\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 \varphi(u)=\left\{ \begin{array}{ll} 0 &u\leq u_{i}\\ \mbox{a smooth increasing function}&u_i 2u_{i}\,, \end{array}\right.\label{rr} 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, f'(\psi(+\infty))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 $$f'(\psi(\xi_*))=D\,.\label{velo}$$ 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 $$\| v(\cdot, t)\|_{H^2_{\omega}}\leq \| v_0(\cdot)\|_{H^2_{\omega}} e^{-\beta t}. \label{2.10}$$ Consequently, \sup_{-\infty 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 0 \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 $$\int^{+\infty}_{\xi_2}\omega(\xi) q w v\, d\xi=0\,. \label{3.8}$$ \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 $$\label{ar} A_\beta(\xi)=\left\{ \begin{array}{ll} {1 \over 2}(\beta{(\xi-\xi_*)\over \langle\xi-\xi_*\rangle}(D-f'(\psi(\xi))) -\langle\xi-\xi_*\rangle f'(\psi(\xi))_\xi)\,, & \xi_0<\xi<+\infty\\ {1 \over 2}\beta{(\xi-\xi_*) \over \langle\xi-\xi_*\rangle}(D-f'(\psi(\xi)))\,,& -\infty<\xi<\xi_0\,. \end{array}\right.$$ Then there exists a positive constant $\beta$ such that $$A_{\beta}(\xi)\geq \left\{ \begin{array}{ll} \beta-{1\over 4}f'(\psi(\xi))_\xi\,, &\xi_0<\xi<+\infty\\ \beta\,, &-\infty<\xi<\xi_0\,. \end{array}\right.\label{ar1}$$ \end{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 $$v_t(\xi,t)+(f'(\psi)-D)v_{\xi}= \epsilon v_{\xi\xi}+qw+F(v_{\xi}, \psi), \label{4.1}$$ 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 $$\| v(t)\|_2^2+\epsilon \int_0^t \| v_\xi(\tau)\|_2^2d\tau \leq cN_{\alpha}^2\,. \label{4.10}$$ 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}. \begin{thebibliography}{00} \bibitem{CF} R. 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Math, 43(1983) 1086-1118. \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}