epsilon\int e^{\alpha \langle\xi-\xi_*\rangle}vv_{\xi\xi}\, d\xi } &&\\
& =& \int e^{\alpha \langle\xi-\xi_*\rangle}(qw+F(v_\xi,\psi))v\, d\xi\,.
\end{eqnarray*}
Integrating by parts and using Lemma \ref{l1}, we arrive at our main estimate
\begin{eqnarray}
\lefteqn{ {1\over 2}{d\over dt}\int^{+\infty}_{-\infty} 
 e^{\alpha \langle\xi-\xi_*\rangle}v^2\, d\xi +
\beta\int^{+\infty}_{-\infty}e^{\alpha \langle\xi-\xi_*\rangle}v^2 \, d\xi
} && \hspace{9cm}\nonumber\\
\lefteqn{ +{1\over 2}\int^{\xi_0}_{-\infty} -| f'(\psi)_\xi|e^{\alpha
 \langle\xi-\xi_*\rangle}v^2\, d\xi+ {1\over 4} \int^{+\infty}_{\xi_0}
|f'(\psi)_\xi|e^{\alpha \langle\xi-\xi_*\rangle}v^2\, d\xi } &&\nonumber\\
\lefteqn{ +\epsilon \int^{+\infty}_{-\infty} e^{\alpha 
\langle\xi-\xi_*\rangle}v^2_\xi\, d\xi+\left|\epsilon
\int^{+\infty}_{-\infty} \alpha e^{\alpha \langle\xi-\xi_*\rangle}
{\xi-\xi_*\over \langle\xi-\xi_*\rangle}vv_\xi\, d\xi\right| } &&\nonumber\\
&\leq& \left|\int^{+\infty}_{-\infty}
e^{\alpha \langle\xi-\xi_*\rangle}(qw+F(v_\xi, \psi)) v\, d\xi\right|\,.
\label{main}
\end{eqnarray}

To estimate the last term on the left hand side of (\ref{main}), 
we make use of Schwarz's inequality to obtain
$$
\left|\epsilon\alpha \int^{+\infty}_{-\infty}
 {\xi-\xi_*\over \langle\xi-\xi_*\rangle}\omega(\xi)
vv_\xi \, d\xi\right| 
\leq {\epsilon\over 2}\int^{+\infty}_{-\infty}
\omega(\xi)v_\xi^2 \, d\xi+{\alpha^2\epsilon\over 2}
\int^{+\infty}_{-\infty} \omega(\xi)v^2 \, d\xi\,,
$$ 
where $\omega(\xi)=e^{\alpha\langle\xi-\xi_*\rangle}$. 
 Choose $\alpha$ such that ${\beta  \over 2}\geq {\alpha^2\epsilon\over 2}$.
Then our main estimate becomes
\begin{eqnarray*}
\lefteqn{ {1\over 2}{d\over dt} \int^{+\infty}_{-\infty}
 \omega(\xi)v^2\, d\xi +{\beta\over 2}
 \int^{+\infty}_{-\infty}\omega(\xi)v^2\, d\xi
  +{1\over 2} \int_{-\infty}^{\xi_0} -|f'(\psi)_\xi|
 \omega(\xi)v^2 \, d\xi } &&\hspace{9cm}\\
 \lefteqn{ + {1\over 4}\int_{\xi_0}^{+\infty} | f'(\psi)_\xi|
 \omega(\xi)v^2 \, d\xi
 +{\epsilon \over 2} \int^{+\infty}_{-\infty}
 \omega(\xi)v_\xi^2\, d\xi }& \\
 &\leq& \left|\int^{+\infty}_{-\infty}
 \omega(\xi)(qw+F(v_\xi, \psi)) v\, d\xi\right|\,.
\end{eqnarray*}

Now we use the characteristic-energy method to
estimate the third term on the left hand side of (\ref{main}),
the bad term arising from the non-monotonicity of the profile.
The idea is to integrate  (\ref{3.1}) for $v$ along the
 characteristic direction to get $v^2$, and then plug it in the integration.
The key condition here is that $|f'(\psi(\xi))_\xi|$ is small
due to $q\ll 1$. See (\ref{2.4}).

Let
\[
S(\xi)=(f'(\psi(\xi))-D)^{-1}\,.
\]
Then (\ref{2.9}) implies
\begin{equation}
0<S(\xi)<{1\over m}\,,\quad  -\infty <\xi <\xi_0\ .
\label{3.4}
\end{equation}
Multiplying (\ref{3.1}) by $vS$, and then integrating from $-\infty$ to
$\xi$, we obtain
\begin{eqnarray*}
 {1\over 2} v^2(\xi,t)& =& \int^\xi_{-\infty} S(\eta)
 (-v v_t+\epsilon v v_{\eta\eta} 
 +q\ w\ v + F(v_{\eta},\psi)v)\,d\eta\\
 & =& \int^\xi_{-\infty} S(\eta)(-v v_t 
 +qwv+ F(v_{\eta},\psi)v)\,d \eta+\epsilon
 \int^\xi_{-\infty} S(\eta)(-v^2_\eta)\,d\eta\\
 &&+\epsilon S(\xi)(v v_\xi)+\epsilon
 \int^\xi_{-\infty} S^2(\eta)f'(\psi)_\eta
 (-v v_\eta)\, d\eta\\
 & =&\epsilon S(\xi)v v_\xi +\int^\xi_{-\infty} 
 S(\eta) (-vv_t +qwv + F(v_{\eta},\psi)v
 - \epsilon v^2_\eta)\,d\eta\\
 && +\epsilon \int^\xi_{-\infty} 
 S^2(\eta)f'(\psi)_\eta (-vv_\eta)d\eta\,.
\end{eqnarray*}
Multiplying the above inequality by $\omega(\xi)f'(\psi(\xi))_\xi$ 
and integrating from $-\infty$ to $\xi_0$, then using Schwarz's inequality 
and Fubini's theorem, we have
\begin{eqnarray*}
\lefteqn{  \int^{\xi_0}_{-\infty}{1\over 2} \omega(\xi)v^2(\xi, t)
 f'(\psi(\xi))_\xi\, d\xi }&&  \\
&  \leq & {1\over 8}\int^{\xi_0}_{-\infty} f'(\psi(\xi))_\xi 
 \omega(\xi){v^2\over 2}\, d\xi
 +8\delta_2\epsilon \int^{\xi_0}_{-\infty}
 \omega(\xi) v_\xi^2\, d\xi \\
 &&  + \int^{\xi_0}_{-\infty} \int^{\xi_0}_\eta
 f'(\psi(\xi))_\xi\omega(\xi)S(\eta) 
 (-vv_t + qwv+F(v_{\eta},\psi)v )\,d\xi\, d\eta  \\
 &&  +\int^{\xi_0}_{-\infty} \int^{\xi_0}_\eta
 f'(\psi(\xi))_\xi\omega(\xi) S^2(\eta)f'(\psi(\eta))_\eta
 (-\epsilon vv_\eta)\, d\xi\, d\eta\,.
\end{eqnarray*}
Now use the smallness assumptions (\ref{2.3}), (\ref{2.4}), and (\ref{22.5})
to obtain 
\begin{eqnarray*}
\lefteqn{ \int^{\xi_0}_{-\infty}{1\over 2} \omega(\xi)v^2(\xi, t)
 f'(\psi(\xi))_\xi\, d\xi }&& \\
&\leq& {1\over 4}  \int^{\xi_0}_{-\infty}
 f'(\psi(\xi))_\xi \omega(\xi){v^2\over 2}\, d\xi
  + C (\delta_1 + \delta_2)\epsilon \int^{\xi_0}_{-\infty}
 \omega(\xi)v^2_\xi\, d\xi \\
&&+ C \delta_1{d \over dt} \int^{\xi_0}_{-\infty}
 S(\eta)\omega(\eta)(-{1\over 2} v^2(\eta ,t))\,d\eta
 + C \delta_1 \int^{\xi_0}_{-\infty}\omega(\eta)
 qwv \, d\eta.
\end{eqnarray*}
Using the argument that $ a\leq \frac{1}{4}a+b$ implies
$ a\leq \frac{4}{3}b$, the above inequality implies that
\begin{eqnarray*}
\lefteqn{ \int^{\xi_0}_{-\infty} f'(\psi(\xi))_\xi \omega(\xi)
 {v^2(\xi,t)\over 2}\, d\xi} &&\\
 &\leq&  C\delta_1 {d \over dt}-\int^{\xi_0}_{-\infty}
 \omega(\xi) {1\over 2} v^2(\xi, t)\,d\xi \\
 &&+C (\delta_1+\delta_2) \epsilon \int^{\xi_0}_{-\infty}
 \omega(\xi)v^2_\xi\, d\xi
 +C \delta_1 \int^{\xi_0}_{-\infty} \omega(\xi)
 qwv\, d\xi\,.
\end{eqnarray*}
Plugging this estimate into (\ref{main}), 
noticing that $\delta_1, \delta_2$ are small,
and using assumption (\ref{22.5}), we have 
\begin{eqnarray*}
\lefteqn{ {d\over dt}\int^{+\infty}_{-\infty} {1\over 2}\omega(\xi)
 v^2\, d\xi + {1\over 2} \int^{+\infty}_{-\infty} 
 |f'(\psi(\xi))_\xi|\omega(\xi) {v^2(\xi,t)\over 2}\, d\xi }&&\hspace{8cm}\\
\lefteqn{  +{\beta\over 2} \int^{+\infty}_{-\infty}\omega(\xi)v^2 \, d\xi
 + \epsilon \int^{+\infty}_{-\infty} \omega(\xi)
 {v_\xi^2\over 2}\, d\xi }&&\\
 &  \leq& C \left|\int^{+\infty}_{-\infty}
 \omega(\xi) q w v\, d\xi\right|\,.
\end{eqnarray*}
                   
To estimate $|\int^{+\infty}_{-\infty} \omega(\xi) q v w\, d\xi|$, 
we divide the real line into three subintervals
$(-\infty ,\xi_1)$, $(\xi_1 , \xi_2)$, $(\xi_2 ,+\infty)$,
and estimate the integral on each subinterval.

\paragraph{On the interval $(\xi_2, +\infty)$.}
According to assumption (\ref{2.6}),
\[
\varphi(u)=\varphi(\psi)=0\quad\mbox{for } \xi > \xi_2\, .
\]
Hence
\begin{eqnarray*}
 w(\xi ,t) &=& (z - Z)(\xi,t) \\
 & =& \exp(k \int^{+\infty}_\xi \varphi(u(\eta,t))\,d\eta) -
 \exp(k\int^{+\infty}_\xi \varphi(\psi(\eta))\,d\eta)\\
 &=& 0\,, \quad\mbox{for }\xi>\xi_2\,.
\end{eqnarray*}
So that
\begin{equation}
\int^{+\infty}_{\xi_2}\omega(\xi) q w v\, d\xi=0\,.
\label{3.8}
\end{equation}

\paragraph{On the interval $(\xi_1, \xi_2)$.}
\begin{eqnarray*}
 |w(\xi,t)|&=&|\exp(k\int^{+\infty}_{\xi}\varphi(u(\eta, t))\,d\eta) -
  \exp(k\int^{+\infty}_{\xi}\varphi (\psi(\eta))\, d\eta)|\\
 & =&|\exp(k\int^{\xi_2}_\xi \varphi (u(\eta,t))d\eta) -
 \exp(k\int^{\xi_2}_\xi \varphi (\psi(\eta))\, d\eta)|\\
 & =& C \left|\int^{\xi_2}_\xi (\varphi(u(\eta,t)) - 
 \varphi(\psi (\eta)))\,d\eta\right|\\
 &  \leq& C \int^{\xi_2}_\xi |v_\eta|\, d\eta \\
 &\leq& C (\int^{\xi_2}_{\xi_1}\omega(\eta)
 |v_\eta|^2 \,d\eta)^{1/2}\,.
\end{eqnarray*}
Using the Schwarz inequality, (\ref{2.8}), and the above estimate
for $w$, we have
\begin{eqnarray*}
 \left|\int^{\xi_2}_{\xi_1}\omega(\xi) q w v\, d\xi \right|
 &  \leq& \frac{1}{2} \int^{\xi_2}_{\xi_1}\omega(\xi) q v^2\, d\xi +
 \frac{1}{2} \int^{\xi_2}_{\xi_1}\omega(\xi) q w^2\, d\xi\\
 &  \leq& Cq \int^{\xi_2}_{\xi_1}\omega(\xi) |f'(\psi)_\xi|v^2\, d\xi
 +Cq\int^{\xi_2}_{\xi_1} \omega(\xi)|v_\xi|^2\, d\xi\,.
\end{eqnarray*}
Since $q\ll \epsilon \ll 1$, the terms
on the right-hand side of (\ref{main}) are under control.

\paragraph{On the interval $(-\infty, \xi_1)$.}
According to (\ref{2.7}) and the result on the above subinterval, we have
\[
 |w(\xi, t)|=|w(\xi_1,t)|e^{-k(\xi_1-\xi)}
 \leq C(\int^{\xi_2}_{\xi_1}\omega(\eta) 
 |v_\eta|^2\, d\eta)^{1/2}.
\]
An application of the Schwarz inequality yields 
\begin{eqnarray*}
\left|\int^{\xi_1}_{-\infty}\omega(\xi) q w v\, d\xi\right|
 & \leq& Cq \int^{\xi_1}_{-\infty} \omega(\xi) v^2 e^{-k|\xi|}\, d\xi
 +C q \int^{\xi_2}_{\xi_1} \omega(\xi)v^2_\xi\, d\xi\\
 &:=& I + II.
\end{eqnarray*}
Since $q\ll \epsilon \ll 1$, $II$ is under control in (\ref{main}).
For $I$, we find characteristic-energy estimates as we did for
$$ \int^{\xi_0}_{-\infty} \omega(\xi)f'(\psi(\xi))_\xi 
 {v^2(\xi,t)\over 2}\, d\xi\,$$
The result is
\begin{eqnarray*}
\lefteqn{ \int^{\xi_1}_{-\infty} e^{-k|\xi|} \omega(\xi)v^2 (\xi,t)\,d\xi
}&&\\
&\leq& C \int^{\xi_1}_{-\infty} \omega(\xi)|f'(\psi(\xi))_\xi|
 v^2\,d\xi+C{d \over dt}\int^{\xi_1}_{-\infty}
 \omega(\xi)v^2 (\xi,t)\,d\xi\\
&&  + C \int^{\xi_1}_{-\infty} \omega(\xi)v^2_\xi\, d\xi
 +C\int^{\xi_1}_{-\infty}\omega(\xi) q w v\, d\xi\,.
\end{eqnarray*}
Therefore,
\begin{eqnarray*}
\left|\int^{\xi_1}_{-\infty}\omega(\xi) q w v\, d\xi\right|
&\leq& C q\bigg\{ C \int^{\xi_2}_{-\infty}\omega(\xi)
 v^2_\xi\, d\xi+C{d \over dt}\int^{\xi_{1}}_{-\infty}
 \frac{1}{2}\omega(\xi)v^{2}(\xi,t)\,d\xi\\
&& + \int^{\xi_1}_{-\infty} \omega(\xi)
 |f'(\psi)_\xi| v^2\, d\xi\bigg\}\,.
\end{eqnarray*}

Plugging the estimates of 
$ |\int\omega(\xi) q w v\, d\xi|$
over the three intervals into (\ref{main}) and 
noticing that $q\ll \epsilon \ll 1$, we have
\begin{eqnarray*}
0&\geq&  {1\over 2}{d \over dt} \int^{+\infty}_{-\infty} \omega(\xi)
 v^2(\xi,t)\,d\xi +{1\over 4}\int^{+\infty}_{-\infty}
 |f'(\psi(\xi))_\xi|\omega(\xi) v^2\,d\xi\\
&& +{\beta\over 2} \int^{+\infty}_{-\infty}\omega(\xi)v^2 \, d\xi
 +{1\over 4}\epsilon \int^{+\infty}_{-\infty}
 \omega(\xi) v^2_\xi\, d\xi\,.
\end{eqnarray*}
Similarly, we have estimates for the derivatives $v_\xi$ and $v_{\xi\xi}$
of $v$.
$$
{1\over 2}{d \over dt} \int^{+\infty}_{-\infty} \omega(\xi)
 v_\xi^2(\xi,t)\,d\xi+{\beta\over 2} \int^{+\infty}_{-\infty}
 \omega(\xi)v_\xi^2 \, d\xi
 +{1\over 4}\epsilon \int^{+\infty}_{-\infty}
 \omega(\xi) v^2_{\xi\xi}\, d\xi\leq 0\,, $$
and
$$ 
 {1\over 2}{d \over dt} \int^{+\infty}_{-\infty} \omega(\xi)
 v_{\xi\xi}^2(\xi,t)\,d\xi+{\beta\over 2} \int^{+\infty}_{-\infty}
 \omega(\xi)v_{\xi\xi}^2 \, d\xi
 +{1\over 4}\epsilon \int^{+\infty}_{-\infty}
 \omega(\xi) v^2_{\xi\xi\xi}\, d\xi\leq 0\,.
$$
Combining these estimates, we have
\begin{eqnarray*}
\lefteqn{ {d \over dt}(\int^{+\infty}_{-\infty} \omega(\xi)v^2\,d\xi+
 \int^{+\infty}_{-\infty} \omega(\xi)v_\xi^2\,d\xi+
 \int^{+\infty}_{-\infty} \omega(\xi)v_{\xi\xi}^2\, d\xi) }&&\\
& \leq& - \beta(\int^{+\infty}_{-\infty} \omega(\xi)v^2\, d\xi+
 \int^{+\infty}_{-\infty} \omega(\xi)v_\xi^2\,d\xi+
 \int^{+\infty}_{-\infty} \omega(\xi)v_{\xi\xi}^2\, d\xi)\,.
\end{eqnarray*}
By Gronwall's inequality, we have
\[
  \| v(\cdot, t) \|_{H^2_{\omega}}\leq \| v(\cdot, 0) 
  \|_{H^2_{\omega}} e^{-\beta t/2}\,.
\]
Hence
\begin{eqnarray*}
 |u(x,t)-\psi(x-Dt)|&=&|v_x(x,t)|
 = \big(2\int_{-\infty}^{x} v_xv_{xx}(y,t)\,dy\big)^{1/ 2} \\
& \leq& \big( \int_{-\infty}^{+\infty}v_x^2(x,t)\,dx+\int_{-\infty}^{+\infty}
 v_{xx}^2(x,t)\,dx\big)^{1/2}\\
 &\leq& \| v(\cdot, t) \|_{H^2_{\omega}}\\
& \leq&  C e^{-\beta t/2}\,,
\end{eqnarray*}
which completes the proof of Theorem~\ref{th2.1}.

\paragraph{Remark:} The above inequality guarantees that the {\it a
priori} assumption (\ref{22.5}) is satisfied.

\section{Proof of stability: Algebraic decay}

To prove Theorem \ref{th2.2}, we use the iteration introduced by 
Kawashima and Matsumura \cite{KM}, and weighted energy
estimates. First we state a lemma similar to   Lemma~\ref{l1}.
 

\begin{lemma} \label{l2}
Let $\xi_*$ be defined by (\ref{velo}), and 
\begin{equation} \label{ar}
A_\beta(\xi)=\left\{
\begin{array}{ll}
{1 \over 2}(\beta{(\xi-\xi_*)\over \langle\xi-\xi_*\rangle}(D-f'(\psi(\xi)))
-\langle\xi-\xi_*\rangle f'(\psi(\xi))_\xi)\,, &  \xi_0<\xi<+\infty\\
{1 \over 2}\beta{(\xi-\xi_*) \over \langle\xi-\xi_*\rangle}(D-f'(\psi(\xi)))\,,& 
-\infty<\xi<\xi_0\,.
\end{array}\right.
\end{equation}
Then there exists a positive constant $\beta$ such that
\begin{equation}
A_{\beta}(\xi)\geq \left\{
\begin{array}{ll}
\beta-{1\over 4}f'(\psi(\xi))_\xi\,, &\xi_0<\xi<+\infty\\
\beta\,, &-\infty<\xi<\xi_0\,.
\end{array}\right.\label{ar1}
\end{equation}
\end{lemma}
The proof of this lemma is similar the proof of Lemma~\ref{l1}.\medskip

From the (\ref{3.1}) it follows that the 
anti-derivative $v$ of the perturbation $u-\psi$ satisfies
\begin{equation}
v_t(\xi,t)+(f'(\psi)-D)v_{\xi}=
\epsilon v_{\xi\xi}+qw+F(v_{\xi}, \psi), \label{4.1}
\end{equation}
where $w(\xi,t)=z(\xi,t)-Z(\xi)$ and
$|F(v_{\xi},\psi)|\leq C|v_{\xi}|^2$ for small values of $|v_{\xi}|$.
 
Let
\[
|v(\cdot, t)|_{\beta}^2=\int^{+\infty}_{-\infty}
\langle\xi-\xi_*\rangle^{\beta}v^2(\xi,t)\,d\xi\,.
\]
Multiplying (\ref{4.1}) by $(1+t)^{\gamma}\langle\xi-\xi_*\rangle^{\beta}v$,
 integrating by parts, and using Lemma~\ref{l2}, we obtain our main estimate,
\begin{eqnarray}
\lefteqn{ {1 \over 2}(1+t)^{\gamma}|v(\cdot, t)|_{\beta}^2+
 \beta \int_0^t (1+\tau)^{\gamma}|v(\cdot, t)|_{\beta-1}^2\,d\tau  }
 &&\hspace{10cm}\label{main1} \\
 \lefteqn{ +{1 \over 4}\int_0^t\int_{\xi_0}^{+\infty}(1+\tau)^{\gamma}
 |f'(\psi(\xi))_\xi|v^2 \langle\xi-\xi_*\rangle^{\beta}d\xi\, d\tau }
  \nonumber\\
 \lefteqn{ -{1 \over 2}\int_0^t\int^{\xi_0}_{-\infty}(1+\tau)^{\gamma}
 |f'(\psi(\xi))_\xi|v^2 \langle\xi-\xi_*\rangle^{\beta}d\xi\, d\tau 
 +\epsilon \int^t_0(1+\tau)^{\gamma}|v_\xi(\cdot,\tau)|_{\beta}^2 
 \,d\tau } \nonumber \\
 &\leq& c|v_0|_{\beta}^2+c\gamma
 \int_0^t(1+\tau)^{\gamma-1}|v(\cdot, \tau)|_{\beta}^2\, d\tau\nonumber \\
&&+ c\beta\int_0^t\int_{-\infty}^{+\infty}(1+\tau)^{\gamma}
 \langle\xi-\xi_*\rangle^{\beta-2}\xi|vv_\xi|\,d\xi\, d\tau  \nonumber \\
&&+c\int_0^t\int_{-\infty}^{+\infty}(1+\tau)^{\gamma}
\langle\xi-\xi_*\rangle^{\beta}v(qw+F(v_\xi,\psi))\,d\xi\, d\tau\,.\nonumber
\end{eqnarray}
The third term on the right hand side of  (\ref{main1})
can be estimated using twice the Schwarz inequality. Notice that
\begin{eqnarray*}
\lefteqn{ \beta \int_{-\infty}^{+\infty}\langle\xi-\xi_*\rangle^{\beta-1}(\xi)
 |vv_\xi|\,d\xi }&& \\
&\leq& {\beta \over 2}\int_{-\infty}^{+\infty}
\langle\xi-\xi_*\rangle^{\beta-1}v^2\, d\xi+\beta c\int_{-\infty}^{+\infty}
\langle\xi-\xi_*\rangle^{\beta-1}v_\xi^2\, d\xi\\
&\leq& {\beta \over 2}\int_{-\infty}^{+\infty} 
\langle\xi-\xi_*\rangle^{\beta-1}v^2\, d\xi+{\epsilon \over 2}
 \int_{-\infty}^{+\infty}\langle\xi-\xi_*\rangle^{\beta}v_\xi^2\, d\xi
 +\beta c\int_{-\infty}^{+\infty}v_\xi^2\, d\xi\,.
\end{eqnarray*}

To estimate the fourth term the left-hand side of (\ref{main1}), 
we use the characteristic energy method again. 
Since the non-monotonicity spike is small under our assumption
$q\ll \epsilon \ll 1$,
we use the method as in the previous section, 
with the weight function $\langle\xi-\xi_*\rangle^{\beta}$ instead
of $\exp(\alpha \langle\xi-\xi_*\rangle)$.

Since the term
\[
 \int_0^t \int_{-\infty}^{\xi_0}qwv\langle\xi-\xi_*\rangle^{\beta}
 (1+\tau)^{\gamma}d\xi\, d\tau\,,
\]
with  $q\ll \epsilon \ll 1$,
can be treated similarly as in the previous section,
we omit the details of the calculations
and just give the result here.

Combining the estimates for the term in (\ref{main1})
we obtain
\begin{eqnarray}
 \lefteqn{ {1 \over 2}(1+t)^{\gamma}|v(\cdot, t)|_{\beta}^2+
 {\beta  \over 2}\int_0^t (1+\tau)^{\gamma}
 |v(\cdot, t)|_{\beta-1}^2\,d\tau } && \label{4.8} \\
\lefteqn{ +{1 \over 8}\int_0^t\int_{\xi_0}^{+\infty}(1+\tau)^{\gamma}
|f'(\psi(\xi))_\xi|v^2 \langle\xi-\xi_*\rangle^{\beta}\,d\xi\, d\tau }
&&\nonumber\\
\lefteqn{ +{\epsilon \over 4} \int^t_0(1+\tau)^{\gamma}
 |v_\xi(\cdot,\tau)|_{\beta}^2\, d\tau }&&\nonumber \\
&\leq& c|v_0|_{\beta}^2+c\gamma \int_0^t
 (1+\tau)^{\gamma-1}|v(\cdot, \tau)|_{\beta}^2\, d\tau
+c\beta\int_0^t(1+\tau)^{\gamma}\| v_\xi(\tau)\|^2\,d\tau\,.\nonumber
\end{eqnarray}
Observing that the process for obtaining the above inequality
also applies for $\beta=0$, we have
\begin{eqnarray}
\lefteqn{ {1 \over 2}(1+t)^{\gamma}|v(\cdot, t)|^2 
 +  {1 \over 4}\int_0^t\int_{\xi_0}^{+\infty}(1+\tau)^{\gamma}
 |f'(\psi(\xi))_\xi|v^2\, d\xi\, d\tau }&&\nonumber \\
\lefteqn{+  {\epsilon \over 2} \int^t_0(1+\tau)^{\gamma}
 |v_\xi(\cdot,\tau)|^2\, d\tau } &&  \label{4.9}\\
&\leq& c\bigg(\| v_0\|^2+\gamma \int_0^t
(1+\tau)^{\gamma-1}\| v(\cdot, \tau)\|^2\, d\tau \bigg)\,. \nonumber
\end{eqnarray}
Using the condition $N_{\alpha}=|v_0|_{\alpha}+\| v_{0,\xi}\|_1\ll 1$,
in the case of $\beta=0,\gamma=0$ we have that
\begin{equation}
\| v(t)\|_2^2+\epsilon \int_0^t \| v_\xi(\tau)\|_2^2d\tau
\leq cN_{\alpha}^2\,.  \label{4.10}
\end{equation}

Now, we prove the iteration lemma.

\begin{lemma} \label{l3} For $\gamma$ in   $[0,\alpha]$, we have
\begin{eqnarray}
\lefteqn{ (1+t)^{\gamma}|v(t)|^2_{\alpha-\gamma}+ (\alpha-\gamma)
 \int_0^t(1+\tau)^{\gamma}|v(\tau)|_{\alpha-\gamma-1}^2\, d\tau
  +\int_0^t(1+\tau)^{\gamma}|v_\xi(\tau)|_{\alpha-\gamma}^2\,
d\tau }&&\hspace{11cm}\nonumber \\
&\leq& cN_{\alpha}^2\,.\label{4.11}
\end{eqnarray}
 Furthermore,
\begin{eqnarray}
(1+t)^{\gamma}\| v(t)\|^2+\epsilon \int_0^t(1+\tau)^{\gamma}
\| v_\xi(\tau)\|^2d\tau\leq cN_{\alpha}^2. \label{4.12}
\end{eqnarray}
\end{lemma}
\paragraph{Proof.}
First, we prove this lemma for $\gamma$ integer in $[0,[\alpha]]$, by
using the following steps.

\noindent Step 1.  Let $\beta=0, \gamma=0$ in (\ref{4.8}) and use
(\ref{4.10}) to get (\ref{4.11}) with $\gamma=0$.
Therefore, the lemma is proved for $\alpha<1$.
 
\noindent Step 2. If $\alpha\geq 1$, we use that (\ref{4.11}) holds for
$\gamma=0$.

Let $\beta=0, \gamma=1$ in (\ref{4.8}) and use (\ref{4.11}) with $\gamma=0$ 
to get (\ref{4.12}) with $\gamma=1$.
 
Let $\beta=\alpha-1, \gamma=1$ in (\ref{4.8}) and use (\ref{4.11}) 
with $\gamma=0$
and (\ref{4.12}) with $\gamma=1$ to get (\ref{4.11}) with $\gamma=1$.
 
Therefore, the lemma is proved for $\alpha<2$.
 
\noindent Step 3.  $\alpha\geq 2$.
Let $\beta=0, \gamma=2$ in (\ref{4.8}) and use (\ref{4.9}) with $\gamma=1$
to get (\ref{4.12}) with $\gamma=2$.
 
Let $\beta=\alpha-2, \gamma=1$ in (\ref{4.8}) and use (\ref{4.11}) with $\gamma=1$
 and (\ref{4.12}) with $\gamma=2$ to get (\ref{4.11}) with $\gamma=2$.
 
 The lemma is proved for $\alpha<3$.
And by an inductive argument we can prove this lemma for any $\alpha$.
 
Similarly, for $l=0,1,2$ we have
\[
 (1+t)^{\gamma}\| \partial_x^l v(t)\|^2+\epsilon 
 \int_0^t(1+\tau)^{\gamma}
 \|  \partial_x^{l+1}v(\tau)\|^2\,d\tau\leq cN_{\alpha}^2
\]
Hence
\[
 (1+t)^{\gamma}\| v(t)\|_2^2+\epsilon\int_0^t
 (1+\tau)^{\gamma}\| v_x(\tau)\|_2^2d\tau\leq cN_{\alpha}^2\,,
\]
which concludes the proof for $\gamma$ integer in  $[0,[\alpha]]$.\medskip

For $\gamma\in([\alpha],\alpha]$, from (\ref{4.8}) 
with $\beta=0$ it follows  that
\begin{eqnarray*}
\lefteqn{ (1+t)^{\gamma}\| v(\cdot, t)\|^2
 +\epsilon \int^t_0(1+\tau)^{\gamma}
 \| v_\xi(\cdot,\tau)\|^2\, d\tau }&& \\
& \leq& c\| v_0\|^2+c\gamma \int_0^t
 (1+\tau)^{\gamma-1}\| v(\cdot, \tau)\|^2\, d\tau\\
& \leq& c\| v_0\|^2+c\gamma \int_0^t
 (1+\tau)^{[\gamma]}\| v(\cdot, \tau)\|^2\, d\tau\,.
\end{eqnarray*}
Combining the above results on integer exponents, we arrive at our 
conclusion for any $\gamma\in([\alpha],\alpha]$. \hfill$\Box$\smallskip

Finally, we obtain the estimate
\begin{eqnarray*}
 v_x^2(x, t)& =& 2\int^x_{-\infty}v_xv_{xx}(y, t)\,dy\\
 &  \leq& 2 \big(\int_{-\infty}^{+\infty}
 v_x^2(y,t)dy\big)^{1/2}\big(\int_{-\infty}^{+\infty}
 v_{xx}^2(y,t)\,dy\big)^{1/2}\\
& \leq& \int_{-\infty}^{+\infty}
 v_x^2(y,t)\,dy+\int_{-\infty}^{+\infty}v_{xx}^2(y,t)\,dy\\
 &\leq& cN_{\alpha}^2(1+t)^{-\gamma}\,.
\end{eqnarray*}
Hence,
\[
\sup_x|u(x,t)-\psi(x-Dt)|=\sup_x|v_x(x,t)|
\leq CN_{\alpha}(1+t)^{-\gamma/2}\,,
\]
which is the statement in Theorem~\ref{th2.2}.

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\end{thebibliography}
{\sc Tong Li}\\
Department of Mathematics\\
University of Iowa\\Iowa City, IA 52242, USA\\
Tele: (319)335-3342 Fax: (319)335-0627\\
E-mail address: tli@math.uiowa.edu

\end{document}