Decay properties of solutions toward a multiwave pattern to the Cauchy problem for the scalar conservation law with degenerate flux and viscosity
Name\corref{cor1
In this paper, we study the precise decay rate in time to solutions of the Cauchy problem for the one-dimensional conservation law with a nonlinearly degenerate viscosity $ \partial_tu +\partial_x \bigl(f(u) \bigr) = \mu \, \partial_x \left( \, \left| \, \partial_xu \, \right|^{p-1 \partial_xu \, \right) $ where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval. As the corresponding Riemann problem admits a Riemann solution as a multiwave pattern which consists %of of the rarefaction waves and the contact discontinuity, it has already been proved by Yoshida that the solution to the Cauchy problem tends toward the linear combination of the rarefaction waves and contact wave for $p$-Laplacian type viscosity as the time goes to infinity. We investigate that the decay rate in time of the corresponding solutions toward the multiwave pattern. Furthermore, we also investigate that the decay rate in time of the solution for the higher order derivative. The proof is given by $L^1$, $L^2$-energy and time-weighted $L^q$-energy methods under the use of the precise asymptotic properties of the interactions between the nonlinear waves.
1502.04750.tex
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%\journal{J. Differential Equations}

\begin{document}

\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newdefinition{rmk}{Remark}
\newproof{pf}{Proof}
\newproof{pot}{Proof of Theorem \ref{thm2}}


\begin{frontmatter}

%% Title, authors and addresses

%% use the tnoteref command within \title for footnotes;
%% use the tnotetext command for the associated footnote;
%% use the fnref command within \author or \address for footnotes;
%% use the fntext command for the associated footnote;
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\title{Decay properties of solutions 
toward a multiwave pattern 
to the Cauchy problem 
for the scalar conservation law 
with degenerate flux and viscosity}

%% use optional labels to link authors explicitly to addresses:
%% \author[label1,label2]{<author name>}
%% \address[label1]{<address>}
%% \address[label2]{<address>}
\author[rvt]{Natsumi Yoshida}
\ead{14v00067@gst.ritsumei.ac.jp}


\address{BKC Research Organization of Social Sciences, 
Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan
/Osaka City University Advanced Mathematical Institute, 
Sumiyoshi, Osaka 558-8585, Japan.}
\address{}

\begin{abstract}
In this paper, we study the precise decay rate in time to solutions 
of the Cauchy problem for 
the one-dimensional conservation law 
with a nonlinearly degenerate viscosity 
$
\partial_tu +\partial_x \bigl(f(u) \bigr)
  = \mu \, 
    \partial_x \left( \, 
    \left| \, \partial_xu \, \right|^{p-1} \partial_xu \, 
    \right)
$
where the far field states are prescribed.
Especially, we deal with the case 
when the flux function is convex or concave 
but linearly degenerate on some interval. 
As the corresponding Riemann problem admits a Riemann solution as 
a multiwave pattern which consists %of 
of the rarefaction waves and the contact discontinuity, 
it has already been proved by Yoshida 
that the solution to the Cauchy problem 
tends toward the linear combination of 
the rarefaction waves and contact wave for $p$-Laplacian type viscosity 
as the time goes to infinity. 
We investigate that the decay rate in time 
of the corresponding solutions toward the multiwave pattern. 
Furthermore, we also investigate that the decay rate in time 
of the solution for the higher order derivative. 
The proof is given by 
$L^1$, $L^2$-energy and time-weighted $L^q$-energy methods 
under the use of the precise asymptotic properties of 
the interactions between the nonlinear waves. 
\end{abstract}
\begin{keyword} 
viscous conservation law \sep decay estimates \sep asymptotic behavior
\sep nonlinearly degenerate viscosity \sep rarefaction wave and viscous contact wave
\end{keyword}

\end{frontmatter}
%\newtheorem{thm}{Theorem}
%\newtheorem{lem}[thm]{Lemma}
%\newdefinition{rmk}{Remark}
%\newproof{pf}{Proof}
%\newproof{pot}{Proof of Theorem \ref{thm2}}
%


% The thanks line in the title should be filled in if there is
% any support acknowledgement for the overall work to be included
% This \thanks is also used for the received by date info, but
% authors are not expected to provide this.


%This work is 
%supported in part by Grant-in-Aid for Scientific Research (B) 23340036, Japan}}
%         \ \\
% Department of Pure and Applied Mathematics,\\
% Graduate School of Information Science and Technology,\\
% Osaka University, Japan}

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\pagestyle{myheadings}
\thispagestyle{plain}
\markboth{N. YOSHIDA}{NATSUMI YOSHIDA}


\section{Introduction and main theorems}
In this paper, 
we are concerned with the asymptotic behavior 
and the time-decay of 
solutions 
for the one-dimensional scalar conservation law 
with a nonlinearly degenerate viscosity 
($p$-Laplacian type viscosity with $p>1$) 
\begin{eqnarray}
 \left\{\begin{array}{ll}
  \partial_tu +\partial_x \bigl(f(u) \bigr)
  = \mu \, 
    \partial_x \left( \, 
    \left| \, \partial_xu \, \right|^{p-1} \partial_xu \, 
    \right)
  \qquad &(t>0, x\in \mathbb{R}), \\[5pt]
  u(0,x) = u_0(x) \qquad &( x \in \mathbb{R} ).
 \end{array}
 \right.\,
\end{eqnarray}
%for $p>1$. 
Here, $u=u(t,x)$ denotes the unknown function of $t>0$ and $x\in \mathbb{R}$, 
the so-called conserved quantity, 
$f=f(u)$ is the flux function depending only on $u$, 
$\mu$ is the viscosity coefficient, 
$u_0$ is the given initial data 
and $u$ is assumed to be 
asymptotically far field constant states $u_{\pm }$ 
as 
$
x\rightarrow \pm \infty,
$
that is, 
$$
\displaystyle{\lim_{x\to \pm \infty}} u(t,x) =u_{\pm}  \in \mathbb{R} 
\qquad \bigl( t \ge 0 \bigr).
$$
We suppose the given flux $f=f(u)$ is a $C^1$-function 
satisfying $f(0)=f'(0)=0$, 
$\mu$ is a positive constant 
and far field states $u_{\pm }$ satisfy $u_-<u_+$ 
without loss of generality. 

We are interested in the asymptotic behavior 
and its precise estimates in time of 
the global solution to the Cauchy problem (1.1). 
It can be expected that the large-time behavior 
is closely related to the weak solution (``Riemann solution'') of the corresponding 
Riemann problem 
(cf. \cite{lax}, \cite{smoller})
for the non-viscous hyperbolic part of (1.1): 
\begin{eqnarray}
 \left\{\begin{array} {ll}
 \partial _t u + \partial _x \bigl( f(u) \bigr)=0 
 \qquad &(t>0, x\in \mathbb{R}),\\[5pt]
u(0,x)=u_0 ^{\rm{R}} (x)\qquad &(x \in \mathbb{R}),
 \end{array}
  \right.\,
\end{eqnarray}
where $u_0 ^{\rm{R}}$ is the Riemann data defined by
$$
u_0 ^{\rm{R}} (x)=u_0 ^{\rm{R}} (x\: ;\: u_- ,u_+)
          := 
          \left\{\begin{array} {ll}
          u_-  & \; (x < 0),\\[5pt]
          u_+  & \; (x > 0).
          \end{array}\right.
$$
In fact, for $p=1$ in (1.1), the usual linear viscosity case: 
\begin{eqnarray}
 \left\{\begin{array}{ll}
  \partial_tu +\partial_x \bigl(f(u) \bigr)= \mu \, \partial_x^2 u
  \qquad &(t>0, x\in \mathbb{R}), \\[5pt]
  u(0,x) = u_0(x) \qquad &( x \in \mathbb{R} ),\\[5pt]
  \displaystyle{\lim_{x\to \pm \infty}} u(t,x) =u_{\pm}  
  \qquad &\bigl( t \ge 0 \bigr),   
 \end{array}
 \right.\,
\end{eqnarray}
when the smooth flux function $f$ is genuinely nonlinear 
on the whole space $\mathbb{R}$,
i.e., $f''(u)\ne 0\ (u\in \mathbb{R})$, 
Il'in-Ole{\u\i}nik \cite{ilin-oleinik} showed 
the following: if  $f''(u) > 0\ (u\in \mathbb{R})$, that is, the Riemann solution 
consists of a single rarefaction wave solution, 
the global solution in time of the Cauchy problem 
(1.3) tends toward the rarefaction wave;
if  $f''(u) < 0\ (u\in \mathbb{R})$, that is,
the Riemann solution consists of a single shock wave solution, 
the global solution of the
Cauchy problem (1.3) does the corresponding smooth traveling wave solution 
(``viscous shock wave'') of (1.3) with a spacial shift 
(cf. \cite{ilin-kalashnikov-oleinik}). 
Hattori-Nishihara \cite{hattori-nishihara} also proved that the asymptotic decay rate in time, 
of the solution toward the single rarefaction wave%, is  
%$(1+t)^{-{\frac{1}{2}}\, \left(1-\frac{1}{p} \right)}$ 
%in the $L^p$-norm $\bigl(1\leq p\leq \infty \bigr)$ for large $t>0$ 
(see also \cite{hashimoto-kawashima-ueda}, \cite{hashimoto-matsumura}, \cite{nakamura}). 
More generally, in the case of the flux functions 
which are not uniformly genuinely nonlinear, 
when the Riemann solution consists of a single shock wave 
satisfying Ole{\u\i}nik's shock condition, 
Matsumura-Nishihara \cite{matsu-nishi3} showed the asymptotic stability 
of the corresponding viscous shock wave. 
Moreover, Matsumura-Yoshida \cite{matsumura-yoshida} 
considered the circumstances 
where the Riemann solution generically forms 
a complex pattern of multiple nonlinear waves which consists of 
rarefaction waves %, shock waves 
and waves of contact discontinuity (refer to \cite{liep-rosh}), 
%there had been no results about the asymptotics toward 
%the multiwave pattern. 
and investigated %focused their attention on 
that the case where the flux function $f$ is smooth
and genuinely nonlinear (that is, $f$ is convex function or concave function) 
on the whole $\mathbb{R}$ except 
a finite interval $I := (a,b) \subset \mathbb{R}$, and 
linearly degenerate on $I$, that is, 
\begin{equation}
\left\{
\begin{array}{ll}
  f''(u) >0 & \; \bigl(u \in (-\infty ,a\, ]\cup [\, b,+\infty )\bigr),\\[5pt]
  f''(u) =0 & \; \bigl(u \in (a,b)\bigr).
\end{array}\right.
\end{equation}
Under the conditions (1.4), 
the corresponding Riemann solution does form 
multiwave pattern which consists of the contact discontinuity 
with the jump from $u=a$ to $u=b$ and the rarefaction waves, 
depending on the choice of $a$, $b$, $u_-$ and $u_+$.
Thanks to that the cases in which the interval $(a,b)$ is disjoint 
from the interval $(u_-,u_+)$ 
are similar as in the case 
the flux function $f$ is genuinely nonlinear 
on the whole space $\mathbb{R}$, 
and the case $u_-<a<u_+<b$ is 
the same as that for $a<u_-<b<u_+$, 
we may only consider the 
typical cases 
\begin{equation}
a<u_-<b<u_+ \quad \mbox{or} \quad u_-<a<b<u_+.   
\end{equation}
Under the conditions (1.4) and (1.5), 
they have shown the unique global solution in time 
to the Cauchy problem (1.3) tends uniformly in space toward 
the multiwave pattern of the combination of 
the viscous contact wave 
and the rarefaction waves 
as the time goes to infinity. 
It should be noted that the rarefaction wave which connects the 
far field states $u_-$ and $u_+$ 
$\bigl(u_\pm \in (-\infty,a\, ]$ or $u_\pm \in [\, b,\infty)\bigr)$
is explicitly given by
\begin{equation}
u=u^r \left( \frac{x}{t}\: ;\:  u_- , u_+ \right)
:= \left\{
\begin{array}{ll}
  u_-  & \; \bigl(\, x \leq \lambda(u_-)\,t \, \bigr),\\[7pt]
  \displaystyle{ (\lambda)^{-1}\left( \frac{x}{t}\right) } 
  & \; \bigl(\, \lambda(u_-)\,t \leq x \leq \lambda(u_+)\,t\,  \bigr),\\[7pt]
   u_+ & \; \bigl(\, x \geq \lambda(u_+)\,t \, \bigr),
\end{array}
\right.
\end{equation} 
where $\lambda(u):=f'(u)$,
and the viscous contact wave 
which connects 
$u_-$ and $u_+$ ($u_\pm \in [\,a,b\,]$) is given by an exact
solution of the linear 
convective heat equation 
\begin{equation}
%$$
  \partial _t u + \tilde{\lambda} \, \partial_x u = \mu \, \partial_x^2 u
  \qquad \biggl(
  \,\tilde{\lambda}:= \frac{f(b)-f(a)}{b-a},\ 
  t>0,x \in  \mathbb{R} 
  \biggr)
%$$
\end{equation}
 which has the form
$$
u=U\left(\frac{x-\tilde{\lambda} \, t}{\sqrt{t}}\: ;\: u_- ,u_+ \right)
$$
where $U\left(\frac{x}{\sqrt{t}}\,;\,u_- ,u_+ \right)$ is explicitly defined by
\begin{equation}
 U\left(\frac{x}{\sqrt{t}}\: ;\: u_- ,u_+ \right)
 :=u_- +\frac{u_+ - u_-}{\sqrt{\pi}}
   \lint ^{\frac{\mathlarger{x}}{\mathlarger{\sqrt{4\mu t}}}}_{-\infty} 
 \mathrm{e}^{-\xi^2}\, \mathrm{d}\xi 
\quad \, (t>0, x\in \mathbb{R}). 
\end{equation}
Yoshida \cite{yoshida} also obtained 
the almost optimal decay properties 
for the asymptotics toward the multiwave pattern. 
In fact, owing to \cite{yoshida}, 
the decay rate in time is 
$(1+t)^{-{\frac{1}{2}}\, \left(\frac{1}{2}-\frac{1}{p} \right)}$ 
in the $L^p$-norm $\bigl(2\leq p<+\infty \bigr)$ 
and 
$(1+t)^{-{\frac{1}{4}}\, +\epsilon}$ for any $\epsilon >0$ 
in the $L^\infty$-norm 
if the initial perturbation from the corresponding asymptotics 
satisfies $H^1$. 
Furthermore, 
if the perturbation satisfies $H^1\cap L^1$, 
the decay rate in time is 
$(1+t)^{-{\frac{1}{2}}\, \left(1-\frac{1}{p} \right)\, +\epsilon }$ 
for any $\epsilon >0$ 
in the $L^p$-norm $\bigl(1\leq p<+\infty \bigr)$ 
and
$(1+t)^{-{\frac{1}{2}}\, +\epsilon}$ for any $\epsilon >0$
in the $L^\infty$-norm. 

For $p>1$, 
there are few results for the asymptotic behavior 
for the problem (1.1)
(the related problems are studied in 
\cite{gur-mac}, \cite{nagai-mimura1}, \cite{nagai-mimura2} 
and so on). 
In the case where the flux function is genuinely nonlinear 
on the whole space $\mathbb{R}$, 
Matsumura-Nishihara \cite{matsu-nishi3} 
proved that if the far field states satisfy $u_- = u_+=:\tilde{u}$, 
then the solution tends toward the constant state $\tilde{u}$, 
and if the far field states $u_- < u_+$, 
then the solution tends toward a single rarefaction wave. 
Furthermore, Yoshida \cite{yoshida''} recently showed 
their precise decay estimates. 
In the case where the flux function is given as (1.4) 
and the far field states as (1.5), 
Yoshida \cite{yoshida'} also showed 
that the similar asymptotics as the one in \cite{matsumura-yoshida} 
which tends toward the multiwave pattern 
of the combination of 
the rarefaction waves 
which connect the 
far field states $u_-$ and $u_+$ 
$\bigl(u_\pm \in (-\infty,a\, ]$ or $u_\pm \in [\, b,\infty)\bigr)$, 
and the viscous contact wave which connects 
$u_-$ and $u_+$ ($u_\pm \in [\,a,b\,]$).  
In more detail, 
the viscous contact wave is 
said to be ``contact wave for $p$-Laplacian type viscosity'' 
and explicitly given by 
\begin{align}
U\left(\, \frac{x-\tilde{\lambda} \, t}
          {t^{\frac{1}{p+1}}}\: ;\: u_- ,u_+ \right) 
= u_- + \displaystyle{
        \lint^{\frac{\mathlarger{x-\tilde{\lambda} \, t}}
        { \mathlarger{t}^{{\scriptscriptstyle \frac{1}{p+1}}} }}_{-\infty}
        %\int^{ \left( x-\tilde{\lambda} \, t \right) \, t^{- \frac{1}{p+1} } }
        %_{-\infty}
             \Bigl( \left( \, 
                A-B \xi^2 \, 
                \right)\vee 0 \, \Bigr)^{\frac{1}{p-1}}
                \, \mathrm{d}\xi },
\end{align}
\begin{eqnarray*}
\left\{\begin{array}{ll}
\displaystyle{
A=A_{p,\mu,u_{\pm}}
:= \left( \, 
   \frac{(p-1)\, \left( \, u_{+} - u_{-} \, \right)}
   {8\, \mu\, p(p+1) 
   \biggl( \, \displaystyle{\int^{\frac{\pi}{2}}_{0} 
   \bigl( \sin \theta \bigr)^{\frac{p+1}{p-1}}\, \mathrm{d}\theta }
   \biggr)^2 }
   \, \right)^{\frac{p-1}{p+1}}
}, \\[25pt]
\displaystyle{ 
B=B_{p,\mu}:= \frac{p-1}{2\, \mu \, p(p+1)} 
}, \\[10pt] 
\displaystyle{ 
2A^{\frac{p+1}{2(p-1)}}B^{-\frac{1}{2}}
      \int^{\frac{\pi}{2}}_{0} 
      \bigl( \sin \theta \bigr)^{\frac{p+1}{p-1}}\, \mathrm{d}\theta 
      = u_{+} - u_{-}}.
\end{array}
\right.\,
\end{eqnarray*}
It should be noted that the wave (1.9) is 
constructed by the Barenblatt-Kompanceec-Zel'dovi{\v{c}} solution 
(see also \cite{carillo-toscani}, \cite{huang-pan-wang}, \cite{kamin}) 
\begin{equation}
v(t,x)
%v\left(\, t,\, \frac{x}{(1+t)^{\frac{1}{p+1}}} \right)
:= \frac{1}{(1+t)^{\frac{1}{p+1}}}\, 
   \Bigggl( \left( 
   A-B \left(\frac{x}{(1+t)^{\frac{1}{p+1}}} \right)^2 \, 
   \right)\vee 0 \Bigggr)^{\frac{1}{p-1}}
\end{equation}
of the following Cauchy problem of the porous medium equation 
\begin{eqnarray}
 \left\{\begin{array}{ll}
  \partial_t v
  = %\mu \, 
%    \partial_x^2 \left( \, v^p \, \right)
    \mu \, 
    \partial_x^2 
    \left( \, 
    \left| \, v \, \right|^{p-1} v \, 
    \right)
  \qquad &(t>-1, x\in \mathbb{R}), \\[5pt]
  v(-1,x) = (u_{+} - u_{-})\, \delta(x) 
  \qquad &( x \in \mathbb{R}\, ;\,  u_{-}<u_{+}),\\[5pt]
  \displaystyle{\lim_{x\to \pm \infty}} v(t,x) =0  
  \qquad &\bigl( t \ge -1 \bigr),  
 \end{array}
 \right.\,
\end{eqnarray}
where $\delta(x)$ is the Dirac $\delta$-distribution. 

%However, the decay rate of any asymptotics of the problem (1.1) 
%has been not known.  

\medskip
The aim of the present paper is to obtain 
the precise time-decay estimates for the asympotics of the 
previous study in \cite{yoshida'}. 

\medskip

\noindent
{\bf Stability Theorem } (Yoshida \cite{yoshida'}){\bf .}\quad{\it
Let the flux function $f$ satisfy {\rm(1.4)} and %<== Let ... satisfy ...
the far field states $u_{\pm }$ {\rm(1.5)}. 
Assume that the initial data satisfies %<== Assume (that) ... satisfies ...
$u_0-u_0 ^{\rm{R}} \in L^2$ and 
$\partial _x u_0 \in L^{p+1}$. 
Then the Cauchy problem {\rm(1.1)} with $p>1$ has a 
unique global %weak 
solution in time $u=u(t,x)$ 
satisfying 
\begin{eqnarray*}
\left\{\begin{array}{ll}
u-u_0 ^{\rm{R}} \in C^0\bigl( \, [\, 0,\infty)\, ;L^2 \bigr)
                \cap L^{\infty}\bigl( \, \mathbb{R}^{+} \, ;L^{2} \bigr),\\[5pt]
\partial _x u \in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr),\\[5pt]
                  %\cap L^{p+1}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       {\mathbb{R}}_{x} \bigr)
                  %\cap L^{p+2}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       \left\{x \in \mathbb{R}\, |\, u>0 \right\} \bigr),\\[5pt]
\partial _t u 
\in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr)
    %\cap \left( \, L^{p+1}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       {\mathbb{R}}_{x} \bigr)
%    \oplus L^{2}\bigl(\, {\mathbb{R}^{+}_{t}} \times {\mathbb{R}}_{x} \bigr) \, 
%     \right)
,\\[5pt]
%\partial_x \bigl(f(u) \bigr) 
%%\lambda (u) \, \partial _x u 
%\in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr)
%               \cap L^{p+1}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       {\mathbb{R}}_{x} \bigr)
%               %\cup L^{2}\bigl(\, {\mathbb{R}^{+}_{t}} \times {\mathbb{R}}_{x} \bigr)
%%L^{\infty}\bigl( \, \mathbb{R}^{+} \, ;L^{2} \bigr)
%,\\[5pt]
\partial_x \left( \, \left| \, \partial_x u \, \right|^{p-1} \partial_xu \, \right)
\in L^{2}\bigl(\, {\mathbb{R}^{+}_{t}} \times {\mathbb{R}}_{x} \bigr),
\end{array} 
\right.\,
\end{eqnarray*}
and the asymptotic behavior 
$$
\lim _{t \to \infty}\sup_{x\in \mathbb{R}}
|\,u(t,x)-U_{multi}(\, t,x \: ;\: u_-,u_+)\,| = 0, 
$$
%U_{multi}(t,\cdot \: ;u_-,u_+)
where $U_{multi}(t,x)=U_{multi}(\, t,x \: ;\: u_-,u_+)$ is defined as follows: 
in the case $a<u_-<b<u_+$,
$$
%U\left(\, \frac{x-\tilde{\lambda} \, t}{t^{\frac{1}{p+1}}}\: ;\: u_- ,u_+ \right)
U_{multi}(t,x):=
U\left(\, \frac{x-\tilde{\lambda} \, t}{t^{\frac{1}{p+1}}}\: ;\: u_- , b \right)
 + u^r\left(\, \frac{x}{t}\: ;\: b , u_+ \right) - b
$$
and, in the case $u_-<a<b<u_+$,
$$
U_{multi}(t,x):=
u^r\left(\, \frac{x}{t}\: ;\: u_- ,a \right) -a
 + U\left(\, \frac{x-\tilde{\lambda} \, t}{t^{\frac{1}{p+1}}}\: ;\: a ,b \right)
 + u^r\left(\, \frac{x}{t}\: ;\: b , u_+ \right) - b.
$$
}

Now we are ready to state our main results. 

\medskip

\noindent
{\bf Theorem 1.1} (Main Theorem I){\bf .}\quad{\it
Under the same assumptions in Stability Theorem, 
%for the flux function $f$, the far field states $u_{\pm }\in \mathbb{R}$ 
%and the initial data $u_0$, 
the unique global solution in time $u$ 
of the Cauchy problem {\rm(1.1)} %$u=u(t,x)$ 
satisfying 
\begin{eqnarray*}
\left\{\begin{array}{ll}
u-u_0 ^{\rm{R}} \in C^0\bigl( \, [\, 0,\infty)\, ;L^2 \bigr)
                \cap L^{\infty}\bigl( \, \mathbb{R}^{+} \, ;L^{2} \bigr),\\[5pt]
\partial _x u \in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr),\\[5pt]
                  %\cap L^{p+1}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       {\mathbb{R}}_{x} \bigr)
                  %\cap L^{p+2}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       \left\{x \in \mathbb{R}\, |\, u>0 \right\} \bigr),\\[5pt]
\partial _t u 
\in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr)
    %\cap \left( \, L^{p+1}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       {\mathbb{R}}_{x} \bigr)
%    \oplus L^{2}\bigl(\, {\mathbb{R}^{+}_{t}} \times {\mathbb{R}}_{x} \bigr) \, 
%     \right)
,\\[5pt]
%\partial_x \bigl(f(u) \bigr) 
%%\lambda (u) \, \partial _x u 
%\in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr)
%               \cap L^{p+1}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       {\mathbb{R}}_{x} \bigr)
%               %\cup L^{2}\bigl(\, {\mathbb{R}^{+}_{t}} \times {\mathbb{R}}_{x} \bigr)
%%L^{\infty}\bigl( \, \mathbb{R}^{+} \, ;L^{2} \bigr)
%,\\[5pt]
\partial_x \left( \, \left| \, \partial_x u \, \right|^{p-1} \partial_xu \, \right)
\in L^{2}\bigl(\, {\mathbb{R}^{+}_{t}} \times {\mathbb{R}}_{x} \bigr),
\end{array} 
\right.\,
\end{eqnarray*}
satisfies the following time-decay estimates 
\begin{eqnarray*}
\left\{\begin{array} {ll}
\left|\left|\,
u(t) - 
U_{multi}(\, t,\cdot \: ;\: u_-,u_+)\,
\right|\right|_{L^q }
\leq C( \, p, \, q, \, u_0 \, ) \, 
     (1+t)^{-\frac{1}{3p+1}\left(1-\frac{2}{q}\right)},\\[7pt]
\left|\left|\,
u(t) - 
U_{multi}(\, t,\cdot \: ;\: u_-,u_+)
\,\right|\right|_{L^{\infty} }
\leq C( \, \epsilon, \, p, \, q, \, u_0, \, \partial _x u_0 \, ) \, 
     (1+t)^{-\frac{1}{3p+1}+\epsilon},
\end{array}
  \right.\,
\end{eqnarray*}
for %$2 \le  p< \infty $
$q \in [\, 2, \infty) $ and any $\epsilon>0$. 
}

\medskip

\noindent
{\bf Theorem 1.2} (Main Theorem I\hspace{-.1em}I){\bf .}\quad{\it
Under the same assumptions in Theorem 1.1, 
%for the flux function $f$, the far field states $u_{\pm }\in \mathbb{R}$ 
%and the initial data $u_0$, 
if the initial data further satisfies $u_0-u_0 ^{\rm{R}} \in L^1$, %L^1(\mathbb{R})
then it holds that the unique global solution in time $u$ 
of the Cauchy problem {\rm(1.1)} %$u=u(t,x)$ 
satisfies the following time-decay estimates 
\begin{eqnarray*}
\left\{\begin{array} {ll}
\left|\left| \,
u(t) - 
U_{multi}(\, t,\cdot \: ;\: u_-,u_+)
\, \right|\right|_{L^q }
\leq C( \, p, \, q, \, u_0 \, ) \, 
     (1+t)^{-\frac{1}{2p}\left(1-\frac{1}{q}\right)},\\[7pt]
\left|\left|\,
u(t) - 
U_{multi}(\, t,\cdot \: ;\: u_-,u_+)
\,\right|\right|_{L^{\infty} }
\leq C( \, \epsilon, \, p, \, q, \, u_0, \, \partial _x u_0 \, ) \, 
     (1+t)^{-\frac{1}{2p}+\epsilon},
\end{array}
  \right.\,
\end{eqnarray*}
for %$1 \le  p< \infty $ 
$q \in [\, 1, \infty) $ and any $\epsilon>0$. 
Furthermore, the solution satisfies 
the following time-decay estimates for the higher order derivative
\begin{align*}
\begin{aligned}
&\bigl|\bigl|\,
 \partial _x u(t) \,
 \bigr|\bigr|_{L^{p+1} }, \, \; \; 
 \left|\left|\,
 \partial _x u(t) - 
 \displaystyle{\partial _x U_{multi}(\, t,\cdot \: ;\: u_-,u_+)}
 \,\right|\right|_{L^{p+1} } \\[5pt]
& \leq \left\{\begin{array} {ll} 
      C( \, p, \, u_0, \, \partial _x u_0 \, ) \, 
      (1+t)^{-\frac{p}{(p+1)^2}} \, \: \: \; \; \; \quad \qquad 
      \left( \, 1 < p < 
      \displaystyle{\frac{7 + \sqrt{73} }{12} }
      \, \right),\\[15pt] 
      C( \, \epsilon, \, p, \, u_0, \, \partial _x u_0 \, ) \, 
      (1+t)^{-\frac{3}{2(p+1)(3p-2)} + \epsilon} \, \, \: \; \; \; 
      \left( \,  
      \displaystyle{\frac{7 + \sqrt{73} }{12} }
      \leq p \, \right),
      \end{array}
      \right.\,
\end{aligned}
\end{align*}
for any $\epsilon>0$. 
}

\medskip

\noindent
{\bf Theorem 1.3} (Main Theorem I\hspace{-.1em}I\hspace{-.1em}I ){\bf .}\quad{\it
Under the same assumptions in Theorem 1.2, 
%for the flux function $f$, the far field states $u_{\pm }\in \mathbb{R}$ 
%and the initial data $u_0$, 
if the initial data further satisfies 
$\partial _x u_0 \in L^{r+1} \, (r>p)$, 
then it holds that the unique global solution in time $u$ 
of the Cauchy problem {\rm(1.1)} %$u=u(t,x)$ 
satisfies the following time-decay estimates 
for the higher order derivative
\begin{align*}
\begin{aligned}
&\bigl|\bigl|\,
 \partial _x u(t) \,
 \bigr|\bigr|_{L^{r+1} }, \, \; \; 
 \left|\left|\,
 \partial _x u(t) - 
 \displaystyle{\partial _x U_{multi}(\, t,\cdot \: ;\: u_-,u_+)}
 \,\right|\right|_{L^{r+1} } \\[5pt]
& \leq \left\{\begin{array} {ll} 
       C( \, p, \, r, \, u_0, \, \partial _x u_0 \, ) \, 
       (1+t)^{-\frac{4p(r-p)+7p+3}{6p(p+1)(r+1)}} \\[15pt] 
       \, \, \, \: \; \; \; \quad \quad \quad \quad \qquad 
       \left( \, 
       1 < p < 
       \displaystyle{\frac{7 + \sqrt{73} }{12} }, \; 
       r > \displaystyle{\frac{-4p^2+7p+3}{2p}} > p 
       \, \right),\\[15pt] 
       C( \, \epsilon, \, p, \, r, \, u_0, \, \partial _x u_0 \, ) \, 
       (1+t)^{-\frac{r}{(p+1)(r+1)}+ \epsilon} \\[15pt] 
       \, \, \: \; \; \; \quad \quad \quad \quad \quad \qquad 
       \left( \, 
       1 < p < 
       \displaystyle{\frac{7 + \sqrt{73} }{12} }, \; 
       p < r \le \displaystyle{\frac{-4p^2+7p+3}{2p}}
       \, \right),\\[15pt] 
       C( \, \epsilon, \, p, \, r, \, u_0, \, \partial _x u_0 \, ) \, 
       (1+t)^{-\frac{p+2r}{2p(3p-2)(r+1)}+ \epsilon} \\[15pt] 
       \, \, \, \: \: \; \; \quad \quad \qquad \qquad \qquad \qquad \qquad 
       \left( \, 
       \displaystyle{\frac{7 + \sqrt{73} }{12}} \leq p
       \, \right),
       \end{array}
       \right.\,
\end{aligned}
\end{align*}
for any $\epsilon>0$. 
}

\medskip

This paper is organized as follows. 
In Section 2, we shall prepare the basic properties of 
the rarefaction wave and the 
contact wave for $p$-Laplacian type viscosity.  
In Section 3, 
we reformulate the problem in terms of the deviation from 
the asymptotic state 
(similarly in \cite{matsumura-yoshida}, \cite{yoshida}, \cite{yoshida'}). 
Following the arguments in \cite{matsu-nishi2}, 
we also prepare some uniform boundedness and energy estimates 
of the deviation 
as the solution to the reformulated problem. 
We further introduce the precise properties of the interactions 
between the nonlinear waves. 
In order to obtain the time-decay estimates (Theorem 1.1 and Theorem 1.2), 
in Section 4 and Section 5, %<=================����������������
we establish the uniform energy estimates in time %<=======������������
by using a very technical time-weighted energy method. %<===������������
%and careful estimates of the interactions %<===============������������
%between the nonlinear waves. 
%<============================������������
Finally, in Section 6, %<==================================������������
we prove the time-decay $L^{r+1}$-estimate %<========================������������
for the higher order derivative, Theorem 1.3. %<=================���������������������\��

\smallskip

{\bf Some Notation.}\quad 
We denote by $C$ generic positive constants unless 
they need to be distinguished. 
In particular, use $C(\alpha, \beta, \cdots )$ 
or $C_{\alpha, \beta, \cdots }$ 
when we emphasize the dependency on $\alpha, \beta, \cdots $. 
Use $\mathbb{R}^{+}$ as 
$
\mathbb{R}^{+}:=(0,\infty),
$
and the symbol ``$\vee $'' as
$$
a\vee b:= \max \{a,b\}. 
$$
We also use the Friedrichs mollifier $\rho_\delta \ast $, 
where, 
$\rho_\delta(x):=\frac{1}{\delta}\rho \left( \frac{x}{\delta}\right)$
with 
\begin{align*}
\begin{aligned}
&\rho \in C^{\infty}_0(\mathbb{R}),\quad 
\rho (x)\geq 0\:  (x \in \mathbb{R}), \\
&\mathrm{supp} \{\rho \} \subset 
\left\{x \in \mathbb{R}\: \left|\:  |\, x \, |\le 1 \right. \right\},\quad  
\int ^{\infty}_{-\infty} \rho (x)\, \mathrm{d}x=1, 
\end{aligned}
\end{align*}
and $\rho_\delta \ast f$ denote the convolution. 
For function spaces, 
$L^p = L^p(\mathbb{R})$ and $H^k = H^k(\mathbb{R})$ 
denote the usual Lebesgue space and 
$k$-th order Sobolev space on the whole space $\mathbb{R}$ 
with norms $||\cdot||_{L^p}$ and $||\cdot||_{H^k}$, respectively. 
We also define 
the bounded $C^{m}$-class $\mathscr{B}^{m}$ as follows 
$$
f\in \mathscr{B}^{m}(\Omega)
\, \Longleftrightarrow  \, 
f\in C^m(\Omega), 
\; 
\sup _{\Omega}\, \sum _{k=0}^{m} \, \bigl| \, D^kf\, \bigr|<\infty 
$$
for $m< \infty$ and 
$$
f\in \mathscr{B}^{\infty }(\Omega)
\, \Longleftrightarrow  \, 
\forall n\in \mathbb{N},\, f\in C^n(\Omega), 
\; 
\sup _{\Omega}\, \sum _{k=0}^{n} \, \bigl| \, D^kf\, \bigr|<\infty 
$$
where $\Omega \subset \mathbb{R}^d$ and 
$D^k$ denote the all of $k$-th order derivatives. 

\section{Preliminaries} 
In this section, 
we shall arrange the several lemmas concerned with 
the basic properties of 
the rarefaction wave %and the viscous contact wave 
for accomplishing the proof of our main theorems. 
Since the rarefaction wave $u^r$ is not smooth enough, 
we need some smooth approximated one 
as in the previous results in 
\cite{hashimoto-matsumura}, \cite{liu-matsumura-nishihara}, 
\cite{matsu-nishi1}, \cite{matsumura-yoshida}, 
\cite{yoshida}, \cite{yoshida'}, \cite{yoshida''}. 
We start with the well-known arguments on $u^r$ 
and the method of constructing its smooth approximation. 
We first consider the rarefaction wave solution $w^r$ %in distribution sense 
to the Riemann problem 
for the non-viscous Burgers equation %(hyperbolic equation)
%with prescribed far field states $w_\pm \in \mathbb{R} \: (w_-<w_+)$: 
\begin{equation}
\label{riemann-burgers}
  \left\{\begin{array}{l}
  \partial _t w + 
  \displaystyle{ \partial _x \left( \, \frac{1}{2} \, w^2 \right) } = 0 
  \, \, \; \; \qquad \quad \qquad ( t > 0,\,x\in \mathbb{R}),\\[7pt]
  w(0,x) = w_0 ^{\rm{R}} ( \, x\: ;\: w_- ,w_+):= \left\{\begin{array}{ll}
                                                  w_+ & \, \: \; \quad (x>0),\\[5pt]
                                                  w_- & \, \: \; \quad (x<0),
                                                  \end{array}
                                                  \right.
  \end{array}
  \right.
\end{equation}
where $w_\pm \in \mathbb{R}$ are 
the prescribed far field states. 
If the far field states satisfy $w_-<w_+$, 
then the Riemann problem (2.1) has a unique global weak solution 
$w=w^r\left( \, \frac{x}{t}\: ;\: w_-,w_+\right)$ 
given explicitly by 
\begin{equation}
\label{rarefaction-burgers}
w^r \left( \, \frac{x}{t}\: ;\: w_-,w_+\right) := 
  \left\{\begin{array}{ll}
  w_{-} & \bigl(\, x \leq w_{-}t \, \bigr),\\[5pt]
  \displaystyle{ \frac{x}{t} } & \bigl(\, w_{-} t \leq x \leq w_{+} t \, \bigr),\\[5pt]
  w_+ & \bigl(\, x\geq w_{+} t \, \bigr).
  \end{array}\right.
\end{equation} 
Next, under the condition 
$f''(u)>0\ (u\in \mathbb{R})$ and $u_-<u_+$, 
the rarefaction wave solution 
$u=u^r\left( \, \frac{x}{t}\: ;\: u_-,u_+\right)$ 
of the Riemann problem (1.2) 
for hyperbolic conservation law 
is exactly given by 
\begin{equation}
u^r\left( \, \frac{x}{t} \: ; \:  u_-,u_+\right) 
= (\lambda)^{-1}\biggl( w^r\left( \, \frac{x}{t} \: ; \:  \lambda_-,\lambda_+\right)\biggr)
\end{equation}
which is nothing but (1.6), 
where $\lambda_\pm := \lambda(u_\pm) = f'(u_\pm)$. 
%We make a smooth approximation of the rarefaction wave $u^r$ 
%due to the previous works 
%(cf. \cite{hashimoto-matsumura},\cite{liu-matsumura-nishihara},\cite{matsu-nishi1},\cite{matsumura-yoshida}). 
We also consider 
the Cauchy problem for the following 
non-viscous Burgers equation
\begin{eqnarray}
\label{smoothappm}
\left\{\begin{array}{l}
 \partial _t w 
 + \displaystyle{ \partial _x \left( \, \frac{1}{2} \, w^2 \right) } =0 
 \, \, \; \; \quad \qquad \qquad \qquad \qquad \qquad  
 (\ t>0,\,x\in \mathbb{R}),\\[7pt]
 w(0,x) 
 = w_0(x) 
 := \displaystyle{ \frac{w_-+w_+}{2} + \frac{w_+-w_-}{2}\tanh x } 
 \qquad \quad \; \:  (x\in \mathbb{R}). 
\end{array}
\right.
\end{eqnarray}   
%Then, we define the approximation for 
By using the method of characteristics, 
we easily see that the Cauchy problem (2.4) 
has a unique classical solution 
$$
w=w(\, t,x\: ;\: w_-,w_+)\in \mathscr{B}^{\infty }( \, [\, 0,\infty )\times \mathbb{R})
$$
satisfying the following formula
\begin{eqnarray}
 \left\{\begin{array} {l}
 w(t,x)=w_0\bigl(x_0(t,x)\bigr)=
 \displaystyle{ \frac{\lambda_-+\lambda_+}{2} } 
+ \displaystyle{ \frac{\lambda_+-\lambda_-}{2}\tanh \bigl( x_0(t,x) \bigr)} ,\\[7pt]
 x=x_0(t,x)+w_0\bigl(x_0(t,x)\bigr)\,t.
 \end{array}
  \right.\,
\end{eqnarray}
We define a smooth approximation of $w^r(\, \frac{x}{t}\: ;\: w_-,w_+)$ 
by the solution $w$. 
We also note the assumption of the flux function $f$ to be 
$\lambda'(u)\left( =\frac{\mathrm{d}^2f}{\mathrm{d}u^2}(u)\right)>0$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%��
%%%%%%%%%%%%%%%                     w(t,x)                    %%%%%%%%%%%%%%%%%%%%%��
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%��

Now we summarize the results for the smooth approximation $w(\, t,x\: ;\: w_-,w_+)$ 
in the next lemma. %in Lemma 2.1. 
Since the proof is given by the direct calculation as in \cite{matsu-nishi1}, %of (2.5). 
we omit it. %the details. 

\medskip

\noindent
%     w\bigl(\, t,x\: ;\: f^{\epsilon}(w_-), \, f^{\epsilon}(w_+) \, \bigr)
{\bf Lemma 2.1.}\quad{\it
Assume that the far field states satisfy $w_-<w_+$. 
Then the classical solution $w(t,x)=w(\, t,x\: ;\: w_-,w_+)$
given by {\rm(2.4)} 
satisfies the following properties: 

\noindent
{\rm (1)}\ \ $w_- < w(t,x) < w_+$ and\ \ $\partial_xw(t,x) > 0$  
\quad  $(t>0, x\in \mathbb{R})$.

\smallskip

\noindent
{\rm (2)}\ For any $1\leq q \leq \infty$, there exists a positive 
constant $C_q$ such that
             \begin{eqnarray*}
                 \begin{array}{l}
                    \parallel \partial_x w(t)\parallel_{L^q} \leq 
                    C_q (1+t)^{-1+\frac{1}{q}} 
                    \; \quad \bigl(t\ge 0 \bigr),\\[5pt]
                    \parallel \partial_x^2 w(t) \parallel_{L^q} \leq 
                    C_q (1+t)^{-1} 
                    \, \, \quad \quad \bigl(t\ge 0 \bigr).
                    \end{array}       
              \end{eqnarray*}
              
\smallskip

\noindent
{\rm (3)}\ $\: \displaystyle{\lim_{t\to \infty} 
\sup_{x\in \mathbb{R}}
\left| \,w(t,x)- w^r \left( \frac{x}{t} \right) \, \right| = 0}.$
}

\bigskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%��
%%%%%%%%%%%%%%%                     w(t,x)                    %%%%%%%%%%%%%%%%%%%%%��
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%��

\noindent
We define the approximation for 
the rarefaction wave $u^r\left( \, \frac{x}{t}\: ;\: u_-,u_+\right)$ by 
\begin{equation}
U^r(\, t,x\: ; \: u_-,u_+) := (\lambda)^{-1} \bigl( w(\, t,x\: ;\: \lambda_-,\lambda_+)\bigr).
\end{equation}

Then we have the next lemma 
as in the previous works 
(cf. \cite{hashimoto-matsumura}, \cite{liu-matsumura-nishihara}, 
\cite{matsu-nishi1}, \cite{matsumura-yoshida}, 
\cite{yoshida}, \cite{yoshida'}, \cite{yoshida''}). 

\medskip

\noindent
{\bf Lemma 2.2.}\quad{\it% <----- Lemma 2.2��������������������������������
Assume that the far field states satisfy $u_-<u_+$, 
and the flux fanction $f\in C^3(\mathbb{R})$, $f''(u)>0 \: (u\in [\,u_-,u_+\,])$. 
Then we have the following properties:

\noindent
{\rm (1)}\ $U^r(t,x)$ defined by {\rm (2.6)} is %<----- 2.6��������������������������������
the unique $C^2$-global solution in space-time 
of the Cauchy problem
%\begin{equation*}
$$
\left\{
\begin{array}{l} 
\partial _t U^r +\partial _x \bigl( f(U^r ) \bigr) = 0 
\, \, \, \, \; \; \; \quad \quad \qquad \qquad \qquad \qquad  
(t>0, x\in \mathbb{R}),\\[7pt]
U^r(0,x) 
= \displaystyle{ (\lambda)^{-1} \left( \, \frac{\lambda_- + \lambda_+}{2} 
+ \frac{\lambda_+ - \lambda_-}{2} \tanh x \, \right) } 
\; \: \: \quad \quad( x\in \mathbb{R}),\\[7pt]
\displaystyle{\lim_{x\to \pm \infty}} U^r(t,x) =u_{\pm} 
\, \, \: \: \; \; \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad 
\bigl(t\ge 0 \bigr).
\end{array}
\right.\,     
$$
%\end{equation*}
{\rm (2)}\ \ $u_- < U^r(t,x) < u_+$ and\ \ $\partial_xU^r(t,x) > 0$  
\quad  $(t>0, x\in \mathbb{R})$.

\smallskip

\noindent
{\rm (3)}\ For any $1\leq q \leq \infty$, there exists a positive 
constant $C_q$ such that
             \begin{eqnarray*}
                 \begin{array}{l}
                    \parallel \partial_x %U^r(t,\cdot \: )
                    U^r(t) 
                    \parallel_{L^q} \leq 
                    C_q(1+t)^{-1+\frac{1}{q}} 
                    \quad \bigl(t\ge 0 \bigr),\\[5pt]
                    \parallel \partial_x^2 U^r(t) \parallel_{L^q} \leq 
                    C_q(1+t)^{-1}
                    \, \, \quad \quad \bigl(t\ge 0 \bigr).
                    \end{array}       
              \end{eqnarray*}
              
\smallskip

\noindent
{\rm (4)}\ $\: \displaystyle{\lim_{t\to \infty} 
\sup_{x\in \mathbb{R}}
\left| \,U^r(t,x)- u^r \left( \frac{x}{t} \right) \, \right| = 0}.$


\smallskip

\noindent
{\rm (5)}\ For any $\epsilon \in (0,1)$, there exists a positive 
constant $C_\epsilon$ such that
$$
\left|\, 
U^r(t,x)-u_+ \, \right|
\leq C_\epsilon(1+t)^{-1+\epsilon}\mathrm{e}^{-\epsilon |x-\lambda_+t|}
\quad \bigl(t\ge 0, x \ge \lambda_+t \bigr).
$$

\smallskip

\noindent
{\rm (6)}\ For any $\epsilon \in (0,1)$, there exists a positive 
constant $C_\epsilon$ such that
$$
\left|\, 
U^r(t,x)-u_- \, \right|
\leq C_\epsilon(1+t)^{-1+\epsilon}\mathrm{e}^{-\epsilon |x-\lambda_-t|}
\quad \bigl(t\ge 0, x \le \lambda_-t\bigr).
$$

\smallskip

\noindent
{\rm (7)}\ For any $\epsilon \in (0,1)$, there exists a positive 
constant $C_\epsilon$ such that
$$
\left| \,U^r(t,x) - u^r\left( \frac{x}{t}\right) \, \right| 
\leq C_\epsilon(1+t)^{-1+\epsilon} 
\qquad \bigl(t \ge 1,  \lambda_-t \le x \le \lambda_+t\bigr).
$$

\smallskip

\noindent
{\rm (8)}\ For any $(\epsilon ,q)\in (0,1)\times [\, 1,\infty \, ]$, 
there exists a positive 
constant $C_{\epsilon,q}$ such that
$$
\left|\left|
 \,U^r(t,\cdot \: ) - u^r\left( \frac{\cdot }{t}\right) \, 
\right|\right|_{L^q}  
\leq C_{\epsilon,q}(1+t)^{-1+\frac{1}{q}+\epsilon} 
\qquad \bigl(t \ge 0 \bigr).
$$
}

\noindent
Because the proofs of (1) to (4) are given in \cite{matsu-nishi1}, 
(5) to (7) are in \cite{matsumura-yoshida} 
and (8) is in \cite{yoshida}, 
we omit the proofs here. 
\bigskip

We also prepare the next lemma for the properties of 
the contact wave for $p$-Laplacian type viscosity 
$U\Bigl(\, \frac{x}{t^{\frac{1}{p+1}}}\,;\,u_- ,u_+ \Bigr)$ defined by (1.11). 
In the following, we abbreviate 
%``viscous contact wave for non-Newtonian'' 
``contact wave for $p$-Laplacian type viscosity'' 
to ``viscous contact wave''.  %without confusion
we rewrite the viscous contact wave as %follows: 
\begin{align}
\begin{aligned}
U(t,x)&=U\left(\, \frac{x}{t^{\frac{1}{p+1}}}\: ;\: u_- ,u_+ \right) \\
      &=u_{+}-\Lint^{\infty}_{x}
             \frac{1}{t^{\frac{1}{p+1}}}\, 
             \bigggl( \left( 
             A-B \left(\frac{y}{t^{\frac{1}{p+1}}} \right)^2 \, 
             \right)\vee 0 \bigggr)^{\frac{1}{p-1}}
             \, \mathrm{d}y, 
\end{aligned}
\end{align}
where 
\begin{eqnarray*}
\left\{\begin{array}{ll}
\displaystyle{
A=A_{p,\mu,u_{\pm}}
:= \left( \, 
   \frac{(p-1)\, \left( \, u_{+} - u_{-} \, \right)}
   {8\, \mu\, p(p+1) 
   \biggl( \, \displaystyle{\int^{\frac{\pi}{2}}_{0} 
   \bigl( \sin \theta \bigr)^{\frac{p+1}{p-1}}\, \mathrm{d}\theta }
   \biggr)^2 }
   \, \right)^{\frac{p-1}{p+1}}
}, \\[25pt]
\displaystyle{ 
B=B_{p,\mu}:= \frac{p-1}{2\, \mu \, p(p+1)} 
}, \\[10pt] 
\displaystyle{ 
2A^{\frac{p+1}{2(p-1)}}B^{-\frac{1}{2}}
      \int^{\frac{\pi}{2}}_{0} 
      \bigl( \sin \theta \bigr)^{\frac{p+1}{p-1}}\, \mathrm{d}\theta 
      = u_{+} - u_{-}}
.
\end{array}
\right.\,
\end{eqnarray*}

\noindent
Then, the following properties hold. 

\medskip

\noindent
{\bf Lemma 2.3.}\quad{\it% <----- Lemma 2.3��������������
For any %fixed 
$p>1$ and $u_\pm \in \mathbb{R}$, we have the following: 

\noindent
{\rm (i)}\ $U$ defined by {\rm (1.11)} satisfies 
$$
%\Biggggggl( a \Biggggggm| b \Biggggggr)
U \in 
\mathscr{B}^1\bigl( \, (0,\infty \bigr)\times \mathbb{R}\, ) 
\mathlarger{ \mathlarger{ \mathlarger{ 
\setminus } } } \, 
C^2
\left( \, 
\biggl\{ \, 
(t,x) \in \mathbb{R}^{+} \times \mathbb{R} \, \biggr|\biggl. \, 
x = \pm \sqrt{\frac{A}{B}}\, t^{\frac{1}{p+1}} \, 
\biggr\} \, \right), 
$$  
and is a 
self-similar type strong solution of the Cauchy problem 
%of the following $p$-Laplace evolution equation 
   %\begin{equation*}
   $$
           \left\{
              \begin{array}{l} 
              \partial _t U 
              - \mu \, \partial_x \left( \, 
              \left| \, \partial_x U \, \right|^{p-1} \partial_x U \, \right) 
              = 0 
              \quad \qquad \qquad  (t>0, x\in \mathbb{R}),\\[13pt]
              U(0,x) = u_0 ^{\rm{R}} (\, x\: ;\: u_- ,u_+)
               = 
               \left\{\begin{array} {ll}
               u_-  & \, \; \; \quad \qquad (x < 0),\\[5pt]
               u_+  & \, \; \; \quad \qquad (x > 0),
               \end{array}\right.\\[13pt]
              \displaystyle{\lim_{x\to \pm \infty}} U(t,x) =u_{\pm} 
              \, \: \: \: \; \quad \quad \qquad \qquad \qquad \qquad \bigl(\, t\ge 0\bigr).
              \end{array}
            \right.\,   
    $$          
  %\end{equation*}
{\rm (ii)}\ For $t>0$ and $x\in \mathbb{R}$, 
$$
\left\{
   \begin{array}{l} 
   U(t,x)=u_{-},%\; \partial_xU(t,x) = 0 \: \; \quad \qquad 
   \, \: \; \quad \qquad \qquad \qquad \qquad 
   \left( \, x \leq -\sqrt{\frac{A}{B}}\, t^{\frac{1}{p+1}} \,  \right),
   \\[5pt]
   u_- < U(t,x) < u_+,\; \partial_xU(t,x) > 0 \quad 
   \left( \, 
   -\sqrt{\frac{A}{B}}\, t^{\frac{1}{p+1}} < x < \sqrt{\frac{A}{B}}\, t^{\frac{1}{p+1}}
   \, \right),
   \\[5pt]
   U(t,x)=u_{+},%\; \partial_xU(t,x) = 0 \: \; \quad \qquad 
   \, \: \; \quad \qquad \qquad \qquad \qquad 
   \left( \,  x \geq \sqrt{\frac{A}{B}}\, t^{\frac{1}{p+1}} \,  \right). 
   \end{array}
\right.\, 
$$


\smallskip

\noindent
{\rm (iii)}\ It holds that for any $1 \leq  q < \infty $, 
$$
\|\, \partial _x %U (t,\cdot \: )
U (t)\, \| _{L^q} 
= C_1(\, A,B\: ;\: p ,q\, ) \, t^{-\frac{q-1}{(p+1)q}}\qquad (t > 0) 
$$
where 
$$
C_1(\, A,B\: ;\: p ,q\, ):=
\left( 
2A^{\frac{p+2q-1}{2(p-1)}}B^{-\frac{1}{2}}
\int^{\frac{\pi}{2}}_{0} 
      \bigl( \sin \theta \bigr)^{\frac{q}{p-1}}
\, \mathrm{d}\theta 
\right)^{\frac{1}{q}}.
$$
If $q = \infty$, we have 
$$
\|\, \partial _x U (t) \, \| _{L^\infty} 
= \left(2A \right)^{\frac{1}{p-1}} \, t^{- \frac{1}{p+1}}\qquad (t > 0).
$$

\smallskip

\noindent
{\rm (iv)}\ It holds that for any $1 \leq  q < \frac{p-1}{p-2}$ with $p>2$, 
or any $1 \leq  q < \infty$ with $1 < p \leq 2$, 
$$
\|\, \partial _x^2 U (t) \, \| _{L^q} 
= C_2(\, A,B\: ;\: p ,q\, ) \, t^{-\frac{2q-1}{(p+1)q}}\qquad (t > 0) 
$$
where 
\begin{align*}
\begin{aligned}
& C_2(\, A,B\: ;\: p ,q\, ) \\
& \quad \, 
:=
\bigggl( 
2\left(
\frac{2A^{-\frac{ p-2 }{p-1}} B}{p-1}
\right)^{q}
\left( \frac{B}{A} \right)^{- \frac{q+1}{2}}
\int^{\frac{\pi}{2}}_{0} 
      \bigl( \sin \theta \bigr)^{- \frac{2 ( p-2 )q}{p-1} + 1}
      \bigl( \cos \theta \bigr)^{q}
\, \mathrm{d}\theta
\bigggr)^{\frac{1}{q}}. 
\end{aligned}
\end{align*}
If $1 < p \leq 2$, for $q = \infty$, we have 
$$
\|\, \partial _x^2 U (t) \, \| _{L^\infty} 
= {\frac{2A^{\frac{\left| p-2 \right|}{p-1}} B}{p-1}} 
     \left( \frac{B}{A} \right)^{-\frac{1}{2}}\, 
     t^{- \frac{2}{p+1}}\qquad (t > 0).
$$

\smallskip

\noindent
{\rm (v)}\ It holds that 
$$
\left|\left| \, 
\partial_x \left( \, 
\left| \, \partial_x U \, \right|^{p-1} \partial_x U \, 
\right) (t) \, 
\right| \right| _{L^2} 
= C_3(\, A,B\: ;\: p \, ) \, t^{-\frac{2p+1}{2(p+1)}}\qquad (t > 0) 
$$
where 
$$
C_3(\, A,B\: ;\: p \, ):=
\left( 
2\left(
\frac{2B^{p}}{p-1}
\right)^{2}
\left( \frac{B}{A} \right)^{-\frac{3p-7}{2(p-1)}}
\int^{\frac{\pi}{2}}_{0} 
      \bigl( \sin \theta \bigr)^{\frac{p+3}{p-1}}
      \bigl( \cos \theta \bigr)^{2}
\, \mathrm{d}\theta
\right)^{\frac{1}{2}}.
$$
\smallskip

\noindent
{\rm (vi)}\; $\displaystyle{\lim_{t\to \infty} \sup_{x\in \mathbb{R}}\, 
\bigl|\,U(1+t,x)- U(t,x)\, \bigr| = 0}.$
             
\smallskip

\noindent
{\rm (vii)}\ For any $1\leq q \leq \infty$ with $p>1$, 
there exists a positive 
constant $C_{p,q}$ such that
$$
\left|\left|
 \,U(1+t,\cdot \: ) - U(t,\cdot \: ) \, 
\right|\right|_{L^q}  
\leq C_{p,q}\, t^{-1+\frac{1}{(p+1)q}} 
\qquad \bigl(t > 0 \bigr).
$$

}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Reformulation of the problem}
In this section, 
we reduce our Cauchy problem (1.1) to %the essential
a simpler case 
and reformulate the problem 
in terms of the deviation from the asymptotic state 
(the same as in \cite{matsumura-yoshida}, \cite{yoshida}, \cite{yoshida'}). 
At first, % <-------------------------------------------------------------
without loss of generality, 
we shall consider the case 
where $a<0$, $b=0$ and the flux function $f(u)$ satisfies 
\begin{equation}
\left\{
\begin{array}{ll}
  f''(u) >0 & \; (u \in (-\infty ,a]\cup [0,+\infty )),\\[5pt]
  f(u) =0 & \; (u \in (a,0)),
\end{array}\right.
\end{equation}
under changing the variables and constant as
$x-\tilde{\lambda} \,t \mapsto x$,  $u-b \mapsto u$,
$f(u+b)-f'(b)\,u-f(a) \mapsto f(u)$ and $a-b \mapsto a$
in this order. 
For the far field states 
$u_\pm \in \mathbb{R}$, 
%we are enough only to deal with the essential case 
%we treat only the %essential 
%case 
we only deal with the typical case 
$a<u_-<0<u_+$ for simplicity, since the case 
$u_-<a<0<u_+$ can be treated %is 
technically %quite 
in the same way of the proof as 
$a<u_-<0<u_+$. 
%Actually
Indeed, in the case $u_-<a<0<u_+$,
as we shall see in Section 4  and Section 5, 
%it is easy to see that 
there appears the extra nonlinear interaction terms
between two rarefaction waves 
$u^r(\, \frac{x}{t}\: ;\: u_-,a)$ and $u^r(\, \frac{x}{t}\: ;\: 0,u_+)$
with $\lambda(a)=\lambda(0)=0$ in the remainder term of 
the viscous conservation law for the asymptotics 
$U_{multi}$\,(see the right-hand side of (3.4)). 
These terms can be handled in much easier way % <----- Lemma 2.2��������������
by Lemma 2.2 than that for other essential nonlinear interaction terms 
between the rarefaction and the viscous contact waves. 
Furthermore, we should point out that 
the problem under the assumptions for the flux function (3.1) 
and the far field states $a<u_-<0<u_+$
is essentially the same as that for $a=-\infty$, because 
obtaining the a priori and the uniform energy estimates for the former one can be
given in almost the same way as the latter one. 
Therefore, it is quite natural for us 
to treat only %the essential case 
a simple case 
\begin{equation}
\left\{
\begin{array}{ll}
  f''(u) >0 & \; (u \in [\, 0,\infty )),\\[5pt]
  f(u) =0 & \; (u \in (-\infty,0)),
\end{array}\right.
\end{equation}
and assume $u_-<0<u_+$.
The corresponding stability theorem 
and our main theorems are the following. 

Under the situation, we reformulate the problem 
in terms of the deviation from the asymptotic state.  
We first should note by Lemma 2.2 and Lemma 2.3, 
the asymptotic state $u^r\left( \, \frac{x}{t}\: ;\: u_-,u_+\right)$ 
can be replaced by a following approximated multiwave pattern 
\begin{equation*}
\tilde{U}(t,x) := U(1+t,x) + U^r(t,x),
\end{equation*}
where
$$
U(1+t,x)=U\left( \frac{x}{(1+t)^{\frac{1}{p+1}}}\: ;\: u_- ,0 \right),
\quad U^r(t,x) =U^r(\, t,x\: ; \: 0, u_+).
$$
In fact, 
from Lemma 2.1 (especially (8)) and Lemma 2.3 (especially (vii)), 
it follows that 
for any $\epsilon>0$
\begin{align*}
\begin{aligned}
\left|\left| \, 
\tilde{U}(t,\cdot \: ) - U_{multi}(t,\cdot \: ) \, \right|\right|_{L^q}
&\le
\left|\left| \, U(1+t,\cdot \: ) - U(t,\cdot \: ) \, \right|\right|_{L^q}
+ \left|\left| \, U^r(t,\cdot \: ) 
- u^r\left( \frac{\cdot}{t}\right) \, \right|\right|_{L^q}\\
&\le C_{\epsilon,q}(1+t)^{-\left( 1-\frac{1}{q}\right)+\epsilon}
\qquad (t\ge 0\, ;\, 1\le q\le \infty).
\end{aligned}
\end{align*}
Then it is noted that $\tilde{U}$ is 
monotonically increasing and approximately satisfies 
the equation of (1.1) as
\begin{equation}
 \partial _t\tilde{U} +\partial_x \bigl(f(\tilde{U})\bigr)
 - \mu \, 
 \partial_x 
\left( \, 
\bigl| \, \partial_x \tilde{U} \, \bigr|^{p-1} \partial_x \tilde{U} \, 
\right)
 = -F_{p}(U,U^r),
\end{equation}
where the remainder term $F_{p}(U,U^r)$ is explicitly given by
\begin{align}
\begin{aligned}
F_{p}(U,U^r)
:= 
&\widetilde{F_{p} }(U,U^r)\\
&+ \mu \, \partial_x 
\left( \, 
\bigl| \, \partial_x U + \partial_x U^{r} \, \bigr|^{p-1} 
\bigl( \, \partial_x U + \partial_x U^{r} \, \bigr) - 
\left| \, \partial_x U \, \right|^{p-1} \partial_x U  \, 
\right)\\
:= 
&-\bigl( \, f'(U+U^r)
- f'(U^r) \, \bigr) \, \partial_x U^r 
- f'(U+U^r)\, \partial_x U \\
&+ \mu \, \partial_x 
\left( \, 
\bigl| \, \partial_x U + \partial_x U^{r} \, \bigr|^{p-1} 
\bigl( \, \partial_x U + \partial_x U^{r} \, \bigr) - 
\left| \, \partial_x U \, \right|^{p-1} \partial_x U  \, 
\right)
\end{aligned}
\end{align}
which consists of the interaction terms of the viscous contact wave $U$ and 
the approximation of the rarefaction wave $U^r$, 
and the approximation error of $U^r$ as solution to 
the conservation law for the $p$-Laplacian type viscosity. 
Here we should %underline
note that 
$U$ is monotonically nondecreasing 
and $U^r$ is monotonically increasing, 
that is, 
$\partial_x\tilde{U}(t,x) > 0\ \bigl(t\ge 0, x\in \mathbb{R} \bigr)$
which is frequently used hereinafter. 
Now  putting 
\begin{equation}
u(t,x) = \tilde{U}(t,x) + \phi(t,x)
\end{equation}
and using (3.5),
we can reformulate the problem (1.1) in terms of 
the deviation $\phi $ from 
$\tilde{U}$ as 
\begin{eqnarray}
 \left\{\begin{array}{ll}
  \partial _t\phi + \partial_x \left( f(\tilde{U}+\phi) - f(\tilde{U}) \right) \\[5pt]
  \quad - \mu \, \partial_x 
    \left( \, 
    \bigl| \, \partial_x \tilde{U} + \partial_x \phi \, \bigr|^{p-1} 
    \bigl( \, \partial_x \tilde{U} + \partial_x \phi \, \bigr) - 
    \bigl| \, \partial_x \tilde{U} \, \bigr|^{p-1} \partial_x \tilde{U}  \, 
    \right)\\[2pt]
    \quad \qquad \qquad \qquad \qquad \qquad \qquad = F_{p}(U,U^r)
  \, \, \: \: \quad  (t>0, x\in \mathbb{R}), \\[5pt]
  \phi(0,x) = \phi_0(x) 
  := u_0(x)-\tilde{U}(0,x) 
  \qquad \qquad \quad \; \: \: \; \; \: \; \; \,\, \, \, (x\in \mathbb{R}), \\[5pt]
  \displaystyle{\lim _{x\rightarrow \pm \infty}\, \phi (t,x) = 0 }
  \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \, \bigl( t \geq 0 \bigr).
 \end{array}
 \right.\,
\end{eqnarray}
Then we look for the global solution in time 
$$
\phi \in C^0\bigl( \, [\, 0,\infty)\, ;L^2 \bigr)
         \cap L^{\infty}\bigl( \, \mathbb{R}^{+} \, ;L^{2} \bigr) 
$$ with 
$$
\partial _x \phi \in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr)
                     \cap L^{p+1}
                     \bigl(\, {\mathbb{R}^{+}_{t}} 
                     \times {\mathbb{R}}_{x} \bigr). 
$$
Here we note the fact $\phi_0 \in L^2$ 
and $\partial_x \phi_0 \in L^{p+1}$ by the assumptions on $u_0$ 
and the fact 
$$
\partial_x \tilde{U}(0,\cdot \, )
=\partial_x U(0,\cdot \, )+\partial_x U^r(0,\cdot \, )\in L^{p+1}.
$$
In the following, we always assume that 
the flux function $f\in C^1(\mathbb{R})\cap C^3 (\, [\, 0,\infty))$ satisfies (3.2), 
and  
the far field states satisfy $u_-<0<u_+$. 
Then the corresponding our main theorems for $\phi$ 
we should prove are as follows. 

\medskip

\noindent
{\bf Theorem 3.1.}\quad{\it
Assume that the flux function 
$f\in C^1(\mathbb{R})\cap C^3 (\, [\, 0,\infty))$ satisfies {\rm(3.2)},  
the far field states $u_-<0<u_+$, 
and the initial data 
$\phi_0 \in L^2$ and 
$\partial _x u_0 \in L^{p+1}$. 
Then, the unique global solution in time $\phi$ 
of the Cauchy problem {\rm(3.6)} 
satisfying 
\begin{eqnarray*}
\left\{\begin{array}{ll}
\phi \in C^0\bigl( \, [\, 0,\infty)\, ;L^2 \bigr)
            \cap L^{\infty}\bigl( \, \mathbb{R}^{+} \, ;L^{2} \bigr),\\[5pt]
\partial _x \phi \in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr)
                     \cap L^{p+1}
                          \bigl(\, {\mathbb{R}^{+}_{t}} 
                          \times {\mathbb{R}}_{x} \bigr),\\[5pt]
\partial _x \bigl( \, \tilde{U} + \phi \, \bigr)
              \in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr)
                  %\cap L^{p+1}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       {\mathbb{R}}_{x} \bigr)
                  \cap L^{p+2}\bigl(\, {\mathbb{R}^{+}_{t}} 
                       \times 
                       \left\{x \in \mathbb{R}\, |\, u>0 \right\} \bigr),\\[5pt]
\partial _t \bigl( \, \tilde{U} + \phi \, \bigr) 
%\in L^{2}\bigl(\, {\mathbb{R}^{+}_{t}} \times {\mathbb{R}}_{x} \bigr),\\[5pt]
\in  L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr)
     %\cap L^{p+1}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       {\mathbb{R}}_{x} \bigr)\\
%     \qquad \qquad \qquad \qquad \quad \quad \quad \; \; \; \; \: \: \, 
%     \cap \: L^{p+2}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       \left\{x \in \mathbb{R}\, |\, u>0 \right\} \bigr)
%     \cup L^{2}\bigl(\, {\mathbb{R}^{+}_{t}} \times {\mathbb{R}}_{x} \bigr)
,\\[5pt]
%\partial_x \left( f(\tilde{U}+\phi) \right)
%%\in L^{\infty}\bigl( \, \mathbb{R}^{+} \, ;L^{2} \bigr),\\[5pt]
%\in L^{\infty} \bigl( \, \mathbb{R}^{+} \, ;L^{p+1} \bigr)
%                  \cap L^{p+1}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       {\mathbb{R}}_{x} \bigr)\\
%                  \qquad \qquad \qquad \qquad 
%                  \qquad \quad \quad \; \; \; \; \; \; \: \: \, 
%                  \cap \: L^{p+2}\bigl(\, {\mathbb{R}^{+}_{t}} 
%                       \times 
%                       \left\{x \in \mathbb{R}\, |\, u>0 \right\} \bigr),\\[5pt]
\partial_x \left( \, \bigl| \, 
\partial_x\bigl( \, \tilde{U} + \phi \, \bigr)\, 
\bigr|^{p-1} \partial_x\bigl( \, \tilde{U} + \phi \, \bigr) \, \right)
\in L^{2}\bigl(\, {\mathbb{R}^{+}_{t}} \times {\mathbb{R}}_{x} \bigr),
\end{array} 
\right.\,
\end{eqnarray*}
and 
$$
\displaystyle{\lim_{t\to \infty}}\sup_{x\in \mathbb{R}}\, 
\bigl| \, \phi(t,x)\,\bigr| = 0
$$
satisfies the following time-decay estimates 
\begin{eqnarray*}
\left\{\begin{array} {ll}
\left|\left|\,
\phi (t) 
\, \right|\right|_{L^q }
\leq C( \, p, \, q, \, \phi_0 \, ) \, 
     (1+t)^{-\frac{1}{3p+1}\left(1-\frac{2}{q}\right)},\\[7pt]
\left|\left|\,
\phi (t) 
\, \right|\right|_{L^{\infty} }
\leq C( \, \epsilon, \, p, \, q, \, \phi_0, \, \partial _x u_0 \, ) \, 
     (1+t)^{-\frac{1}{3p+1}+\epsilon},
\end{array}
  \right.\,
\end{eqnarray*}
for %$2 \le  p< \infty $
$q \in [\, 2, \infty) $ and any $\epsilon>0$. 
}

\medskip

\noindent
{\bf Theorem 3.2.}\quad{\it
Under the same assumptions in Theorem 3.1, 
if the initial data further satisfies $\phi_0 \in L^1$, %L^1(\mathbb{R})
then it holds that the unique global solution in time $\phi$ 
of the Cauchy problem {\rm(3.6)} 
satisfies the following time-decay estimates 
\begin{eqnarray*}
\left\{\begin{array} {ll}
\left|\left| \,
\phi (t) 
\, \right|\right|_{L^q }
\leq C( \, p, \, q, \, \phi_0 \, ) \, 
     (1+t)^{-\frac{1}{2p}\left(1-\frac{1}{q}\right)},\\[7pt]
\left|\left|\,
\phi (t) 
\,\right|\right|_{L^{\infty} }
\leq C( \, \epsilon, \, p, \, q, \, \phi_0, \, \partial _x u_0 \, ) \, 
     (1+t)^{-\frac{1}{2p}+\epsilon},
\end{array}
  \right.\,
\end{eqnarray*}
for %$1 \le  p< \infty $ 
$q \in [\, 1, \infty) $ and any $\epsilon>0$. 
Furthermore, the solution satisfies 
the following time-decay estimates for the higher order derivative
\begin{align*}
\begin{aligned}
&\bigl|\bigl|\,
 \partial _x u(t) \,
 \bigr|\bigr|_{L^{p+1} }, \, \; \; 
 \left|\left|\,
 \partial _x \phi (t) 
 \,\right|\right|_{L^{p+1} } \\[5pt]
& \leq \left\{\begin{array} {ll} 
      C( \, \epsilon, \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t)^{-\frac{p}{(p+1)^2}} \, \quad 
      \left( \, 1 < p \le 
      \displaystyle{\frac{7}{12} + 
      \sqrt{\frac{73}{144} - \frac{(p+1)^2(3p-2)}{3}\, \epsilon } }
      \, \right),\\[15pt] 
      C( \, \epsilon, \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t)^{-\frac{3}{2(p+1)(3p-2)} + \epsilon} 
      \left( \,  
      \displaystyle{\frac{7}{12} + 
      \sqrt{\frac{73}{144} - \frac{(p+1)^2(3p-2)}{3}\, \epsilon } }
      < p \, \right),
      \end{array}
      \right.\,
\end{aligned}
\end{align*}
for any $0<\epsilon \ll 1$. 
}

\medskip

\noindent
{\bf Theorem 3.3.}\quad{\it
Under the same assumptions in Theorem 3.2, 
if the initial data further satisfies 
$\partial _x u_0 \in L^{r+1} \, (r>p)$, 
then it holds that the unique global solution in time $\phi$ 
of the Cauchy problem {\rm(3.6)} 
satisfies the following time-decay estimates 
for the higher order derivative
\begin{align*}
\begin{aligned}
&\bigl|\bigl|\,
 \partial _x u(t) \,
 \bigr|\bigr|_{L^{r+1} }, \, \; \; 
 \left|\left|\,
 \partial _x \phi(t) 
 \,\right|\right|_{L^{r+1} } \\[5pt]
& \leq \left\{\begin{array} {ll} 
       C( \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
       (1+t)^{-\frac{4p(r-p)+7p+3}{6p(p+1)(r+1)}} \\[10pt] 
       %\, \, \: \; \; 
       \left( \, 
       1 < p \le 
       \displaystyle{\frac{7}{12} + 
       \sqrt{\frac{73}{144} - \frac{p(p+1)(3p-2)(r+1)}{2(r-p+1)}\, \epsilon } }, \; 
       r > \displaystyle{\frac{-4p^2+7p+3}{2p}} > p 
       \, \right),\\[15pt] 
       C( \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
       (1+t)^{-\frac{p+2r}{2p(3p-2)(r+1)}+ \epsilon} \\[15pt] 
       %\, \: \; \; \quad 
       \left( \, 
       1 < p \le 
       \displaystyle{\frac{7}{12} + 
       \sqrt{\frac{73}{144} - \frac{p(p+1)(3p-2)(r+1)}{2(r-p+1)}\, \epsilon } }, \; 
       p < r \le \displaystyle{\frac{-4p^2+7p+3}{2p}}
       \, \right),\\[15pt] 
       C( \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
       (1+t)^{-\frac{6p(r-p)+7p+2r+3}{2(3p+1)(3p-2)(r+1)}+ \epsilon} \\[10pt] 
       \, \, \, \, \: \: \; \; \; \; \quad \quad \quad \quad \qquad \qquad \qquad 
       \left( \, 
       \displaystyle{\frac{7}{12} + 
       \sqrt{\frac{73}{144} - \frac{p(p+1)(3p-2)(r+1)}{2(r-p+1)}\, \epsilon } } < p
       \, \right),
       \end{array}
       \right.\,
\end{aligned}
\end{align*}
for any $0<\epsilon \ll 1$. 
}

\medskip
In order to accomplish the proofs of 
Theorem 3.1, Theorem 3.2 and Theorem 3.3, 
we will need some estimates about boundedness 
of the perturbation $\phi$ and $u$.  
We shall arrange some lemmas for them. 

By using the maximum principle 
(cf. \cite{ilin-kalashnikov-oleinik}, \cite{ilin-oleinik}), 
we first have the following uniform boundedness 
of the perturbation $\phi$ (and also $u$), that is, 

\medskip

\noindent
{\bf Lemma 3.1} (uniform boundedness){\bf .}\quad {\it
It holds that 
\begin{align}
\begin{aligned}
&\sup_{t\in [\, 0,\infty ), x\in \mathbb{R}}|\,\phi(t,x)\,| \\
&\le 
\sup_{x\in \mathbb{R}}|\,u_0 (x)\,| 
+ \sup_{t\in [\, 0,\infty ), x\in \mathbb{R}}|\,U (t,x)\,| 
+ \sup_{t\in [\, 0,\infty ), x\in \mathbb{R}}|\,U^r (t,x)\,| \, \\ 
&= 
\Vert \,\phi_0 \, \Vert_{L^{\infty}} 
+ 2 \, \bigl(\, | \, u_- \, | + | \, u_+ \, | \, \bigr), 
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
&\sup_{t\in [\, 0,\infty), x\in \mathbb{R}}|\, u(t,x)\,| \\
&\le \| \,\phi_0\, \|_{L^{\infty}} 
     + 2 \, \bigl(\, | \, u_- \, | + | \, u_+ \, | \, \bigr)
     + | \, u_- \, | \vee | \, u_+ \, | 
     =: \widetilde{C}. 
     \quad \quad \; \; \: \: \, \, 
\end{aligned}
\end{align}
}

\medskip

\noindent
Secondly, 
we also have the uniform estimates of $\phi$ as follows
(for the proof of it, 
see \cite{yoshida'}). 

\medskip

\noindent
{\bf Lemma 3.2} (uniform estimates){\bf .}\quad {\it
The unique global solution in time $\phi$ of the Cauchy problem {\rm (3.3)} 
satisfies the following uniform energy inequalities. 

\smallskip

\noindent
{\rm (1)}\ 
There exists a positive constant 
$
C_{p}(\phi_0)=C_{p}\bigl(\, \| \, \phi _0 \, \| _{L^2} \, \bigr)
$
such that 
\begin{align*}
\begin{aligned}
&\| \, \phi (t)\, \| _{L^2}^2
+\int _0^{\infty} G(t) \, \mathrm{d}t \\
&+\int _0^{\infty} \int _{-\infty}^{\infty} 
\bigl( \partial _x \phi \bigr)^2 
\left( 
\bigl| \partial _x \phi \bigr|^{p-1} 
+ \bigl| \partial _x U \bigr|^{p-1} + \bigl|  \partial _x U^r  \bigr|^{p-1}  
\right) \, \mathrm{d}x \mathrm{d}t 
\leq C_{p}(\phi_0)
 \quad \bigl(t \ge 0\bigr),
\end{aligned}
\end{align*}
where $G=G(t)$ is exactly given by 
\begin{align*}
\begin{aligned}
G(t):=\left( \, \int_{ \tilde{U} \geq 0} 
     \phi^2 \, \partial _x \tilde{U}\, \mathrm{d}x\right) (t)   
     &+\left( \, \int_{\tilde{U} + \phi \geq 0, \tilde{U} < 0} 
     \bigl( \, \tilde{U}+\phi \, \bigr)^2 \partial _x \tilde{U}
     \, \mathrm{d}x\right) (t)\\
    &+\left( \, \int_{\tilde{U} + \phi < 0, \tilde{U} \geq 0} 
     \bigl( \, \tilde{U}+|\, \phi \, |\,  \bigr)^2 \partial _x \tilde{U}
     \, \mathrm{d}x\right) (t).
\end{aligned}
\end{align*}
\smallskip

\noindent
{\rm (2)}\ 
There exists a positive constant 
$
C_{p}(\phi_0, \partial _x u_0)
=C_{p}\bigl(\, \| \, \phi _0 \, \| _{L^2}, 
 \| \, \partial _x u_0 \, \| _{L^{p+1}} \, \bigr)
$
such that 
\begin{align*}
\begin{aligned}
\| \, \partial _x u (t) 
\, \| _{L^{p+1}}^{p+1} 
&+\int _0^{\infty} \int _{-\infty}^{\infty} 
 \bigl| \, \partial _x u \, \bigr|^{2(p-1)} 
 \left( \, \partial _x^2 u \, \right)^2 
 \, \mathrm{d}x \mathrm{d}t \\
&\qquad \quad \leq 
 C_{p}(\phi_0, \partial _x u_0)
  \quad \bigl(t \ge 0\bigr).
\end{aligned}
\end{align*}
\smallskip

\noindent
{\rm (3)}\ 
There exists a positive constant 
$
C_{p}(\phi_0, \partial _x u_0)
=C_{p}\bigl(\, \| \, \phi _0 \, \| _{L^2}, 
 \| \, \partial _x u_0 \, \| _{L^{p+1}} \, \bigr)
$
such that 
\begin{equation*}
\int _0^{\infty}
 \| \, 
 \partial _x u (t) 
 \, \| _{L^{p+2}\left( \left\{x \in \mathbb{R}\, |\, u>0 \right\} \right)}^{p+2}
 \, \mathrm{d}t 
\leq C_{p}(\phi_0, \partial _x u_0)
 \quad \bigl( t \ge 0 \bigr).
\end{equation*}
}


\medskip

We also prepare the precise properties 
for the nonlinear interaction terms 
of the viscous contact wave $U$ 
and the approximation of rarefaction wave $U^r$. 
Namely, due to Lemma 2.2 and Lemma 2.3, 
we can easily see the fact that 
for any $t\ge 0$ there uniquely exists $x=X(t)\in \mathbb{R}$ such that 
\begin{equation}
\tilde{U}\bigl(t,X(t)\bigr) 
= U\bigl(t,X(t)\bigr)+U^r\bigl(t,X(t)\bigr)= 0
\quad \bigl( t \ge 0 \bigr).
\end{equation}
that is,
\begin{align*}
\begin{aligned}
U^r\bigl(t,X(t)\bigr)
&= -U\bigl(t,X(t)\bigr) \\ 
&= \Lint^{\infty}_{X(t)}
  \frac{1}{(1+t)^{\frac{1}{p+1}}}\, 
  \bigggl( \left( 
  A-B \left(\frac{y}{(1+t)^{\frac{1}{p+1}}} \right)^2 
  \right)\vee 0 \bigggr)^{\frac{1}{p-1}}
  \, \mathrm{d}y \quad \bigl( t \ge 0 \bigr). 
\end{aligned}
\end{align*}
More precisely, we have the following lemma. 

\medskip

\noindent
{\bf Lemma 3.3.}\quad {\it
%The mapping
The function 
$$
X:[\, 0,\infty)\ni t \longmapsto X(t)\in \mathbb{R}
$$
defined by {\rm(3.9)} 
has following asymptotic properties. 
\smallskip

\noindent
{\rm (i)}\ 
There exists a positive time $T_0$ such that 
for some $\delta \in \left(\, 0, \sqrt{\frac{A}{B}} \, \right)$, 
\begin{equation*}
\left(\, \sqrt{\frac{A}{B}} - \delta \, \right)\, (1+t)^{\frac{1}{p+1}} 
< X(t) < \sqrt{\frac{A}{B}}\, (1+t)^{\frac{1}{p+1}}
\qquad \bigl(\, t \ge T_0 \, \bigr).
\end{equation*}
\smallskip

\noindent
{\rm (ii)}\
For any $\epsilon \in (0,1)$, 
there exists a positive constant $C_{p, \epsilon}$ such that 
\begin{equation*}
\left|\, 
(\lambda)^{-1} \left(\frac{X(t)}{1+t} \right) -
\lint^{\infty}_{\frac{\mathlarger{X(t)}}{ \mathlarger{(1+t)^{\frac{1}{p+1}} }}}
                \Bigl ( \left( \, 
                A-B \, \xi^2 \, 
                \right)\vee 0 \Bigr)^{\frac{1}{p-1}}
                \, \mathrm{d}\xi 
\, \right| \le C_{p, \epsilon}(1+t)^{-1 + \epsilon} 
\qquad \bigl(\, t \ge T_0 \, \bigr).
\end{equation*}
\smallskip

\noindent
{\rm (iii)}\ 
There exists a positive constant $C_{p}$ such that 
$$
\left| \, 
\sqrt{\frac{A}{B}} - \frac{X(t)}{ (1+t)^{\frac{1}{p+1}} } 
\, \right|
\leq %\left( \frac{A}{B} \right)^{\frac{1}{2}}
%     \left( {\frac{p}{p-1}} \right)^{\frac{p-1}{p}}
     C_{p} (1+t)^{-{\frac{p-1}{p+1}}}
\qquad \bigl(\, t \ge T_0 \, \bigr).
$$
}

\bigskip 

\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Time-decay estimates with $2 \le q \le \infty$ }
In this section, we show the time-decay estimates with $2 \le q \le \infty$ 
(not assuming $L^1$-integrability to the initial perturbation), 
that is, Theorem 3.1. 
To do that, we shall obtain the time-weighted $L^q$-energy estimates 
to $\phi$ with $2 \le q< \infty$ (cf. \cite{yoshida}). 

\medskip

\noindent
{\bf Proposition 4.1.}\quad {\it
Suppose the same assumptions in Theorem 3.1. 
For any $q \in [ \, 2, \infty)$, 
there exist positive constants $\alpha$ and $C_{\alpha,p,q}$, 
such that the unique global solution in time $\phi$ of the Cauchy problem {\rm(3.6)} 
satisfies the following $L^q$-energy estimate 
 \begin{align}
 \begin{aligned}
 &(1+t)^\alpha  \| \,\phi(t) \,\|_{L^q}^q 
  + \int ^t_0 (1+\tau )^\alpha  G_{q}(\tau) 
    \, \mathrm{d}\tau \\
 & \quad 
  + \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} 
    \bigl( \, \partial _x \phi \, \bigr)^2 \\
 & \qquad \qquad \qquad \qquad \quad \; \, 
    \times 
    \left( \, 
    \bigl| \partial _x \phi \bigr|^{p-1} 
    + \bigl| \partial _x U \bigr|^{p-1} 
    + \bigl| \partial _x U^r \bigr|^{p-1}  
    \, \right) \, \mathrm{d}x \mathrm{d}\tau  \\
 & \quad 
  + \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} \, 
    \left| \, 
    \bigl| \partial _x \phi + \partial _x U + \partial _x U^r \bigr|^{p-1} 
    - \bigl| \partial _x U + \partial _x U^r \bigr|^{p-1}  
    \, \right| \\
 & \qquad \qquad \qquad \qquad \quad \; \, 
   \times 
    \left| \, 
    \bigl( \, \partial _x \phi + \partial _x U + \partial _x U^r \, \bigr)^2 
    - \bigl( \, \partial _x U + \partial _x U^r \, \bigr)^2 \, \right| 
    \, \mathrm{d}x \mathrm{d}\tau  \\
 &\leq C_{\alpha,p,q} \| \,\phi_0 \,\|_{L^q}^q 
       + C\left( \, \alpha, \, p, \, q, \, \phi_0 \, \right) \, 
       (1+t)^{\alpha - \frac{q-2}{3p+1}}
       \quad \bigl(\, t \ge T_0 \, \bigr),
 \end{aligned}
 \end{align}
 where $G_q=G_q(t)$ is explicitly given by 
\begin{align*}
\begin{aligned}
G_q(t)&:=\left( \, \int_{\tilde{U}+\phi \geq 0, \tilde{U} \geq 0} 
        | \, \phi \, |^q \partial _x \tilde{U}\, \mathrm{d}x \, \right) (t) \\
      &\, \quad \;   +\left( \, \int_{\tilde{U}+\phi < 0, \tilde{U} \geq 0}
         | \, \phi \, |^{q-1}\, \tilde{U} \, 
         \partial _x \tilde{U} \, \mathrm{d}x \, \right) (t)
         + \left( \, \int_{\tilde{U}+\phi < 0,\tilde{U} \geq 0}
           {\tilde{U}}^q\, \partial _x \tilde{U} \, \mathrm{d}x \, \right) (t)\\
      &\, \quad \;  +\left( \int_{\tilde{U} + \phi \geq 0, \tilde{U} < 0} 
        \left( \, | \, \phi \, |^{q-1} 
        \bigl(\, q \, \tilde{U} + (q-1)\, | \, \phi \, | \, \bigr) + | \, \tilde{U} \, |^q\right)
        \partial _x \tilde{U}\, \mathrm{d}x \, \right) (t). \\
\end{aligned}
\end{align*}
}

\medskip

The proof of Proposition 4.1 is provided by the
following two lemmas.

\medskip

\noindent
{\bf Lemma 4.1.}\quad {\it
For any $2\leq q < \infty$, 
there exist positive constants $\alpha$ and $C_q$ such that 
\begin{align}
\begin{aligned}
&(1+t)^\alpha  \| \,\phi(t) \,\|_{L^q}^q 
  + q \, (q-1)
    \int ^t_0 (1+\tau )^\alpha  G_{q}(\tau) 
    \, \mathrm{d}\tau \\
 & 
  + C_{q} 
    \int ^t_0 (1+\tau )^\alpha 
    \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} 
    \bigl( \, \partial _x \phi \, \bigr)^2 \\
 & \qquad \qquad \qquad \qquad \quad \quad \quad \; \, 
   \times 
    \left( \, 
    \bigl| \partial _x \phi \bigr|^{p-1} 
    + \bigl| \partial _x U \bigr|^{p-1} 
    + \bigl| \partial _x U^r \bigr|^{p-1} 
    \, \right) \, \mathrm{d}x \mathrm{d}\tau  \\
 & 
  + C_{q} 
    \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} \, 
    \left| \, 
    \bigl| \partial _x \phi + \partial _x \tilde{U} \bigr|^{p-1} 
    - \bigl| \partial _x \tilde{U} \bigr|^{p-1}  
    \, \right| \\
 & \qquad \qquad \qquad \qquad \quad \quad \quad \; \, 
   \times 
    \left| \, 
    \bigl( \, \partial _x \phi + \partial _x \tilde{U} \, \bigr)^2 
    - \bigl( \, \partial _x \tilde{U} \, \bigr)^2 \, \right| 
    \, \mathrm{d}x \mathrm{d}\tau  \\
 &\leq \| \,\phi_0 \,\|_{L^q}^q 
       + \alpha 
       \int ^t_0 (1+\tau )^{\alpha -1} 
       \| \,\phi(\tau ) \,\|_{L^p}^p 
       \, \mathrm{d}\tau \\
%&\qquad \qquad \; \; \,  \\
 &\, \: \quad \quad \quad \quad \, + 
    q \int ^t_0 (1+\tau )^\alpha 
    \| \,\phi(\tau ) \,\|_{L^\infty}^{p-1} 
    \left|\left|\, \widetilde{F_{p} }(U,U^r)(\tau ) \, \right| \right|_{L^1}
    \, \mathrm{d}\tau \\
 &\, \: \quad \quad \quad \quad \, + 
    \mu \, q \int ^t_0 (1+\tau )^\alpha 
    \| \,\phi(\tau ) \,\|_{L^\infty}^{p-2} \\
 &\, \: \quad \quad \quad \quad \quad \, \times    
    \int _{-\infty}^{\infty} 
    \bigl| \, \partial _x \phi \, \bigr|
    \Bigl( \, 
    \bigl( \, \partial_x U + \partial_x U^{r} \, \bigr)^{p} 
    - \left( \, \partial_x U \, \right)^{p} \, 
     \Bigr) 
     \, \mathrm{d}x\mathrm{d}\tau \quad \bigl( t \ge 0 \bigr). 
 \end{aligned}
 \end{align}
}

\bigskip

\noindent
{\bf Lemma 4.2.}\quad {\it
Assume $p>1$ and $2\leq q < \infty$. 
We have the following interpolation inequalities.  

 \noindent
 {\rm (1)}\ \ For any $2\leq r < \infty$, there exists a positive 
 constant $C_{p,q,r}$  such that 
 \begin{align*}
 \begin{aligned}
 &\Vert \, \phi (t) \, \Vert _{L^r }
  \leq 
  C_{p,q,r} \left( \, \int _{-\infty}^{\infty} | \, \phi \, |^2 \, \mathrm{d}x \, \right)
  ^{\frac{pr+p+q-1}{(3p+q-1)r}} \\
 & \qquad \quad \quad \quad \quad \; \: \, 
         \times 
         \left( \, \int _{-\infty}^{\infty} | \, \phi \, |^{q-2} 
         \bigl| \, \partial _x \phi \, \bigr|^{p+1} \, \mathrm{d}x \, \right)
         ^{\frac{r-2}{(3p+q-1)r}} \quad \bigl( t \ge 0 \bigr). 
 \end{aligned}
 \end{align*}

 \noindent
 {\rm (2)}\ \ 
 There exists a positive 
 constant $C_{p,q}$ 
 such that 
 \begin{align*}
 \begin{aligned}
 &\Vert \, \phi (t)  \, \Vert _{L^\infty }
  \leq 
  C_{p,q} \left( \, \int _{-\infty}^{\infty} | \, \phi \, |^2 \, \mathrm{d}x \, \right)
  ^{\frac{p}{3p+q-1}} \\
 & \qquad \quad \quad \quad \quad \; \, 
         \times 
         \left( \, \int _{-\infty}^{\infty} | \, \phi \, |^{q-2} 
         \bigl| \, \partial _x \phi \, \bigr|^{p+1} \, \mathrm{d}x \, \right)
         ^{\frac{1}{3p+q-1}} \quad \bigl( t \ge 0 \bigr). 
 \end{aligned}
 \end{align*}
}

\medskip

In what follows, we first prove Lemma 4.1 
and finally give the proof of Proposition 4.1 
(the proof of Lemma 4.2 is given in \cite{yoshida''}, 
so we omit here). 

\medskip

{\bf Proof of Lemma 4.1}.
Multiplying the equation in (3.6) by 
$\left|\phi \right|^{q-2} \phi$ with $2\leq q < \infty$, 
we obtain the divergence form 
 \begin{align}
 \begin{aligned}
 &\partial_t\left(\frac{1}{q} \left|\, \phi \, \right|^q \right) \\
 &+\partial _x 
   \biggl( \, 
   \left|\, \phi \, \right|^{q-2} \phi \, 
   \left( f(\tilde{U}+\phi)-f(\tilde{U}) \right) 
   \biggr) \\ 
 &+\partial _x \left( 
   -(q-1)\int _0^{\phi} 
   \left( f(\tilde{U}+\eta)-f(\tilde{U}) \right)\left|
   \, \eta \, \right|^{q-2}
   \, \mathrm{d}\eta 
   \, \right) \\
 &+\partial _x \biggl( \, 
   -\mu \, \left|\, \phi \, \right|^{q-2} \phi \, %\partial _x \phi \, 
   \biggr.\\
 & \quad \times \biggl. 
    \Bigl( \, 
    \bigl|\, \partial _x \tilde{U} + \partial _x \phi \, \bigr|^{p-1} 
    \bigl( \, \partial _x \tilde{U} + \partial _x \phi \, \bigr) 
    - \bigl|\, \partial _x \tilde{U} \, \bigr|^{p-1} 
      \bigl( \, \partial _x \tilde{U} \, \bigr) \, 
   \Bigr)
   \, \biggr) \\
 &+(q-1)\int _0^{\phi} 
   \left( \lambda(\tilde{U}+\eta)-\lambda(\tilde{U}) \right)\left| 
   \, \eta \, \right|^{q-2}
   \, \mathrm{d}\eta \, \bigl( \, \partial _x \tilde{U} \, \bigr) \\ 
 &+\mu \, (q-1) \, \left|\, \phi \, \right|^{q-2} \partial _x \phi \, \\
 & \quad \times 
   \Bigl( \, 
   \bigl|\, \partial _x \tilde{U} + \partial _x \phi \, \bigr|^{p-1} 
   \bigl( \, \partial _x \tilde{U} + \partial _x \phi \, \bigr) 
   - \bigl|\, \partial _x \tilde{U} \, \bigr|^{p-1} 
     \bigl( \, \partial _x \tilde{U} \, \bigr) \, 
   \Bigr)\\
 &=\left|\, \phi \, \right|^{q-2} \phi \, F_{p}(U,U^r). 
 \end{aligned}
 \end{align}
Integrating (4.3) with respect to $x$, we have
 \begin{align}
 \begin{aligned}
 &\frac{1}{q} \frac{\mathrm{d}}{\mathrm{d}t} 
 \,\Vert \, \phi(t) \, \Vert_{L^q}^q \\
 &+\int ^{\infty }_{-\infty } (q-1) \int _0^{\phi} 
 \left( \lambda(\tilde{U}+\eta)-\lambda(\tilde{U}) \right)
 \left| \, \eta \, \right|^{q-2}
 \, \mathrm{d}\eta \, \bigl( \, \partial _x \tilde{U} \, \bigr) 
 \, \mathrm{d}x \\
 &+\mu \, (q-1) \, \int ^{\infty }_{-\infty }
   \left|\, \phi \, \right|^{q-2} \partial _x \phi \, \\
 & \quad \times 
   \Bigl( \, 
   \bigl|\, \partial _x \tilde{U} + \partial _x \phi \, \bigr|^{p-1} 
   \bigl( \, \partial _x \tilde{U} + \partial _x \phi \, \bigr) 
   - \bigl|\, \partial _x \tilde{U} \, \bigr|^{p-1} 
     \bigl( \, \partial _x \tilde{U} \, \bigr) \, 
   \Bigr) 
   \, \mathrm{d}x \\
% &+\frac{4\, \mu \, (p-1)}{p^2} 
% \, \left|\left| \,
% \partial _x \bigl( | \, \phi \, |^{\frac{p}{2}} \bigr)(t,\cdot \, ) 
% \,\right|\right|_{L^2}^2 
 &=\int ^{\infty}_{-\infty}\left|\, \phi \, \right|^{q-2} \phi 
  \, F_{p}(U,U^r) \, \mathrm{d}x. 
 \end{aligned}
 \end{align} 
In order to estimate the second term on the left-hand side of 
(4.4), 
noting the shape of the flux function $f$, 
we divide the integral region of $x$ depending on the signs of
$\tilde U +\phi$, $\tilde U$ and $\phi$ as 
\begin{align*}
\begin{aligned}
&\int ^{\infty }_{-\infty } \biggl(\,  \int _0^{\phi} 
\left( \lambda \bigl(\tilde{U}+\eta \bigr)-\lambda \bigl(\tilde{U} \bigr) \right) 
\left| \, \eta \, \right|^{q-2}
\, \mathrm{d}\eta \biggr)
\, \bigl( \, \partial _x \tilde{U} \, \bigr) \, \mathrm{d}x 
\\
\qquad &= \int_{\tilde{U}+\phi \geq 0, \tilde{U} \geq 0, \phi \geq 0}
          +\int_{\tilde{U}+\phi \geq 0, \tilde{U} \geq 0, \phi \le 0}
          +\int_{\tilde{U}+\phi \geq 0, \tilde{U} < 0}
          + \int_{\tilde{U}+\phi < 0, \tilde{U} \geq 0}
\end{aligned}
\end{align*}
where we used the fact that the integral is clearly zero on the 
domain $\tilde{U}+\phi \le 0$ and $\tilde{U}\le 0$.
By Lagrange's mean-value theorem, we easily get as 
\begin{align*}
\begin{aligned}
&\left( \int ^{\infty }_{-\infty } \biggl(\,  \int _0^{\phi} 
\left( \lambda \bigl(\tilde{U}+\eta \bigr)-\lambda \bigl(\tilde{U} \bigr) \right) 
\left| \, \eta \, \right|^{q-2}
\, \mathrm{d}\eta \biggr)
\, \bigl( \, \partial _x \tilde{U} \, \bigr) \, \mathrm{d}x \right) (t)
\sim  G_{q}(t) 
\end{aligned}
\end{align*}
where $G_{q}=G_{q}(t)$ is defined in Proposition 4.1 
(cf. \cite{matsumura-yoshida}, \cite{yoshida}, \cite{yoshida'}). 
Next, by using the uniform boundedness, Lemma 3.1, 
and the following absolute equality 
with $p>1$, 
for any $a,b \in \mathbb{R}$, 
\begin{align*}
\begin{aligned}
&\left(\, |\, a\, |^{p-1}a - |\, b\, |^{p-1}b \, \right) 
 \left(\, a-b \, \right) \\
&= \frac{1}{2} \, 
   \left(\, |\, a\, |^{p-1} + |\, b\, |^{p-1} \, \right) 
   \left(\, a-b \, \right)^2 
   + \frac{1}{2} \, 
     \left(\, |\, a\, |^{p-1} - |\, b\, |^{p-1} \, \right) 
     \left(\, a^2-b^2 \, \right)\\
& \geq  \frac{1}{4} \, 
   \left(\, |\, a\, |^{p-1} + |\, b\, |^{p-1} + |\, a-b\, |^{p-1} \, \right) 
   \left(\, a-b \, \right)^2 
   + \frac{1}{2} \, 
     \left(\, |\, a\, |^{p-1} - |\, b\, |^{p-1} \, \right) 
     \left(\, a^2-b^2 \, \right),
\end{aligned}
\end{align*}
we have 
 \begin{align}
 \begin{aligned}
 &\frac{1}{q} \frac{\mathrm{d}}{\mathrm{d}t} 
 \,\Vert \, \phi(t) \, \Vert_{L^q}^q 
  +C_{p,q}^{-1} \, G_{q}(t) \\
 &+\frac{\mu \, (q-1)}{4} \, 
   \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} 
    \bigl( \, \partial _x \phi \, \bigr)^2 
    \left( \, 
    \bigl| \partial _x \phi \bigr|^{p-1} 
 + \bigl| \partial _x U \bigr|^{p-1} + \bigl| \partial _x U^r \bigr|^{p-1} \, 
   \right) \, \mathrm{d}x \\
 &+\frac{\mu \, (q-1)}{2} \, 
   \int ^{\infty }_{-\infty }
   | \, \phi \, |^{q-2} \, 
    \left| \, 
    \bigl| \partial _x \phi + \partial _x U^r \bigr|^{p-1} 
    - \bigl| \partial _x U^r \bigr|^{p-1}  
    \, \right| \\
 & \qquad \qquad \qquad \qquad \quad \; \; \, \, 
   \times 
    \left| \, 
    \bigl( \, \partial _x \phi + \partial _x U^r \, \bigr)^2 
    - \bigl( \, \partial _x U^r \, \bigr)^2 \, \right| 
    \, \mathrm{d}x \\
 &\leq \left| \, \int ^{\infty}_{-\infty}\left|\, \phi \, \right|^{q-2} \phi 
       \, F_{p}(U,U^r) \, \mathrm{d}x \, \right|. 
 \end{aligned}
 \end{align}
We note the right-hand side of (4.5) can be estimated as 
 \begin{align}
 \begin{aligned}
 &\left| \, 
  \int ^{\infty}_{-\infty} 
  \left|\, \phi \, \right|^{q-2} \phi 
  \, F_{p}(U,U^r) \, \mathrm{d}x \, 
  \right| 
 \leq \left| \, 
      \int ^{\infty}_{-\infty} 
      \left|\, \phi \, \right|^{q-1} \, 
      \widetilde{F_{p} }(U,U^r) 
      \, \mathrm{d}x \, 
      \right| \\
 & \qquad \; 
   +\int _{-\infty}^{\infty} 
    \left|\, \phi \, \right|^{q-2}
    \bigl| \, \partial _x \phi \, \bigr|
    \Bigl( \, 
    \bigl( \, \partial_x U + \partial_x U^{r} \, \bigr)^{p} 
    - \left( \, \partial_x U \, \right)^{p} \, 
      \Bigr) 
      \, \mathrm{d}x.
 \end{aligned}
 \end{align}
Thus, substituting (4.6) into (4.5), multiplying the inequality by 
$(1+t)^{\alpha}$ with $\alpha>0$ 
and integrating over $(0,t)$ with respect to the time, 
we complete the proof of Lemma 4.1.
%
%\medskip
%
%{\bf Proof of Lemma 4.2}.\ 
%Noting that $\phi (t, \cdot \: ) \in L^2$ 
%and $\partial_x \phi (t, \cdot \: ) \in L^{p+1}$ 
%imply 
%$\displaystyle{\lim _{x\rightarrow \pm \infty}\phi(t,x)}=0$ 
%for $t \ge 0$, 
%we have 
%\begin{align}
%\begin{aligned}
% | \, \phi \, |^{s} 
% &\leq s \int _{-\infty}^{\infty} 
%      | \, \phi \, |^{s-1}  \, 
%      \bigl| \, \partial _x \phi \, \bigr|
%      \, \mathrm{d}x\\
%\end{aligned}
%\end{align}
%By the Cauchy-Schwarz inequality, we have 
%\begin{align}
%\begin{aligned}
% | \, \phi \, |^{s} 
% &\leq s \left( \, 
%       \int _{-\infty}^{\infty} 
%       \left( \, | \, \phi \, |^{s-1-\frac{q-2}{p+1}} \, \right)^{\frac{p+1}{p}}
%       \, \mathrm{d}x
%       \right)^{\frac{p}{p+1}} \\
% & \qquad \times 
%       \left( \, 
%       \int _{-\infty}^{\infty} 
%       | \, \phi \, |^{q-2} 
%       \bigl| \, \partial _x \phi \, \bigr|^{p+1}
%       \, \mathrm{d}x
%       \right)^{\frac{1}{p+1}} 
%\end{aligned}
%\end{align}
%Taking $s=\frac{3p+q-1}{p+1}$, we get 
%\begin{align}
%\begin{aligned}
% \Vert \, \phi \, \Vert _{L^\infty }^{\frac{3p+q-1}{p+1}}
% &\le \frac{3\, p+q-1}{p+1} 
%     \left( \, 
%     \int _{-\infty}^{\infty} 
%     | \, \phi \, |^{2} 
%     \, \mathrm{d}x
%     \, \right)^{\frac{p}{p+1}} \\
% & \qquad \times 
%     \left( \, 
%     \int _{-\infty}^{\infty} 
%     | \, \phi \, |^{q-2} 
%     \bigl| \, \partial _x \phi \, \bigr|^{p+1}
%     \, \mathrm{d}x
%     \right)^{\frac{1}{p+1}},
%\end{aligned}
%\end{align}
%and 
%\begin{align}
%\begin{aligned}
% \Vert \, \phi \, \Vert _{L^r }^r
% &\le \Vert \, \phi \, \Vert _{L^\infty }^{r-2}
%      \Vert \, \phi \, \Vert _{L^2 }^2\\
% &\le \left( \frac{3\, p+q-1}{p+1} \right)^{\frac{(p+1)(r-2)}{3p+q-1}}
%      \left( \, 
%     \int _{-\infty}^{\infty} 
%     | \, \phi \, |^{2} 
%     \, \mathrm{d}x
%     \, \right)^{\frac{pr+p+q-1}{3p+q-1}} \\
% & \qquad \times 
%     \left( \, 
%     \int _{-\infty}^{\infty} 
%     | \, \phi \, |^{q-2} 
%     \bigl| \, \partial _x \phi \, \bigr|^{p+1}
%     \, \mathrm{d}x
%     \right)^{\frac{r-2}{3p+q-1}}.
%\end{aligned}
%\end{align}
%Thus, we complete the proof of Lemma 4.2. 

\medskip

{\bf Proof of Proposition 4.1}.\
By using Lemma 4.1 and Lemma 4.2, 
we shall estimate the second term, the third term and the fourth term 
on the right-hand side of (4.2) as follows: 
for any $\epsilon>0$, 
\begin{align}
\begin{aligned}
 &\alpha \int ^t_0 (1+\tau )^{\alpha -1} 
  \| \,\phi(\tau) \,\|_{L^q}^q 
  \, \mathrm{d}\tau \\
 &\le C_{\alpha ,p,q} \int ^t_0 (1+\tau )^{\alpha -1} 
      \left( \, 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{q-2} 
      \bigl| \, \partial _x \phi \, \bigr|^{p+1}
      \, \mathrm{d}x
      \right)^{\frac{q-2}{3p+q-1}} \\
 &\qquad \qquad \qquad \qquad \qquad \quad \times 
      \left( \, 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{2} 
      \, \mathrm{d}x
      \, \right)^{\frac{pq+p+q-1}{3p+q-1}}
      \, \mathrm{d}\tau \\
 &\le \int ^t_0 
      \left( \, 
      (1+\tau )^{\alpha} 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{q-2} 
      \bigl| \, \partial _x \phi \, \bigr|^{p+1}
      \, \mathrm{d}x
      \, \right)^{\frac{q-2}{3p+q-1}} \\
 &\qquad \quad \quad \quad \quad \times 
      C_{\alpha ,p,q}\,  (1+\tau )^{\alpha - 1 -\frac{\alpha (q-2)}{3p+q-1}}
         \| \,\phi(\tau) \,\|_{L^2}^{\frac{2(pq+p+q-1)}{3p+q-1}}
         \, \mathrm{d}\tau \\
 &\le \epsilon \int ^t_0 (1+\tau )^{\alpha} 
      \left( \, 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{q-2} 
      \bigl| \, \partial _x \phi \, \bigr|^{p+1}
      \, \mathrm{d}x
      \right) \, \mathrm{d}\tau \\
 &\quad \quad \quad \quad \quad 
         +C_{\alpha ,p,q}(\epsilon ) 
         \int ^t_0 (1+\tau )^{\alpha - {\frac{3p+q-1}{3p+1}}}
         \| \,\phi(\tau ) \,\|_{L^2}^{\frac{2(pq+p+q-1)}{3p+1}} 
         \, \mathrm{d}\tau, 
\end{aligned}
\end{align}
 \begin{align}
 \begin{aligned}
 &q \int ^t_0 (1+\tau )^{\alpha} 
  \| \,\phi(\tau ) \,\|_{L^\infty}^{q-1} 
  \left|\left| \, \widetilde{F_{p} }(U,U^r)(\tau ) \, \right| \right|_{L^1}
  \, \mathrm{d}\tau \\
 &\le C_{p,q} \int ^t_0 (1+\tau )^{\alpha} 
      \left( \, 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{q-1} 
      \bigl| \, \partial _x \phi \, \bigr|^{p+1}
      \, \mathrm{d}x
      \right)^{\frac{q-1}{3p+q-1}} \\
 &\qquad \quad \quad \quad \quad \times 
      \left( \, 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{2} 
      \, \mathrm{d}x
      \, \right)^{\frac{p(q-1)}{3p+q-1}}
      \left|\left| \, \widetilde{F_{p} }(U,U^r)(\tau ) \, \right| \right|_{L^1}
      \, \mathrm{d}\tau \\
 &\le \int ^t_0 
      \left( \, 
      (1+\tau )^{\alpha} 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{q-2} 
      \bigl| \, \partial _x \phi \, \bigr|^{p+1}
      \, \mathrm{d}x
      \, \right)^{\frac{q-2}{3p+q-1}} \\
 &\quad \: \: \: \times 
         C_{p,q}\,  (1+\tau )^{\alpha -\frac{\alpha (q-1)}{3p+q-1}}
         \| \,\phi(\tau) \,\|_{L^2}^{\frac{2p(q-1)}{3p+q-1}}
         \left|\left| \, \widetilde{F_{p} }(U,U^r)(\tau ) \, \right| \right|_{L^1}
         \, \mathrm{d}\tau \\
 &\le \epsilon \int ^t_0 (1+\tau )^{\alpha} 
      \left( \, 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{q-1} 
      \bigl| \, \partial _x \phi \, \bigr|^{p+1}
      \, \mathrm{d}x
      \right) 
      \, \mathrm{d}\tau \\
 &\quad 
         + C_{p,q}(\epsilon) \int ^t_0 
          (1+\tau )^{\alpha} \| \,\phi(\tau ) \,\|
          _{L^2}^{{\frac{2}{3}}(q-1)}
          \left|\left| \, \widetilde{F_{p} }(U,U^r)(\tau ) \, \right| \right|_{L^1}
          ^{\frac{3p+q-1}{3p}} 
          \, \mathrm{d}\tau, 
 \end{aligned}
 \end{align}
  \begin{align}
 \begin{aligned}
 & \mu \, q \int ^t_0 (1+\tau )^\alpha 
   \| \,\phi(\tau ) \,\|_{L^\infty}^{p-2} \\
 & \quad \quad \, \times    
    \int _{-\infty}^{\infty} 
    \bigl| \, \partial _x \phi \, \bigr|
    \Bigl( \, 
    \bigl( \, \partial_x U + \partial_x U^{r} \, \bigr)^{p} 
    - \left( \, \partial_x U \, \right)^{p} \, 
     \Bigr) 
     \, \mathrm{d}x\mathrm{d}\tau \\
 &\le C_{p,q} \int ^t_0 (1+\tau )^{\alpha} 
      \left( \, 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{q-1} 
      \bigl| \, \partial _x \phi \, \bigr|^{p+1}
      \, \mathrm{d}x
      \right)^{\frac{q-2}{3p+q-1}} \\
 &\quad \quad \, \times 
      \left( \, 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{2} 
      \, \mathrm{d}x
      \, \right)^{\frac{p(q-2)}{3p+q-1}} \\
 &\quad \quad \, \times 
      \left( \, 
      \int _{-\infty}^{\infty} 
      \Bigl( \, 
      \bigl( \, \partial_x U + \partial_x U^{r} \, \bigr)^{p} 
      - \left( \, \partial_x U \, \right)^{p} \, 
      \Bigr)^{\frac{p+1}{p}} 
      \, \mathrm{d}x 
      \, \right)
      \mathrm{d}\tau \\
 &\le \int ^t_0 
      \left( \, 
      (1+\tau )^{\alpha} 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{q-2} 
      \bigl| \, \partial _x \phi \, \bigr|^{p+1}
      \, \mathrm{d}x
      \, \right)^{\frac{q-2}{3p+q-1}} \\
 &\quad \: \: \: \times 
         C_{p,q}\,  (1+\tau )^{\alpha -\frac{\alpha (q-2)}{3p+q-1}}
         \| \,\phi(\tau) \,\|_{L^2}^{\frac{2p(q-2)}{3p+q-1}} \\
 &\quad \: \: \: \times 
         \left( \, 
      \int _{-\infty}^{\infty} 
      \Bigl( \, 
      \bigl( \, \partial_x U + \partial_x U^{r} \, \bigr)^{p} 
      - \left( \, \partial_x U \, \right)^{p} \, 
      \Bigr)^{\frac{p+1}{p}} 
      \, \mathrm{d}x 
      \, \right)
         \, \mathrm{d}\tau \\
 &\le \epsilon \int ^t_0 (1+\tau )^{\alpha} 
      \left( \, 
      \int _{-\infty}^{\infty} 
      | \, \phi \, |^{q-1} 
      \bigl| \, \partial _x \phi \, \bigr|^{p+1}
      \, \mathrm{d}x
      \right) 
      \, \mathrm{d}\tau \\
 &\quad \, 
         + C_{p,q}(\epsilon) \int ^t_0 
          (1+\tau )^{\alpha} \| \,\phi(\tau ) \,\|
          _{L^2}^{\frac{2p(q-2)}{3p+1}}\\
 &\quad \, \, \: \: \: \times 
      \left( \, 
      \int _{-\infty}^{\infty} 
      \Bigl( \, 
      \bigl( \, \partial_x U + \partial_x U^{r} \, \bigr)^{p} 
      - \left( \, \partial_x U \, \right)^{p} \, 
      \Bigr)^{\frac{p+1}{p}} 
      \, \mathrm{d}x 
      \, \right)^{\frac{3p+q-1}{3p+1}}
          \, \mathrm{d}\tau. 
 \end{aligned}
 \end{align}
Substituting (4.7), (4.8) and (4.9) into (4.2), 
we have 
\begin{align}
\begin{aligned}
 &(1+t)^\alpha  \| \,\phi(t) \,\|_{L^q}^q 
  + \int ^t_0 (1+\tau )^\alpha  G_q(\tau ) \, \mathrm{d}\tau \\
 & \quad 
   + \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} 
    \bigl( \, \partial _x \phi \, \bigr)^2 \\
 & \qquad \qquad \qquad \qquad \quad \quad \; \, \, 
    \times 
    \left( 
    \bigl| \partial _x \phi \bigr|^{p-1} 
 + \bigl| \partial _x U \bigr|^{p-1} + \bigl| \partial _x U^r \bigr|^{p-1}  
   \right) \, \mathrm{d}x \mathrm{d}\tau  \\
 & \quad 
  + \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} \, 
    \left| \, 
    \bigl| \partial _x \phi + \partial _x \tilde{U} \bigr|^{p-1} 
    - \bigl| \partial _x \tilde{U} \bigr|^{p-1}  
    \, \right| \\
 & \qquad \qquad \qquad \quad \quad \quad \; \, \, 
   \times 
    \left| \, 
    \bigl( \, \partial _x \phi + \partial _x \tilde{U} \, \bigr)^2 
    - \bigl( \, \partial _x \tilde{U} \, \bigr)^2 \, \right| 
    \, \mathrm{d}x \mathrm{d}\tau  \\
&\leq C_{\alpha,p,q} \| \,\phi_0 \,\|_{L^p}^p 
 + C_{\alpha ,p,q}(\epsilon ) 
         \int ^t_0 (1+\tau )^{\alpha - {\frac{3p+q-1}{3p+1}}}
         \| \,\phi(\tau ) \,\|_{L^2}^{\frac{2(pq+p+q-1)}{3p+1}} 
         \, \mathrm{d}\tau \\
 &\qquad \quad \; \; \: \,  
 + C_{p,q}(\epsilon) \int ^t_0 
          (1+\tau )^{\alpha} \| \,\phi(\tau ) \,\|
          _{L^2}^{{\frac{2}{3}}(q-1)}
          \left|\left| \, \widetilde{F_{p} }(U,U^r)(\tau ) \, \right| \right|_{L^1}
          ^{\frac{3p+q-1}{3p}} 
          \, \mathrm{d}\tau \\
 &\qquad \quad \; \; \: \,  
 + C_{p,q}(\epsilon) \int ^t_0 
          (1+\tau )^{\alpha} \| \,\phi(\tau ) \,\|
          _{L^2}^{\frac{2p(q-2)}{3p+1}}\\ 
 &\qquad \quad \quad \, \, \: \: \: \; \; \; \; \; \times 
      \left( \, 
      \int _{-\infty}^{\infty} 
      \Bigl( \, 
      \bigl( \, \partial_x U + \partial_x U^{r} \, \bigr)^{p} 
      - \left( \, \partial_x U \, \right)^{p} \, 
      \Bigr)^{\frac{p+1}{p}} 
      \, \mathrm{d}x 
      \, \right)^{\frac{3p+q-1}{3p+1}}
          \, \mathrm{d}\tau. 
 \end{aligned}
 \end{align}
By using the $L^2$-boundedness of $\phi$, Lemma 3.2, 
we first get 
 \begin{align}
 \begin{aligned}
 \| \, \phi (t)\, \| _{L^2}^2 
 \leq C_{p}(\phi_0). 
 \end{aligned}
 \end{align}
By using Lemma 2.2, Lemma 2.3 and Lemma 3.3, 
we also get 

\medskip

\noindent
{\bf Lemma 4.3.}\quad {\it
For any fixed $p \in (1, \infty)$, 
we have the following time-decay estimates. 

\noindent
{\rm (1)}\ \ For any $\delta \in (0, 1)$, there exists positive 
constants $C_{p}$, $C_{\delta }$ and $T_{0}$ such that 
\begin{align*}
\begin{aligned}
\left|\left| \, \widetilde{F_{p} }(U,U^r)(t) \, \right| \right|_{L^1} 
\leq C_{p} \left( \, (1+t)^{-{\frac{2p}{p+1}}} 
     + C_{\delta }(1+t)^{-2(1-\delta )} \, \right) 
     \quad \bigl( \, t \ge T_{0} \, \bigr).
\end{aligned}
\end{align*}

\noindent
{\rm (2)}\ \ There exists a positive constant $C_{p}$ 
such that 
\begin{align*}
\begin{aligned}
\left( \, 
\int _{-\infty}^{\infty} 
      \Bigl( \, 
      \bigl( \, \partial_x U + \partial_x U^{r} \, \bigr)^{p} 
      - \left( \, \partial_x U \, \right)^{p} \, 
      \Bigr)^{\frac{p+1}{p}} 
      \, \mathrm{d}x 
\, \right) (t)
\leq C_{p}(1+t)^{-1}
      \quad \bigl( t \ge 0 \bigr).
\end{aligned}
\end{align*}
}

\medskip

\noindent
We estimate the each terms on the right-hand side of (4.10) 
as follows: 
\begin{align}
\begin{aligned}
&C_{\alpha ,p,q} \, 
         \int ^t_0 (1+\tau )^{\alpha - {\frac{3p+q-1}{3p+1}}} 
         \| \,\phi(\tau ) \,\|_{L^2}^{\frac{2(pq+p+q-1)}{3p+1}} 
         \, \mathrm{d}\tau \\
&\leq C_{\alpha ,p,q} 
      \bigl( \, C_{p}(\phi_0) \, \bigr)^{\frac{pq+p+q-1}{3p+1}} 
      \int ^t_0 (1+\tau )^{\alpha - {\frac{3p+q-1}{3p+1}}} 
      \, \mathrm{d}\tau \\
&\leq C_{\alpha ,p,q} 
      \bigl( \, C_{p}(\phi_0) \, \bigr)^{\frac{pq+p+q-1}{3p+1}} 
      (1+t )^{\alpha - {\frac{q-2}{3p+1}}}, 
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
&C_{p,q} \int ^t_0 
          (1+\tau )^{\alpha} \| \,\phi(\tau ) \,\|
          _{L^2}^{{\frac{2}{3}}(q-1)}
          \left|\left| \, \widetilde{F_{p} }(U,U^r)(\tau ) \, \right| \right|_{L^1}
          ^{\frac{3p+q-1}{3p}} 
          \, \mathrm{d}\tau \\
&\leq C_{p,q} 
      \bigl( \, C_{p}(\phi_0) \, \bigr)^{{\frac{1}{3}}(q-1)} 
      \int ^t_0 (1+\tau )^{\alpha - {\frac{2(3p+q-1)}{3(p+1)}}} 
      \, \mathrm{d}\tau \\
&\leq C_{p,q} 
      \bigl( \, C_{p}(\phi_0) \, \bigr)^{{\frac{1}{3}}(q-1)} 
      (1+t )^{\alpha - {\frac{3p+2q-5}{3(p+1)}}}, 
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
&C_{p,q} \int ^t_0 
          (1+\tau )^{\alpha} \| \,\phi(\tau ) \,\|
          _{L^2}^{\frac{2p(q-2)}{3p+1}}\\ 
&\quad \quad \, \, \: \: \: \; \; \; \; \; \times 
      \left( \, 
      \int _{-\infty}^{\infty} 
      \Bigl( \, 
      \bigl( \, \partial_x U + \partial_x U^{r} \, \bigr)^{p} 
      - \left( \, \partial_x U \, \right)^{p} \, 
      \Bigr)^{\frac{p+1}{p}} 
      \, \mathrm{d}x 
      \, \right)^{\frac{3p+q-1}{3p+1}} (\tau )
      \, \mathrm{d}\tau \\
&\leq C_{p,q} 
      \bigl( \, C_{p}(\phi_0) \, \bigr)^{{\frac{p(q-2)}{3p+1}}} 
      \int ^t_0 (1+\tau )^{\alpha - {\frac{3p+q-1}{3p+1}}} 
      \, \mathrm{d}\tau \\
&\leq C_{p,q} 
      \bigl( \, C_{p}(\phi_0) \, \bigr)^{{\frac{p(q-2)}{3p+1}}} 
      (1+t )^{\alpha - {\frac{q-2}{3p+1}}}. 
\end{aligned}
\end{align}
Substituting (4.12), (4.13) and (4.14) into (4.10), 
we get (4.1). 
Thus the proof of Proposition 4.1 is complete. 
In particular, it follows that 
\begin{align}
\begin{aligned}
\| \,\phi(t) \,\|_{L^q} 
\leq C( \, p, \,q, \, \phi_0 \, ) \, 
     (1+t)^{-{\frac{1}{3p+1}} \left( 1 - {\frac{2}{q}} \right)}
\end{aligned}
\end{align}
for $2 \leq q < \infty$. 

\medskip

{\bf Proof of Theorem 3.1.}\ 
We already have proved the decay estimate 
of $\| \,\phi(t) \,\|_{L^q}$ with $2 \le q < \infty$. 
Therefore we only 
show the $L^{\infty}$-estimate. 
We first note by Lemma 2.2 and Lemma 2.3 that 
\begin{align}                
\begin{aligned}
&\bigl|\bigl| 
 \, \partial _x \phi (t)\, 
 \bigr|\bigr| _{L^{p+1}}^{p+1}\\
&\leq \bigl|\bigl| 
      \, \partial _x u (t)\, 
      \bigr|\bigr| _{L^{p+1}}^{p+1} 
      + \bigl|\bigl| 
        \, \partial _x U^r (t)\, 
        \bigr|\bigr| _{L^{p+1}}^{p+1} \\
&\leq C\bigl( \, \| \, \phi_0 \, \|_{L^2}, \: 
       \| \, \partial _x u_0 \, \|_{L^{p+1}} 
       \, \bigr) 
       + C_{p}(1+t)^{-1}. 
\end{aligned}
\end{align}
We use the following Gagliardo-Nirenberg inequality: 
\begin{align}
\| \,\phi(t) \,\|_{L^\infty }
\le C_{q,\theta}
\| \,\phi(t) \,\|_{L^q}^{1-\theta}
\| \,\partial _x \phi(t) \,\|_{L^{p+1}}^{\theta}
\end{align}
for any 
$(q,\theta)\in [\, 1,\infty)\times (0,1\, ]$ satisfying 
$$
%\, ;\, 
\frac{p}{p+1} \, {\theta}=(1-\theta) \, \frac{1}{q}. 
$$
Substituting (4.15) and (4.16) into (4.17), 
we have 
\begin{align}
\begin{aligned}
\| \,\phi(t) \,\|_{L^\infty }
&\le C( \, p, \,q, \, \theta, \, \phi_0, \, \partial _x u_0 \, ) \, 
     (1+t)^{-{\frac{1}{3p+1}} \left( 1 - {\frac{2}{q}} \right)(1-\theta)}\\
&\le C( \, p, \, \theta, \, \phi_0, \, \partial _x u_0 \, ) \, 
     (1+t)^{-{\frac{1}{3p+1}} + {\frac{\theta}{p+1}}}
\end{aligned}
\end{align}
for $\theta \in (0,1\, ]$. 
Consequently, we do complete the proof of Theorem 3.1.
\bigskip 

\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Time-decay estimates with $1 \le q \le \infty$ }
In this section, 
we show the time-decay estimates 
with $1 \le q \le \infty$ and 
time-decay estimate for the higher order derivative 
in the $L^{p+1}$-norm, 
in the case where $\phi_0 \in L^1 \cap L^2$ 
with $\partial _x u_0 \in L^{p+1}$, 
that is, Theorem 3.2. 
Then, we 
first establish the $L^1$-estimate 
to the solution $\phi$ of the reformulated Cauchy problem (3.6). 
To do that, we use the Friedrichs mollifier $\rho_\delta \ast $, 
where, 
$\rho_\delta(\phi):=\frac{1}{\delta}\rho \left( \frac{\phi}{\delta}\right)$
with 
\begin{align*}
\begin{aligned}
&\rho \in C^{\infty}_0(\mathbb{R}), \quad
\rho (\phi)\geq 0 \quad (\phi \in \mathbb{R}),
\\
&\mathrm{supp} \{\rho \} \subset 
\left\{\phi \in \mathbb{R}\: \left|\:  |\,\phi \, |\le 1 \right. \right\},\quad  
\int ^{\infty}_{-\infty} \rho (\phi)\, \mathrm{d}\phi=1.
\end{aligned}
\end{align*}

%We record some useful properties as 
Some useful properties of the mollifier are as follows. 
\medskip

\noindent
{\bf Lemma 5.1.}\quad{\it
%For each $\phi \in \mathbb{R}$, we have the following:

\noindent
{\rm (i)}\ 
$\displaystyle{\lim_{\delta \to 0}\, 
\left( \rho_\delta \ast \mathrm{sgn} \right)(\phi)}
= \mathrm{sgn} (\phi)
\qquad (\phi \in \mathbb{R}),$

\noindent
{\rm (ii)}\ 
$\displaystyle{\lim_{\delta \to 0}\, 
\int^{\phi}_0\left( \rho_\delta \ast \mathrm{sgn} \right)(\eta)
\, \mathrm{d}\eta}
=|\, \phi \, |
\qquad (\phi \in \mathbb{R}),$

\medskip

\noindent
{\rm (iii)}\ 
$\Big. \left( \rho_\delta \ast \mathrm{sgn} \right) \Bigr|_{\phi=0} =0,$

\medskip

\noindent
{\rm (iv)}\ 
$\displaystyle{ \frac{\mathrm{d}}{\mathrm{d}\phi}\, 
\left( \rho_\delta \ast \mathrm{sgn} \right)(\phi)}
=2\, \rho_\delta(\phi)
\ge 0
\qquad (\phi \in \mathbb{R}),$

\medskip

\noindent
where 
$$
\left( \rho_\delta \ast \mathrm{sgn} \right)(\phi)
:=\int^{\infty}_{-\infty}
\rho_\delta(\phi-y)\, \mathrm{sgn}(y)\, \mathrm{d}y
\qquad (\phi \in \mathbb{R})
$$
and 
\begin{equation*}
\mathrm{sgn}(\phi):=\left\{
\begin{array}{ll}
-1& \quad \; \bigl(\, \phi < 0 \, \bigr),\\[3pt]
\, \, 0& \quad \; \bigl(\, \phi = 0 \, \bigr),\\[3pt]
\, \, 1& \quad \; \bigl(\, \phi > 0 \, \bigr).
\end{array}
\right.
\end{equation*} 
}

\noindent
Making use of Lemma 5.1, 
we obtain the following $L^1$-estimate. 

\medskip

\noindent
{\bf Proposition 5.1.}\quad {\it
Assume that the same assumptions in Theorem 3.2. 
For any $p>1$ and any $\epsilon>0$, 
there exist positive constants $C_{p}$ and $C_{\epsilon}$ 
such that 
the unique global solution in time $\phi$ 
of the Cauchy problem {\rm(3.6)} 
satisfies the following $L^1$-estimate 
\begin{align}
\begin{aligned}
\| \, \phi(t) \, \|_{L^1}
\le \| \,\phi_0 \,\|_{L^1} + C_{p}(1+t)^{-\frac{2p}{p+1}} 
    + C_{\epsilon}(1+t)^{-2(1-\epsilon)} 
\quad \bigl( \, t \ge T_0 \, \bigr) 
\end{aligned}
\end{align}
for any $\epsilon>0$.
}

\medskip

\noindent
{\bf Proof of Proposition 5.1.}\quad
Multiplying the equation in the problem (3.6) by 
$\left( \rho_\delta \ast \mathrm{sgn} \right)(\phi)$, 
we obtain the divergence form 
\medskip
\begin{align}
\begin{aligned}
&\partial_t\left( 
\int^{\phi}_0 
\left( \rho_\delta \ast \mathrm{sgn} \right)(\eta)
\, \mathrm{d}\eta 
\right) \\
&+\partial _x 
  \biggl( \, 
  \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
  \bigl( f(\tilde{U}+\phi)-f(\tilde{U}) \bigr) 
  \biggr) \\ 
&+\partial _x \left( 
  -\int _0^{\phi} 
  \bigl( f(\tilde{U}+\eta)-f(\tilde{U}) \bigr)
  \frac{\mathrm{d}\left( \rho_\delta \ast \mathrm{sgn} \right)}
  {\mathrm{d}\phi}
  (\eta)
  \, \mathrm{d}\eta 
  \, \right) \\
&+\partial _x \biggl( \, 
  -\mu \, \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \biggr. \\
& \quad \times \biggl. 
   \Bigl( \, 
    \bigl|\, \partial _x \tilde{U} + \partial _x \phi \, \bigr|^{p-1} 
    \bigl( \, \partial _x \tilde{U} + \partial _x \phi \, \bigr) 
    - \bigl|\, \partial _x \tilde{U} \, \bigr|^{p-1} 
      \bigl( \, \partial _x \tilde{U} \, \bigr) \, 
   \Bigr)
  \, \biggr) \\
&+\int _0^{\phi} 
  \bigl( \lambda(\tilde{U}+\eta)-\lambda(\tilde{U}) \bigr) 
  \frac{\mathrm{d}\left( \rho_\delta \ast \mathrm{sgn} \right)}
  {\mathrm{d}\phi}(\eta) 
  \, \mathrm{d}\eta \, \bigl( \, \partial _x \tilde{U} \, \bigr) \\ 
&+\mu \, 
  \frac{\mathrm{d}\left( \rho_\delta \ast \mathrm{sgn} \right)}
  {\mathrm{d}\phi}(\phi) \, 
  \partial _x \phi \\
& \quad \times 
  \Bigl( \, 
    \bigl|\, \partial _x \tilde{U} + \partial _x \phi \, \bigr|^{p-1} 
    \bigl( \, \partial _x \tilde{U} + \partial _x \phi \, \bigr) 
    - \bigl|\, \partial _x \tilde{U} \, \bigr|^{p-1} 
      \bigl( \, \partial _x \tilde{U} \, \bigr) \, 
   \Bigr) \\
& = \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
   F_{p}(U,U^r). 
\end{aligned}
\end{align}
Integrating (5.2) with respect to $x$ and $t$, we have
\begin{align}
\begin{aligned}
&\int ^{\infty}_{-\infty} \int^{\phi (t) }_0 
 \left( \rho_\delta \ast \mathrm{sgn} \right)(\eta)
 \, \mathrm{d}\eta \, \mathrm{d}x \\
&+ \int _0^t \int ^{\infty}_{-\infty} \int^{\phi}_0
   \bigl( \lambda(\tilde{U}+\eta)-\lambda(\tilde{U}) \bigr) 
   \frac{\mathrm{d}\left( \rho_\delta \ast \mathrm{sgn} \right)}
   {\mathrm{d}\phi}(\eta)
   \, \mathrm{d}\eta \, \bigl( \, \partial _x \tilde{U} \, \bigr)
   \, \mathrm{d}x \mathrm{d}\tau \\
&+ \frac{\mu}{2} \int _0^t \int ^{\infty}_{-\infty} 
   \frac{\mathrm{d}\left( \rho_\delta \ast \mathrm{sgn} \right)}
   {\mathrm{d}\phi}(\phi) \, 
   \bigl( \, \partial _x \phi \, \bigr)^2 
    \left( \, 
    \bigl| \partial _x \phi \bigr|^{p-1} 
    + \bigl| \partial _x \tilde{U} \bigr|^{p-1}  
    \, \right)
   \, \mathrm{d}x \mathrm{d}\tau \\
&+ \frac{\mu}{2} \int _0^t \int ^{\infty}_{-\infty} 
   \frac{\mathrm{d}\left( \rho_\delta \ast \mathrm{sgn} \right)}
   {\mathrm{d}\phi}(\phi) \, 
   \left| \, 
    \bigl| \partial _x \phi + \partial _x \tilde{U} \bigr|^{p-1} 
    - \bigl| \partial _x \tilde{U} \bigr|^{p-1}  
    \, \right| \\
& \qquad \qquad \qquad \qquad \quad \quad \quad \; \, \, 
   \times 
    \left| \, 
    \bigl( \, \partial _x \phi + \partial _x \tilde{U} \, \bigr)^2 
    - \bigl( \, \partial _x \tilde{U} \, \bigr)^2 \, \right| 
   \, \mathrm{d}x \mathrm{d}\tau \\
&= \int ^{\infty}_{-\infty} \int^{\phi _0}_0 
   \left( \rho_\delta \ast \mathrm{sgn} \right)(\eta)
   \, \mathrm{d}\phi \, \mathrm{d}x \\
&\qquad \qquad  \qquad \; 
 + \int _0^t \int ^{\infty}_{-\infty} 
   \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
   F_{p}(U,U^r)
   \, \mathrm{d}x \mathrm{d}\tau .
\end{aligned}
\end{align}
%We note 
By using Lemma 5.1, we first note that for $t\in [\, 0,\infty)$,
\begin{align}
\left| \, 
\int^{\phi (t)}_0\left( \rho_\delta \ast \mathrm{sgn} \right)(\eta)
\, \mathrm{d}\eta 
\, \right|
\le \left( \rho_\delta \ast \mathrm{sgn} \right)
    (|\,  \phi (t) \,|)\, |\,  \phi (t) \,|
\le |\,  \phi (t) \,|, 
\end{align}
\begin{align}
\displaystyle {\lim_{\delta\rightarrow 0}}
\int ^{\infty}_{-\infty} \int^{\phi (t) }_0
\left( \rho_\delta \ast \mathrm{sgn} \right)(\eta)
\, \mathrm{d}\eta 
= \| \,\phi(t) \,\|_{L^1}. 
\end{align}
We also note the following (the proof is similar to the one in \cite{yoshida}). 
\medskip

\noindent
{\bf Lemma 5.2.}\quad{\it
%It follows that 
It holds that 
\begin{align}
\begin{aligned}
%\displaystyle {\lim_{\delta\rightarrow 0}}
& \int ^{\infty}_{-\infty} \int^{\phi %(t,\cdot \, )
(t) }_0
  \left( \lambda(\tilde{U}+\eta)-\lambda(\tilde{U}) \right) 
  \frac{\mathrm{d}\left( \rho_\delta \ast \mathrm{sgn} \right)}
  {\mathrm{d}\phi}(\eta)
  \, \mathrm{d}\eta \, \bigl( \, \partial _x \tilde{U} \, \bigr)
  \, \mathrm{d}x \\
& \geq C^{-1} \left( \, \int_{\tilde{U}+\phi \geq 0, \tilde{U} \geq 0 } 
              \left|\, \int_{0}^{| \phi |} 
              \eta \, \rho _{\delta} (\eta) \, \mathrm{d}\eta \, \right|
              \bigl( \partial _x \tilde{U} \bigr) 
              \, \mathrm{d}x \, \right)(t)\\
& \quad        
       + C^{-1} \left( \, \int_{\tilde{U}+\phi \geq 0, \tilde{U} < 0 } 
                \left|\, \int_{0}^{| \phi |} 
                \bigl(\tilde{U}+\eta \bigr) \, \rho _{\delta} (\eta) \, 
                \mathrm{d}\eta \, \right|
                \bigl( \partial _x \tilde{U} \bigr) 
                \, \mathrm{d}x \, \right)(t)\\
& \quad 
       + C^{-1} \left( \, \int_{\tilde{U}+\phi < 0, \tilde{U} \geq 0 } 
                \left|\, \int_{0}^{| \phi |} 
                \tilde{U} \, \rho _{\delta} (\eta) \, 
                \mathrm{d}\eta \, \right|
                \bigl( \partial _x \tilde{U} \bigr) 
                \, \mathrm{d}x  \, \right)(t)
                \geq 0
                \qquad \bigl( t \ge 0 \bigr). 
\end{aligned}
\end{align}
}
\medskip

\noindent
So we can get 
\begin{align}
\begin{aligned}
&\| \,\phi (t) \,\|_{L^1} 
\le \| \,\phi_0 \,\|_{L^1} 
     + \displaystyle {\lim_{\delta\rightarrow 0}}
     \int^t_0 \left|\, \int ^{\infty}_{-\infty} 
     \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
     F_{p}(U,U^r)
   \, \mathrm{d}x \, \right|\, \mathrm{d}\tau \\ 
& \qquad \; \; \; \; \; \; \; \, 
     \le \| \,\phi_0 \,\|_{L^1} 
     + \displaystyle {\lim_{\delta\rightarrow 0}}
     \int^t_0 \left|\, \int ^{\infty}_{-\infty} 
     \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
     \widetilde{F_{p}}(U,U^r)
     \, \mathrm{d}x \, \right|\, \mathrm{d}\tau \\
& \qquad \qquad \quad \; \, 
   + \mu \, \displaystyle {\lim_{\delta\rightarrow 0}}
     \int^t_0 \biggl|\, \int ^{\infty}_{-\infty} 
     \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, \biggr.\\
& \quad \; 
     \biggl. \, \times 
     \partial _x \Bigl( \, 
     \bigl|\, \partial _x U + \partial _x U^r \, \bigr|^{p-1} 
     \bigl( \, \partial _x U + \partial _x U^r \, \bigr) 
     - \bigl|\, \partial _x U \, \bigr|^{p-1} 
      \bigl( \, \partial _x U \, \bigr) \, 
     \Bigr)
     \, \mathrm{d}x \, \biggr|\, \mathrm{d}\tau 
\end{aligned}
\end{align}
Noting by the asymptotic prorerties of $X(t)\, (t \ge T_0)$, Lemma 3.3, that 
\begin{align*}
\begin{aligned}
&\left( \, \int ^{X(t)}_{-\infty} + \int ^{\infty}_{X(t)} \, \right) 
 \bigl| \, \lambda(U+U^r)-\lambda(U^r) \, \bigr| \, \partial _x U^r
 \, \mathrm{d}x \\
&\qquad \qquad \qquad \quad \; \; \; \: \, 
\leq C_{p}(1+t)^{-1-\frac{p-1}{p+1}} + C_{\epsilon}(1+t)^{-1-(1-2\epsilon)} 
\quad \left( \, \epsilon \in (0,1) \, \right)
\end{aligned}
\end{align*}
and
$$
\int ^{\infty}_{-\infty} \lambda(U+U^r) \, \partial _x U
\, \mathrm{d}x 
\leq C_{p}(1+t)^{-1-\frac{p-1}{p+1}} + C_{\epsilon}(1+t)^{-1-(1-2\epsilon)} 
\quad \left( \, \epsilon \in (0,1) \, \right), 
$$
we immediately get 
\begin{align}
\begin{aligned}
& \displaystyle {\lim_{\delta\rightarrow 0}}\, 
  \left|\, \int ^{\infty}_{-\infty} 
  \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
  \widetilde{F_{p}}(U,U^r)
  \, \mathrm{d}x \, \right|\, (t) \\
& \leq  \left(\, \int ^{\infty}_{-\infty} 
  \bigl| \, \mathrm{sgn} (\phi) \, \bigr| \, 
  \Bigl| \, 
  \widetilde{F_{p}}(U,U^r)
  \, \Bigr| 
  \, \mathrm{d}x \, \right)\, (t) \\
& \leq C_{p}(1+t)^{-\frac{2p}{p+1}} 
    + C_{\epsilon}(1+t)^{-2(1-\epsilon)} 
  \quad \bigl( \, t \ge T_0 \, \bigr). 
\end{aligned}
\end{align}
Next, we estimate 
\begin{align*}
\begin{aligned}
&\biggl|\, \int ^{\infty}_{-\infty} 
 \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
 \partial _x \Bigl( \, 
 \bigl|\, \partial _x U + \partial _x U^r \, \bigr|^{p-1} 
 \bigl( \, \partial _x U + \partial _x U^r \, \bigr) 
 - \bigl|\, \partial _x U \, \bigr|^{p-1} 
 \bigl( \, \partial _x U \, \bigr) \, 
 \Bigr)
 \, \mathrm{d}x \, \biggr|\\
& \leq p \, \biggl|\, \int ^{\infty}_{-\infty} 
       \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
       \Bigl( \, 
       \bigl(\, \partial _x U + \partial _x U^r \, \bigr)^{p-1} 
       - \bigl(\, \partial _x U \, \bigr)^{p-1} \, 
       \Bigr) \, \partial _x^2 U
       \, \mathrm{d}x \, \biggr| \\
& \quad \, 
  + p \, \biggl|\, \int ^{\infty}_{-\infty} 
       \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
       \bigl(\, \partial _x U + \partial _x U^r \, \bigr)^{p-1} 
       \, \partial _x^2 U^r
       \, \mathrm{d}x \, \biggr|.
\end{aligned}
\end{align*}
By using Lagrange's mean-value theorem, we have
\begin{align*}
\begin{aligned}
& \biggl|\, \int ^{\infty}_{-\infty} 
  \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
  \Bigl( \, 
  \bigl(\, \partial _x U + \partial _x U^r \, \bigr)^{p-1} 
  - \bigl(\, \partial _x U \, \bigr)^{p-1} \, 
  \Bigr) \, \partial _x^2 U
  \, \mathrm{d}x \, \biggr|\\
& \leq \left\{\begin{array} {ll} 
       \displaystyle{
       (p-1) \, \biggl|\, \int ^{\infty}_{-\infty} 
       \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
       \bigl(\, \partial _x U \, \bigr)^{p-2} \, 
       \partial _x U^r \, \partial _x^2 U 
       \, \mathrm{d}x \, \biggr| 
       \quad \quad \quad \; \bigl( \, 1<p<2 \, \bigr)
       }, \\[15pt]
       \displaystyle{
       (p-1) \, \biggl|\, \int ^{\infty}_{-\infty} 
       \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
       \bigl(\, \partial _x U + \partial _x U^r \, \bigr)^{p-2} \, 
       \partial _x U^r \, \partial _x^2 U 
       \, \mathrm{d}x \, \biggr|
       \quad \; \; \bigl( \, p \geq 2 \, \bigr). 
       }
       \end{array}
       \right.\, 
\end{aligned}
\end{align*}
By using Lemma 2.2 and Lemma 2.3, we have for $1<p<2$, 
\begin{align}
\begin{aligned}
& \displaystyle {\lim_{\delta\rightarrow 0}}\, 
  \biggl|\, \int ^{\infty}_{-\infty} 
  \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
  \bigl(\, \partial _x U  \, \bigr)^{p-2} \, 
  \partial _x U^r \, \partial _x^2 U 
  \, \mathrm{d}x \, \biggr|\, (t) \\
& \leq \lint ^{\sqrt{\frac{A}{B}}{(1+t)}^{\frac{1}{p+1}} }_{-\sqrt{\frac{A}{B}}{(1+t)}^{\frac{1}{p+1}} }       
       \bigl(\, \partial _x U  \, \bigr)^{p-2} \, 
       \partial _x U^r \, | \, \partial _x^2 U \,|
       \, \mathrm{d}x \\
& \leq C_{p} (1+t)^{-\left( 1+\frac{p}{p+1} \right)}
       \lint ^{\sqrt{\frac{A}{B}}{(1+t)}^{\frac{1}{p+1}} }_{0} 
       \frac{x}{(1+t)^{\frac{1}{p+1}}} 
       \, \mathrm{d}x \\
& \leq C_{p} (1+t)^{-\frac{2p}{p+1}}
  \quad \bigl( \, t \ge T_0 \, \bigr), 
\end{aligned}
\end{align}
and for $p \geq 2$, 
\begin{align}
\begin{aligned}
& \displaystyle {\lim_{\delta\rightarrow 0}}\, 
  \biggl|\, \int ^{\infty}_{-\infty} 
  \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
  \bigl(\, \partial _x U + \partial _x U^r \, \bigr)^{p-2} \, 
  \partial _x U^r \, \partial _x^2 U 
  \, \mathrm{d}x \, \biggr|\, (t) \\
& \leq C_{p} \int ^{\infty}_{-\infty} 
       \bigl(\, \partial _x U  \, \bigr)^{p-2} \, 
       \partial _x U^r \, | \, \partial _x^2 U \,|
       \, \mathrm{d}x
       + C_{p} \int ^{\infty}_{-\infty} 
         \bigl(\, \partial _x U^r  \, \bigr)^{p-1} \, 
         | \, \partial _x^2 U \,| 
         \, \mathrm{d}x \\
& \leq C_{p} (1+t)^{-\frac{2p}{p+1}} 
       + C_{p} (1+t)^{-\left( p-1+\frac{2}{p+1} \right)} \\
& \quad \, \times 
         \lint ^{\sqrt{\frac{A}{B}}{(1+t)}^{\frac{1}{p+1}} }_{-\sqrt{\frac{A}{B}}{(1+t)}^{\frac{1}{p+1}} }       
         \left( \, 
         A-B \left(\frac{x}{(1+t)^{\frac{1}{p+1}}} \right)^2 \, 
         \right)^{-{\frac{p-2}{p-1}}} 
         \frac{| \, x \, |}{(1+t)^{\frac{1}{p+1}}}
         \, \mathrm{d}x \\
& \leq C_{p} (1+t)^{-\frac{2p}{p+1}} 
       + C_{p} (1+t)^{-\frac{p^2}{p+1}} \\
& \leq C_{p} (1+t)^{-\frac{2p}{p+1}}
  \quad \bigl( \, t \ge T_0 \, \bigr). 
\end{aligned}
\end{align}
Similarly, we have 
\begin{align}
\begin{aligned}
& \displaystyle {\lim_{\delta\rightarrow 0}}\, 
  \biggl|\, \int ^{\infty}_{-\infty} 
  \left( \rho_\delta \ast \mathrm{sgn} \right)(\phi) \, 
  \bigl(\, \partial _x U + \partial _x U^r \, \bigr)^{p-1} 
  \, \partial _x^2 U^r
  \, \mathrm{d}x \, \biggr|\, (t) \\
& \qquad 
  \leq C_{p} (1+t)^{-\frac{2p}{p+1}}
\quad \bigl( \, t \ge T_0 \, \bigr). 
\end{aligned}
\end{align}
Then, substituting (5.8), (5.9), (5.10) and (5.11) into (5.7), 
we have the desired $L^1$-estimate (5.1). 

\medskip
Next, we show %have 
the time-weighted $L^q$-energy estimates to $\phi$. 

\medskip

\noindent
{\bf Proposition 5.2.}\quad {\it
Suppose the same assumptions in Theorem 3.2. 
For any $q \in [ \, 1, \infty)$, 
there exist positive constants $\alpha$ and $C_{\alpha,p,q}$, 
such that the unique global solution in time $\phi$ of the Cauchy problem {\rm(3.6)} 
satisfies the following $L^q$-energy estimate 
 \begin{align}
 \begin{aligned}
 &(1+t)^\alpha  \| \,\phi(t) \,\|_{L^q}^q 
  + \int ^t_0 (1+\tau )^\alpha  G_{q}(\tau) 
    \, \mathrm{d}\tau \\
 & \quad 
  + \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} 
    \bigl( \, \partial _x \phi \, \bigr)^2 \\
 & \qquad \qquad \qquad \qquad \quad \; \, 
    \times 
    \left( \, 
    \bigl| \partial _x \phi \bigr|^{p-1} 
    + \bigl| \partial _x U \bigr|^{p-1} 
    + \bigl| \partial _x U^r \bigr|^{p-1}  
    \, \right) \, \mathrm{d}x \mathrm{d}\tau  \\
 & \quad 
  + \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} \, 
    \left| \, 
    \bigl| \partial _x \phi + \partial _x U + \partial _x U^r \bigr|^{p-1} 
    - \bigl| \partial _x U + \partial _x U^r \bigr|^{p-1}  
    \, \right| \\
 & \qquad \qquad \qquad \qquad \quad \; \, 
   \times 
    \left| \, 
    \bigl( \, \partial _x \phi + \partial _x U + \partial _x U^r \, \bigr)^2 
    - \bigl( \, \partial _x U + \partial _x U^r \, \bigr)^2 \, \right| 
    \, \mathrm{d}x \mathrm{d}\tau  \\
 &\leq C_{\alpha,p,q} \| \,\phi_0 \,\|_{L^q}^q 
       + C\left( \, \alpha, \, p, \, q, \, \phi_0 \, \right) \, 
       (1+t)^{\alpha - \frac{q-1}{2p}}
       \quad \bigl( \, t \ge T_0 \, \bigr). 
 \end{aligned}
 \end{align}
}

\medskip

The proof of Proposition 5.2 is given by the
following two lemmas.

\medskip

\noindent
{\bf Lemma 5.3.}\quad {\it
For any 
$1\leq q < \infty$, 
there exist positive constants $\alpha$ and $C_q$ such that 
\begin{align}
\begin{aligned}
&(1+t)^\alpha  \| \,\phi(t) \,\|_{L^q}^q 
  + q \, (q-1)
    \int ^t_0 (1+\tau )^\alpha  G_{q}(\tau) 
    \, \mathrm{d}\tau \\
 & 
  + C_{q} 
    \int ^t_0 (1+\tau )^\alpha 
    \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} 
    \bigl( \, \partial _x \phi \, \bigr)^2 \\
 & \qquad \qquad \qquad \qquad \quad \quad \quad \; \, 
   \times 
    \left( \, 
    \bigl| \partial _x \phi \bigr|^{p-1} 
    + \bigl| \partial _x U \bigr|^{p-1}  
    + \bigl| \partial _x U^r \bigr|^{p-1}  
    \, \right) \, \mathrm{d}x \mathrm{d}\tau  \\
 & 
  + C_{q} 
    \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} \, 
    \left| \, 
    \bigl| \partial _x \phi + \partial _x \tilde{U} \bigr|^{p-1} 
    - \bigl| \partial _x \tilde{U} \bigr|^{p-1}  
    \, \right| \\
 & \qquad \qquad \qquad \qquad \quad \quad \quad \; \, 
   \times 
    \left| \, 
    \bigl( \, \partial _x \phi + \partial _x \tilde{U} \, \bigr)^2 
    - \bigl( \, \partial _x \tilde{U} \, \bigr)^2 \, \right| 
    \, \mathrm{d}x \mathrm{d}\tau  \\
&\leq \| \,\phi_0 \,\|_{L^q}^q 
      + \alpha 
      \int ^t_0 (1+\tau )^{\alpha -1} 
      \| \,\phi(\tau ) \,\|_{L^p}^p 
      \, \mathrm{d}\tau \\
&\qquad \quad \quad \; \, \,  
 + q \int ^t_0 (1+\tau )^\alpha 
   \| \,\phi(\tau) \,\|_{L^\infty}^{q-1}
   \left| \left| \, 
    \widetilde{F_p}(U,U^r) (\tau ) \, \right| \right|_{L^1} 
    \, \mathrm{d}\tau \\
&\qquad \quad \quad \; \, \, 
 + C_{q} \int ^t_0 (1+\tau )^\alpha 
   \widetilde{\widetilde{F_p}}(\phi, U, U^r) (\tau)
   \, \mathrm{d}\tau 
   \quad \bigl( \, t \ge T_0 \, \bigr), 
\end{aligned}
\end{align}
where 
\begin{align*}
\begin{aligned}
& \widetilde{\widetilde{F_p}}(\phi, U, U^r) (t) \\
&:= \left\{\begin{array} {ll} 
   \displaystyle{ 
   \| \,\phi(t) \,\|_{L^\infty}^{q-1} \, 
   \biggl|\, \int ^{\infty}_{-\infty} 
   \partial _x \Bigl( \, 
   \bigl(\, \partial _x U + \partial _x U^r \, \bigr)^{p} 
   - \bigl(\, \partial _x U \, \bigr)^{p} \, 
   \Bigr) \, \mathrm{d}x \, \biggr| \, (t)}
   \quad \bigl( \, 1<q<2 \, \bigr), \\[15pt]
   \displaystyle{ 
   \| \,\phi(t) \,\|_{L^\infty}^{q-2} \, 
   \biggl|\, \int ^{\infty}_{-\infty} 
   \Bigl( \, 
   \bigl(\, \partial _x U + \partial _x U^r \, \bigr)^{p} 
   - \bigl(\, \partial _x U \, \bigr)^{p} \, 
   \Bigr)^{\frac{p+1}{p}} \, \mathrm{d}x \, \biggr| \, (t)
   \quad \quad \; \: \, \bigl( \, q \geq 2 \, \bigr). 
   }
   \end{array}
   \right.\, 
\end{aligned}
\end{align*}
}

\bigskip

\noindent
{\bf Lemma 5.4.}\quad {\it
Assume $p>1$ and $1\leq q < \infty$. 
We have the following interpolation inequalities.  

 \noindent
 {\rm (1)}\ \ For any $1\leq r < \infty$, there exists a positive 
 constant $C_{p,q,r}$  such that 
 \begin{align*}
 \begin{aligned}
 &\Vert \, \phi (t) \, \Vert _{L^r }
  \leq 
  C_{p,q,r} \left( \, \int _{-\infty}^{\infty} | \, \phi \, |^2 \, \mathrm{d}x \, \right)
  ^{\frac{pr+p+q-1}{(2p+q-1)r}} \\
 & \qquad \quad \quad \quad \quad \; \: \, 
         \times 
         \left( \, \int _{-\infty}^{\infty} | \, \phi \, |^{q-2} 
         \bigl| \, \partial _x \phi \, \bigr|^{p+1} \, \mathrm{d}x \, \right)
         ^{\frac{r-1}{(2p+q-1)r}} \quad \bigl( t \ge 0 \bigr). 
 \end{aligned}
 \end{align*}

 \noindent
 {\rm (2)}\ \ 
 There exists a positive 
 constant $C_{p,q}$ 
 such that 
 \begin{align*}
 \begin{aligned}
 &\Vert \, \phi (t)  \, \Vert _{L^\infty }
  \leq 
  C_{p,q} \left( \, \int _{-\infty}^{\infty} | \, \phi \, |^2 \, \mathrm{d}x \, \right)
  ^{\frac{p}{2p+q-1}} \\
 & \qquad \quad \quad \quad \quad \; \, 
         \times 
         \left( \, \int _{-\infty}^{\infty} | \, \phi \, |^{q-2} 
         \bigl| \, \partial _x \phi \, \bigr|^{p+1} \, \mathrm{d}x \, \right)
         ^{\frac{1}{2p+q-1}} \quad \bigl( t \ge 0 \bigr). 
 \end{aligned}
 \end{align*}
}

\medskip

\noindent
The proofs of Lemma 5.3, Lemma 5.4 and Proposition 5.2 are given in 
the quite similar way as those of Lemma 4.1, Lemma 4.2 and Proposition 4.1, 
so we omit them. 
We particularly note that we have by Proposition 5.2 
\begin{equation}
\| \,\phi (t) \,\|_{L^q}
\leq C( \, p, \, q, \, \phi_0 \, )\, (1+t)^
     {-\frac{1}{2p}\left( 1- \frac{1}{q} \right)} 
\end{equation}
for $1\leq q < \infty$. 

\bigskip

We shall finally obtain the time-decay estimates 
for the higher order derivatives, that is, 
$\partial _x \phi$ and $\partial _x u$, 
and also get the $L^{\infty}$-estimate for $\phi$. 

\medskip

\noindent
{\bf Proposition 5.3.}\quad {\it
Suppose the same assumptions in Theorem 3.2. 
%For any $q \in [ \, 1, \infty)$, 
There exist positive constants $\alpha$ and $C_{\alpha,p}$, 
such that the unique global solution in time $\phi$ of the Cauchy problem {\rm(3.6)} 
satisfies the following $L^{p+1}$-energy estimate 
\begin{align}
\begin{aligned}
 &(1+t)^\alpha  
  \| \, \partial _x u (t)\, \| _{L^{p+1}}^{p+1} 
  + %\mu \, p^2 \, (p+1) 
    \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    \bigl| \, \partial _x u \, \bigr|^{2(p-1)} 
    \left( \, \partial _x^2 u \, \right)^2 
    \, \mathrm{d}x \mathrm{d}\tau \\
 & \qquad \qquad \qquad \quad \quad \quad \, 
  + \int ^t_0 (1+\tau )^\alpha  
    \| \, \partial _x u (\tau) \, \| _{L^{p+2}}^{p+2} 
    \, \mathrm{d}\tau  \\
 &\leq C_{\alpha,p} 
       \| \, \partial _x u_{0} \, \| _{L^{p+1}}^{p+1} 
       + C\left( \, \alpha, \, p, \, \phi_0, \, \partial _x u_{0} \, \right) \, 
       (1+t)^{\alpha - \frac{1}{2p}}
       \quad \bigl( \, t \ge T_0 \, \bigr). 
\end{aligned}
\end{align}
}

\medskip

\noindent
To obtain Proposition 5.3, we first show the following. 

\medskip

\noindent
{\bf Lemma 5.5.}\quad {\it
It follows that 
\begin{align}
\begin{aligned}
 &(1+t)^\alpha  
  \| \, \partial _x u (t)\, \| _{L^{p+1}}^{p+1} \\
 & \quad 
  + \mu \, p^2 \, (p+1) 
    \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    \bigl| \, \partial _x u \, \bigr|^{2(p-1)} 
    \left( \, \partial _x^2 u \, \right)^2 
    \, \mathrm{d}x \mathrm{d}\tau \\
 & \qquad \quad \quad \; \; \; \: \, 
  + p \int ^t_0 (1+\tau )^\alpha  
    \int _{\partial _x u \geq 0} 
    f''(u) \left|\, \partial _x u \, \right|^{p+2} 
    \, \mathrm{d}x \mathrm{d}\tau  \\
 &= \| \, \partial _x u_{0} \, \| _{L^{p+1}}^{p+1} 
    + \alpha \int ^t_0 (1+\tau )^{\alpha -1} 
      \| \, \partial _x u (\tau) \, \| _{L^{p+1}}^{p+1} 
      \, \mathrm{d}\tau \\
 & \qquad \quad \quad \; \; \; \: \, 
  + p \int ^t_0 (1+\tau )^\alpha 
    \int _{\partial _x u < 0} 
    f''(u) \left|\, \partial _x u \, \right|^{p+2} 
    \, \mathrm{d}x \mathrm{d}\tau  
       \quad \bigl( \, t \ge T_0 \, \bigr). 
\end{aligned}
\end{align}
}

\medskip

\noindent
{\bf Proof of Lemma 5.5.}\quad
Multiplying the equation in the problem (1.1), that is, 
$$
\partial_tu +\partial_x \bigl(f(u) \bigr)
  = \mu \, 
    \partial_x \left( \, 
    \left| \, \partial_xu \, \right|^{p-1} \partial_xu \, 
    \right)
$$
by 
$$
- \partial_x \left( \, 
\left| \, \partial_xu \, \right|^{p-1} \partial_xu \, 
\right), 
$$ 
we obtain the divergence form 
\begin{align}
\begin{aligned}
&\partial_t 
 \left(\frac{1}{p+1} \left|\, \partial _x u \, \right|^{p+1} \right) 
 + \partial _x \Bigl( \, 
   - \, \left|\, \partial _x u \, \right|^{p-1} 
   \partial _x u \cdot \partial _t u \, \Bigr) \\
&\qquad \; \, 
 + \partial _x \left( 
 - \, \frac{p}{p+1} \, 
 f'(u) \left|\, \partial _x u \, \right|^{p+1} 
 \, \right) \\
&\qquad \; \, + \frac{p}{p+1} \, 
          f''(u) \left|\, \partial _x u \, \right|^{p+1}\partial _x u 
+\mu \, p\, q \left|\, \partial _x u \, \right|^{2(p-1)} 
   \bigl( \, \partial _x^2 u \, \bigr)^2 
= 0. 
\end{aligned}
\end{align}
Integrating the divergence form (5.17) with respect to $x$, 
we have 
\begin{align}
\begin{aligned}
&\frac{1}{p+1} \, 
\frac{\mathrm{d}}{\mathrm{d}t}\, 
\,\Vert \, \partial _x u (t) \, \Vert_{L^{p+1}}^{p+1} 
+\mu \, p^2 \int ^{\infty }_{-\infty } 
\left|\, \partial _x u \, \right|^{2(p-1)} 
\bigl( \, \partial _x^2 u \, \bigr)^2 
\, \mathrm{d}x \\
&\qquad \qquad \qquad \qquad \quad \; \; \; \: 
 +\frac{p}{p+1} \int ^{\infty }_{-\infty } 
  f''(u) \left|\, \partial _x u \, \right|^{p+1}\partial _x u 
  \, \mathrm{d}x 
= 0. 
\end{aligned}
\end{align}
We separate the integral region 
to the third term on the left-hand side of (5.18) as 
\begin{align}
\begin{aligned}
&\int ^{\infty }_{-\infty } 
 f''(u) \left|\, \partial _x u \, \right|^{p+1}\partial _x u 
 \, \mathrm{d}x \\
&= \int _{\partial _x u \geq 0 } + \int _{\partial _x u < 0 } \\
&= \int _{\partial _x u \geq 0 } 
   f''(u) \left|\, \partial _x u \, \right|^{p+2} \, \mathrm{d}x 
  - \int _{\partial _x u < 0 } 
  f''(u) \left|\, \partial _x u \, \right|^{p+2} \, \mathrm{d}x. 
\end{aligned}
\end{align}
Substituting (5.19) into (5.18), we get the following equality 
\begin{align}
\begin{aligned}
&\frac{1}{p+1} \, 
\frac{\mathrm{d}}{\mathrm{d}t}\, 
\,\Vert \, \partial _x u (t) \, \Vert_{L^{p+1}}^{p+1} 
+\mu \, p^2 \int ^{\infty }_{-\infty } 
\left|\, \partial _x u \, \right|^{2(p-1)} 
\bigl( \, \partial _x^2 u \, \bigr)^2 
\, \mathrm{d}x \\
& \; \: \, +\frac{p}{p+1} \int _{\partial _x u \geq 0 } 
   f''(u) \left|\, \partial _x u \, \right|^{p+2} \, \mathrm{d}x 
= \frac{p}{p+1} \int _{\partial _x u < 0 } 
  f''(u) \left|\, \partial _x u \, \right|^{p+2} \, \mathrm{d}x. 
\end{aligned}
\end{align}
Multiplying (5.20) by 
$(1+t)^{\alpha}$ with $\alpha>0$ 
and integrating over $(0,t)$ with respect to the time, 
we complete the proof of Lemma 5.5. 

\bigskip

\noindent
{\bf Proof of Proposition 5.3.}\quad
We use the following important results (cf. \cite{yoshida'}). 

\medskip

\noindent
{\bf Lemma 5.6.}\quad {\it
For any $s \geq 0$, there exists a positive constant $C_{s}$ 
such that 
\begin{align}
\begin{aligned}
\int _{\partial _x u < 0 } 
f''(u) \left|\, \partial _x u \, \right|^{s} \, \mathrm{d}x 
\leq C_{s}
     \int _{\partial _x u < 0 } 
     \left|\, \partial _x \phi \, \right|^{s} \, \mathrm{d}x. 
\end{aligned}
\end{align}
}

\noindent
In fact, taking care of the relation by using Lemma 2.2 and Lemma 2.3
\begin{align}
\begin{aligned}
\partial _x u = \partial _x \tilde{U} + \partial _x \phi <0 \, 
\Longleftrightarrow  \, \partial _x \phi <0, 
                     \, \partial _x \tilde{U} < \bigl|\, \partial _x \phi \, \bigr|, 
\end{aligned}
\end{align}
we immediately have 
\begin{align}
\begin{aligned}
\int _{\partial _x u < 0 } 
&f''(u) \left|\, \partial _x u \, \right|^{s} \, \mathrm{d}x \\
&\leq 2^{s} \, 
     \left( \, \max_{ | u | \leq \widetilde{C}} f''(u) \, \right)
     \int _{\partial _x \phi < 0, \partial _x \tilde{U} < | \partial _x \phi | } 
     \left|\, \partial _x \phi \, \right|^{s} \, \mathrm{d}x.
\end{aligned}
\end{align}

\medskip
Since $ \partial _x u $ is absolutely continuous, 
we first note that 
for any 
$
x \in \bigl\{ \, x \in \mathbb{R} \, \, \bigr. 
 \bigl| \, \partial _x u \, < \, 0 \, \bigr\}, 
$
there exsists 
$
x_{k} \in \mathbb{R}\cup \{ - \infty \}
$
such that 
$$
\partial _x u (x_{k}) = 0, \; \; 
  \partial _x u (y) \, < \, 0 \; 
  \bigl( \, y \in 
  ( x_{k},x ) \, \bigr).
$$
Therefore, 
it follows that for such $x$ and $x_{k}$ with $q\geq p\, (\, >1\, )$, 
\begin{align}
\left|\, \partial _x u \, \right|^q 
= \left(\, - \partial _x u \, \right)^q 
= q \int_{x_k}^{x} 
   \left(\, - \partial _x u \, \right)^{q-1} \left(\, - \partial _x^2 u \, \right) 
   \, \mathrm{d}y 
\end{align}
By using the Cauchy-Schwarz inequality, we have 

\medskip

\noindent
{\bf Lemma 5.7.}\quad {\it
It holds that 
\begin{align}
\begin{aligned}
& \int _{\partial _x u < 0 } 
  \left|\, \partial _x u \, \right|^{p+2} \, \mathrm{d}x \\
& \le C_{p} 
     \left( \, 
     \int _{\partial _x u < 0 } 
     \left|\, \partial _x u \, \right|^{2(p-1)} 
     \bigl( \, \partial _x^2 u \, \bigr)^2 
     \, \mathrm{d}x \, 
     \right)^{\frac{1}{3p+1}} 
     \left( \, 
     \int _{\partial _x u < 0 } 
     \left|\, \partial _x u \, \right|^{p+1} 
     \, \mathrm{d}x \, 
     \right)^{\frac{3p+2}{3p+1}}. 
\end{aligned}
\end{align}
}
By using Young's inequality to (5.25), we also have 

\medskip

\noindent
{\bf Lemma 5.8.}\quad {\it
It follows that for any $\epsilon>0$, 
there exists a positive constant $C_p({\epsilon})$ such that, 
\begin{align}
\begin{aligned}
& \int _{\partial _x u < 0 } 
  \left|\, \partial _x u \, \right|^{p+2} \, \mathrm{d}x \\
& \le \epsilon 
      \int _{\partial _x u < 0 } 
      \left|\, \partial _x u \, \right|^{2(p-1)} 
      \bigl( \, \partial _x^2 u \, \bigr)^2 
      \, \mathrm{d}x \, 
      + C_p({\epsilon}) \left( \, 
        \int _{\partial _x u < 0 } 
        \left|\, \partial _x u \, \right|^{p+1} 
        \, \mathrm{d}x \, 
        \right)^{\frac{3p+2}{3p}}. 
\end{aligned}
\end{align}
}
By using Lemma 5.6, Lemma 5.7 and Lemma 5.8 
with $\epsilon = \frac{\mu \, p^2 \, (p+1)}{2}$, we have 
\begin{align}
\begin{aligned}
 &(1+t)^\alpha  
  \| \, \partial _x u (t)\, \| _{L^{p+1}}^{p+1} \\
 & \quad 
  + \frac{\mu \, p^2 \, (p+1)}{2} 
    \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    \bigl| \, \partial _x u \, \bigr|^{2(p-1)} 
    \left( \, \partial _x^2 u \, \right)^2 
    \, \mathrm{d}x \mathrm{d}\tau \\
 & \qquad \quad \quad \; \; \; \: \, 
  + p \int ^t_0 (1+\tau )^\alpha  
    \int _{\partial _x u \geq 0} 
    f''(u) \left|\, \partial _x u \, \right|^{p+1} 
    \, \mathrm{d}x \mathrm{d}\tau  \\
 &\leq \| \, \partial _x u_{0} \, \| _{L^{p+1}}^{p+1} \\
 & \quad 
       + \alpha \int ^t_0 (1+\tau )^{\alpha -1} 
         \Bigl( \, 
         \| \, \partial _x \phi (\tau) \, \| _{L^{p+1}}^{p+1} 
         + \| \, \partial _x \tilde{U}(\tau) \, \| _{L^{p+1}}^{p+1} 
         \, \Bigr) 
         \, \mathrm{d}\tau \\ 
 & \quad 
  + C_{p} \int ^t_0 (1+\tau )^\alpha 
    \left( \, 
  \int _{\partial _x u < 0 } 
  \left|\, \partial _x u \, \right|^{p+1} 
  \, \mathrm{d}x \, 
  \right)^{\frac{2}{3p}+1} \mathrm{d}\tau . 
\end{aligned}
\end{align}
By using Proposition 5.2, we get the following time-decay estimates. 

\medskip

\noindent
{\bf Lemma 5.9.}\quad {\it
There exist positive constants $\alpha \gg 1$ and $C_{\alpha,p,q}$, 
such that 
 \begin{align}
 \begin{aligned}
 &\int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    | \, \phi \, |^{q-2} 
    \bigl( \, \partial _x \phi \, \bigr)^2 
    \left( \, 
    \bigl| \partial _x \phi \bigr|^{p-1} 
    + \bigl| \partial _x U \bigr|^{p-1}  
    + \bigl| \partial _x U^r \bigr|^{p-1}  
    \, \right) \, \mathrm{d}x \mathrm{d}\tau  \\
 &\leq C\left( \, \alpha, \, p, \, q, \, \phi_0 \, \right) \, 
       (1+t)^{\alpha - \frac{q-1}{2p}}
       \quad \bigl( \, t \ge T_0 \, \bigr).
 \end{aligned}
 \end{align}
}

\medskip

\noindent
By using Lemma 5.9 
with $\alpha \mapsto \alpha - 1 \gg 1$ and $q=2$, we have 
\begin{align}
\alpha \int ^t_0 (1+\tau )^{\alpha - 1} 
\| \, \partial _x \phi (\tau) \, \| _{L^{p+1}}^{p+1} 
\, \mathrm{d}\tau  
\leq C\left( \, \alpha, \, p, \, \phi_0 \, \right) \, 
     (1+t)^{\alpha - \frac{2p+1}{2p}}. 
\end{align}
We can also estimate by using Lemma 2.2 and Lemma 2.3 as 
\begin{align}
\alpha \int ^t_0 (1+\tau )^{\alpha - 1} 
\| \, \partial _x \tilde{U} (\tau) \, \| _{L^{p+1}}^{p+1} 
\, \mathrm{d}\tau  
\leq C\left( \, \alpha, \, p \, \right) \, 
     (1+t)^{\alpha - \frac{p}{p+1}}. 
\end{align}
By using the uniform boundedness in Lemma 3.2, that is, 
$$
\| \, \partial _x u (t) 
\, \| _{L^{p+1}}^{p+1} 
%&+\int _0^{\infty} \int _{-\infty}^{\infty} 
% \bigl| \, \partial _x u \, \bigr|^{2(p-1)} 
% \left( \, \partial _x^2 u \, \right)^2 
% \, \mathrm{d}x \mathrm{d}t \\
%&\qquad \quad 
\leq 
 C_{p}\bigl(\, \| \, \phi _0 \, \| _{L^2}, 
 \| \, \partial _x u_0 \, \| _{L^{p+1}} \, \bigr) 
$$
and Lemma 5.9 
with $q=2$, we can estimate as 
\begin{align}
\begin{aligned}
&C_{p} \int ^t_0 (1+\tau )^\alpha 
 \left( \, 
 \int _{\partial _x u < 0 } 
 \left|\, \partial _x u \, \right|^{p+1} 
 \, \mathrm{d}x \, 
 \right)^{\frac{2}{3p}+1} \mathrm{d}\tau \\
&\leq C_{p} \int ^t_0 (1+\tau )^\alpha 
      \int _{\partial _x u < 0 } 
      \left|\, \partial _x \phi \, \right|^{p+1} 
      \, \mathrm{d}x 
      \cdot 
      \| \, \partial _x u (\tau) \, \| _{L^{p+1}}^{\frac{2(p+1)}{3p}} 
      \, \mathrm{d}\tau \\
&\leq C\left( \, p, \, \phi_0, \, \partial _x u_0 \, \right) \, 
      \int ^t_0 (1+\tau )^\alpha 
      \int _{-\infty}^{\infty} 
      \left|\, \partial _x \phi \, \right|^{p+1} 
      \, \mathrm{d}x \mathrm{d}\tau \\
&\leq C\left( \, \alpha, \, p, \, \phi_0, \, \partial _x u_0 \, \right) \, 
      (1+t)^{\alpha - \frac{1}{2p}}. 
\end{aligned}
\end{align}
Substituting (5.29), (5.30) and (5.31) into (5.27), 
we complete the proof of Proposition 5.3. 
In particular, we have 
\begin{align}
\| \, \partial _x u (t) \, \| _{L^{p+1}}^{p+1} 
\, \mathrm{d}\tau  
\leq C\left( \, p, \, \phi_0, \, \partial _x u_0 \, \right) \, 
     (1+t)^{- \frac{1}{2p}}, 
\end{align}
and 
\begin{align}
\begin{aligned}
\| \, \partial _x \phi (t) \, \| _{L^{p+1}}^{p+1} 
\, \mathrm{d}\tau  
& \leq \| \, \partial _x u (t) \, \| _{L^{p+1}}^{p+1} 
       + \| \, \partial _x U^r (t) \, \| _{L^{p+1}}^{p+1} \\
& \leq C\left( \, p, \, \phi_0, \, \partial _x u_0 \, \right) \, 
       (1+t)^{- \frac{1}{2p}} 
\end{aligned}
\end{align}
for $1 \leq q < \infty$.

\medskip

{\bf Proof of Theorem 3.2.}\ 
We already have proved the decay estimate 
of $\| \,\phi(t) \,\|_{L^q}$ with $1 \le q < \infty$. 
Therefore we only 
show 
the following time-decay estimate for the higher order derivative
\begin{align}
\begin{aligned}
&\bigl|\bigl|\,
 \partial _x u(t) \,
 \bigr|\bigr|_{L^{p+1} }, \, \; \; 
 \left|\left|\,
 \partial _x \phi (t) 
 \,\right|\right|_{L^{p+1} } \\[5pt]
& \leq \left\{\begin{array} {ll} 
      C( \, \epsilon, \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t)^{-\frac{p}{(p+1)^2}}\\[15pt] 
      \,  \, \, \: \; \; \quad \qquad 
      \left( \, 1 < p \le 
      \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)^2(3p-2)}{3}\, \epsilon } }
      \, \right),\\[15pt] 
      C( \, \epsilon, \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t)^{-\frac{3}{2(p+1)(3p-2)} + \epsilon}\\[15pt] 
      \,  \, \, \; \; \quad \qquad \qquad 
      \left( \,  
      \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)^2(3p-2)}{3}\, \epsilon } }
      < p \, \right)
      \end{array}
      \right.\,
\end{aligned}
\end{align}
for any $0<\epsilon \ll 1$, 
and the $L^{\infty}$-estimate for $\phi$. 

\medskip

We first prove (5.34). 
Substituting (5.32) into (5.31), 
we have 
\begin{align}
\begin{aligned}
&C_{p} \int ^t_0 (1+\tau )^\alpha 
      \int _{\partial _x u < 0 } 
      \left|\, \partial _x \phi \, \right|^{p+1} 
      \, \mathrm{d}x 
      \cdot 
      \| \, \partial _x u (\tau) \, \| _{L^{p+1}}^{\frac{2(p+1)}{3p}} 
      \, \mathrm{d}\tau \\
&\leq C\left( \, p, \, \phi_0, \, \partial _x u_0 \, \right) \, 
      \int ^t_0 (1+\tau )
      ^{\alpha - \frac{1}{2p} \cdot \frac{2}{3p}}
      \int _{-\infty}^{\infty} 
      \left|\, \partial _x \phi \, \right|^{p+1} 
      \, \mathrm{d}x \mathrm{d}\tau. 
\end{aligned}
\end{align}

\medskip

\noindent
By using Lemma 5.9 
with $\alpha \mapsto \alpha - \frac{1}{2p} \cdot \frac{2}{3p} \gg 1$ 
and $q=2$, we also have 
\begin{align}
\alpha \int ^t_0 (1+\tau )^{\alpha - 1} 
\| \, \partial _x \phi (\tau) \, \| _{L^{p+1}}^{p+1} 
\, \mathrm{d}\tau  
\leq C\left( \, \alpha, \, p, \, \phi_0 \, \right) \, 
     (1+t)^{\alpha - \frac{1}{2p} \cdot \frac{2}{3p} - \frac{1}{2p}}. 
\end{align}
Substituting (5.36) into (5.27), 
we have 
\begin{align}
\begin{aligned}
&\bigl|\bigl|\,
 \partial _x u(t) \,
 \bigr|\bigr|_{L^{p+1} }^{p+1}, \, \; \; 
 \left|\left|\,
 \partial _x \phi (t) 
 \,\right|\right|_{L^{p+1} }^{p+1} \\[5pt]
& \leq C( \, p, \, \phi_0, \, \partial _x u_0 \, ) \\
& \quad \times 
       \left( \, 
       (1+t)^{-\frac{2p+1}{2p}} + (1+t)^{-\frac{p}{p+1}}  
       + (1+t)
       ^{-\left( \frac{1}{2p} \cdot \frac{2}{3p} + \frac{1}{2p} \right)} 
       \,  \right) \\[5pt] 
& \leq C( \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
       \left( \, 
       (1+t)^{-\frac{p}{p+1}} 
       + (1+t)
       ^{-\left( \frac{1}{2p} \cdot \frac{2}{3p} + \frac{1}{2p} \right)} 
       \,  \right) 
\end{aligned}
\end{align}
Iterating ``$\infty$''-times the above process, we will get 
\begin{align}
\begin{aligned}
&\bigl|\bigl|\,
 \partial _x u(t) \,
 \bigr|\bigr|_{L^{p+1} }^{p+1}, \, \; \; 
 \left|\left|\,
 \partial _x \phi (t) 
 \,\right|\right|_{L^{p+1} }^{p+1} \\[5pt]
& \leq C( \, \epsilon, \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
       \left( \, 
       (1+t)^{-\frac{p}{p+1}} 
       + (1+t)
         ^{- \mathlarger{ \frac{1}{2p}} 
         \substack{{\infty }\\{\substack{{\lsum }\\{n=0}}}}
         \mathlarger{\left( \frac{2}{3p} \right)^{n}} + \mathlarger{\epsilon } }
       \,  \right) \\[15pt] 
& \leq C( \, \epsilon, \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
       \left( \, 
       (1+t)^{-\frac{p}{p+1}} 
       + (1+t)^{- \frac{3}{2(3p-2)} + \epsilon }
       \,  \right) \\[15pt] 
& \leq \left\{\begin{array} {ll} 
      C( \, \epsilon, \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t)^{-\frac{p}{p+1}}\\[15pt] 
      \, \: \; \; \qquad \qquad 
      \left( \, 1 < p 
      \le \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } } \, \right),\\[15pt] 
      C( \, \epsilon, \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t)^{-\frac{3}{2(3p-2)} + \epsilon}\\[15pt]
      \, \: \; \quad \quad \qquad \qquad 
      \left( \,  
      \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      < p \, \right)
      \end{array}
      \right.\,
\end{aligned}
\end{align}
for any $0<\epsilon \ll 1$. 
%We also note the following: 
%if $1 < p \le \frac{1 + \sqrt{3}}{2}$, 
%then 
%$$
%(1+t)^{-\frac{2p+1}{2p}}  
%\leq (1+t)^{- p}, 
%$$
%if $p > \frac{1 + \sqrt{3}}{2}$, 
%then 
%$$
%(1+t)^{- p} 
%\leq (1+t)^{-\frac{2p+1}{2p}}
%\leq (1+t)^{-\frac{3}{3p-2}}, 
%$$
%if $p > \frac{2 + \sqrt{22}}{6}$, 
%then 
%$$
%(1+t)^{- p} 
%\leq (1+t)^{-\frac{3}{3p-2}}, 
%$$
%and if $1 < p \leq \frac{2 + \sqrt{22}}{6}$, 
%then 
%$$
%(1+t)^{-\frac{3}{3p-2}} 
%\leq (1+t)^{- p}. 
%$$

\medskip

\noindent
Thus, we get (5.34). 

We finally show the $L^{\infty}$-estimate for $\phi$ 
by using the Gagliardo-Nirenberg inequality.  
Substituting (5.14) and (5.34) into (4.17), we get 
\begin{align}
\begin{aligned}
&\bigl|\bigl|\,
 \phi(t) \,
 \bigr|\bigr|_{L^{\infty } } \\[5pt] 
&\leq \left\{\begin{array} {ll} 
      C( \, \epsilon, \, p, \, \theta, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t)
      ^{-\frac{1}{2p} + 
      \left( \frac{2p+1}{2p(p+1)} - \frac{p}{(p+1)^2} \right) \theta} \\[15pt] 
      \, \: \: \; \; \quad \qquad \qquad \qquad \, 
      \left( \, 1 < p \le 
      \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)^2(3p-2)}{3}\, \epsilon } }
      \, \right),\\[10pt] 
      C( \, \epsilon, \, p, \, \theta, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t)
      ^{-\frac{1}{2p} + 
      \left( \frac{2p+1}{2p(p+1)} - \frac{3}{2(p+1)(3p-2)}  + \epsilon \right) \theta}\\[15pt] 
      \, \: \: \; \; \quad \qquad \qquad \qquad \qquad 
      \left( \,  
      \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)^2(3p-2)}{3}\, \epsilon } }
      < p \, \right)
      \end{array}
      \right.\, %\\[5pt] 
%& \leq C( \, p, \, \theta, \, \phi_0, \, \partial _x u_0 \, ) \, 
%       (1+t)
%       ^{-\left( \frac{1}{2p} \cdot \frac{2}{3p} + \frac{1}{2p} \right)} 
\end{aligned}
\end{align}
for $\theta \in (0,1\, ]$ and any $0<\epsilon \ll 1$. 
Consequently, we do complete the proof of Theorem 3.2.
\bigskip 

\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{$L^{r+1}$-estimate for the higher order derivative with $r>p$}
In this section, 
we show the time-decay estimates for the higher order derivative 
in the $L^{r+1}$-norm 
with $r>p$, 
in the case where $\phi_0 \in L^1 \cap L^2$ 
with $\partial _x u_0 \in L^{p+1} \cap L^{r+1}$, 
that is, Theorem 3.3. 

\medskip

\noindent
{\bf Proposition 6.1.}\quad {\it
Suppose the same assumptions in Theorem 3.3. 
For any $r>p$, 
there exist positive constants $\alpha$ and $C_{\alpha,p,r}$, 
such that the unique global solution in time $\phi$ of the Cauchy problem {\rm(3.6)} 
satisfies the following $L^{r+1}$-energy estimate 
\begin{align}
\begin{aligned}
 &(1+t)^\alpha  
  \| \, \partial _x u (t)\, \| _{L^{r+1}}^{r+1} 
  + \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    \bigl| \, \partial _x u \, \bigr|^{p+r-2} 
    \left( \, \partial _x^2 u \, \right)^2 
    \, \mathrm{d}x \mathrm{d}\tau \\
 & \qquad \qquad \qquad \quad \quad \quad \, 
  + \int ^t_0 (1+\tau )^\alpha  
    \| \, \partial _x u (\tau) \, \| _{L^{r+2}}^{r+2} 
    \, \mathrm{d}\tau  \\
 &\leq C_{\alpha,p,r} 
       \| \, \partial _x u_{0} \, \| _{L^{r+1}}^{r+1} \\[5pt] 
 & \; \; 
     + \left\{\begin{array} {ll} 
       C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
       (1+t)^{\alpha-\frac{4p(r-p)+7p+3}{6p(p+1)}} \\[15pt] 
       %\, 
       \left( \, 
       1 < p \le 
       \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }, \; 
       r > \displaystyle{\frac{-4p^2+7p+3}{2p}} > p 
       \, \right),\\[15pt] 
       C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
       (1+t)^{\alpha-\frac{r}{p+1} 
       + \frac{2(r-p+1)}{3p} \epsilon} \\[15pt] 
       %\, 
       \left( \, 
       1 < p \le 
       \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }, \; 
       p < r \leq \displaystyle{\frac{-4p^2+7p+3}{2p}} 
       \, \right),\\[15pt] 
       C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
       (1+t)^{\alpha-\frac{p+2r}{2p(3p-2)}
       + \frac{2(r-p+1)}{3p} \epsilon} \\[15pt] 
       \, \, \: \; \; \quad \quad \qquad \qquad \qquad \qquad 
       \left( \, 
       \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
       <p 
       \, \right)
       \end{array}
       \right.\,%\\
%& \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad 
%  \qquad \qquad \qquad 
%  \bigl( t \ge 0 \bigr). 
\end{aligned}
\end{align}
for $t \ge T_0$ and any $0<\epsilon \ll 1$. 
}

\medskip

The proof of Proposition 6.1 is given by the following 
three lemmas. 
Because the proofs of them are similar to those of 
Lemma 5.5, Lemma 5.6, Lemma 5.7 and Lemma 5.8, 
we state only here. 

\medskip

\noindent
{\bf Lemma 6.1.}\quad {\it
There exist positive constants 
$C_{p,r}$ and $C_{\alpha,p,r}$ such that 
\begin{align}
\begin{aligned}
 &(1+t)^\alpha  
  \| \, \partial _x u (t)\, \| _{L^{p+1}}^{p+1} \\
 & \quad 
  + \mu \, p\, r \, (r+1) 
    \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    \bigl| \, \partial _x u \, \bigr|^{p+r-2} 
    \left( \, \partial _x^2 u \, \right)^2 
    \, \mathrm{d}x \mathrm{d}\tau \\
 & \qquad \quad \quad \; \; \; \: \, 
  + r \int ^t_0 (1+\tau )^\alpha  
    \int _{\partial _x u \geq 0} 
    f''(u) \left|\, \partial _x u \, \right|^{r+2} 
    \, \mathrm{d}x \mathrm{d}\tau  \\
 &\leq \| \, \partial _x u_{0} \, \| _{L^{p+1}}\\
 & \quad \; \; \; \: \, 
       + C_{\alpha,p,r} \int ^t_0 
         (1+\tau )
         ^{\alpha -{\frac{2p+r+1}{3p+1}}} 
         \left( \, 
         \int _{-\infty}^{\infty} 
         \left|\, \partial _x u \, \right|^{p+1} 
         \, \mathrm{d}x \, 
         \right)^{\frac{p+2r+1}{3p+1}}
         \, \mathrm{d}\tau \\
 & \quad \; \; \; \: \, 
  + C_{p,r} \int ^t_0 (1+\tau )^\alpha 
    \left( \, 
         \int _{\partial _x u < 0 } 
         \left|\, \partial _x u \, \right|^{p+1} 
         \, \mathrm{d}x \, 
         \right)^{\frac{p+2r+2}{3p}}
         \, \mathrm{d}\tau  
       \quad \bigl( \, t \ge T_0 \, \bigr). 
\end{aligned}
\end{align}
}

\bigskip

\noindent
{\bf Lemma 6.2.}\quad {\it
Assume $p>1$ and $r>p$. 
We have the following interpolation inequalities.  

 \noindent
 {\rm (1)}\ \ There exists a positive 
 constant $C_{p,r}$  such that 
 \begin{align*}
 \begin{aligned}
 &\Vert \, \partial _x u (t) \, \Vert _{L^{r+1} }
 \leq 
  C_{p,r} \left( \, 
     \int _{-\infty}^{\infty}
     \left|\, \partial _x u \, \right|^{2(p-1)} 
     \bigl( \, \partial _x^2 u \, \bigr)^2 
     \, \mathrm{d}x \, 
     \right)^{\frac{r-p}{(2p+r+1)(r+1)}} \\
 & \: \; \; \quad \quad \quad \qquad \qquad \times 
     \left( \, 
     \int _{\partial _x u < 0 } 
     \left|\, \partial _x u \, \right|^{p+1} 
     \, \mathrm{d}x \, 
     \right)^{\frac{p+2r+1}{(2p+r+1)(r+1)}}
 \quad \bigl( t \ge 0 \bigr). 
 \end{aligned}
 \end{align*}

 \noindent
 {\rm (2)}\ \ 
 There exists a positive 
 constant $C_{p,r}$ 
 such that 
 \begin{align*}
 \begin{aligned}
 &\Vert \, \partial _x u (t) \, \Vert _{L^\infty }
 \leq 
  C_{p,r} \left( \, 
     \int _{-\infty}^{\infty} 
     \left|\, \partial _x u \, \right|^{2(p-1)} 
     \bigl( \, \partial _x^2 u \, \bigr)^2 
     \, \mathrm{d}x \, 
     \right)^{\frac{1}{2p+r+1}} \\
 & \: \quad \quad \quad \qquad \qquad \times 
     \left( \, 
     \int _{\partial _x u < 0 } 
     \left|\, \partial _x u \, \right|^{p+1} 
     \, \mathrm{d}x \, 
     \right)^{\frac{1}{2p+r+1}}
 \quad \bigl( t \ge 0 \bigr). 
 \end{aligned}
 \end{align*}
}

\bigskip

\noindent
{\bf Lemma 6.3.}\quad {\it
Assume $p>1$ and $r>p$. 
We have the following interpolation inequalities.  

 \noindent
 {\rm (1)}\ \ There exists a positive 
 constant $C_{p,r}$  such that 
 \begin{align*}
 \begin{aligned}
 &\Vert \, \partial _x u (t) \, \Vert 
  _{L^{r+1}_{x} \left( \{ \partial _x u < 0 \} \right) }
 \leq 
  C_{p,r} \left( \, 
     \int _{\partial _x u < 0 } 
     \left|\, \partial _x u \, \right|^{2(p-1)} 
     \bigl( \, \partial _x^2 u \, \bigr)^2 
     \, \mathrm{d}x \, 
     \right)^{\frac{r-p}{(2p+r+1)(r+1)}} \\
 & \, \: \: \quad \qquad \qquad \qquad \qquad \qquad \times 
     \left( \, 
     \int _{\partial _x u < 0 } 
     \left|\, \partial _x u \, \right|^{p+1} 
     \, \mathrm{d}x \, 
     \right)^{\frac{p+2r+1}{(2p+r+1)(r+1)}}
 \quad \bigl( t \ge 0 \bigr). 
 \end{aligned}
 \end{align*}

 \noindent
 {\rm (2)}\ \ 
 There exists a positive 
 constant $C_{p,r}$ 
 such that 
 \begin{align*}
 \begin{aligned}
 &\Vert \, \partial _x u (t) \, \Vert 
  _{L^{\infty}_{x} \left( \{ \partial _x u < 0 \} \right) }
 \leq 
  C_{p,r} \left( \, 
     \int _{\partial _x u < 0 } 
     \left|\, \partial _x u \, \right|^{2(p-1)} 
     \bigl( \, \partial _x^2 u \, \bigr)^2 
     \, \mathrm{d}x \, 
     \right)^{\frac{1}{2p+r+1}} \\
 & \, \, \: \: \; \qquad \qquad \qquad \qquad \qquad \times 
     \left( \, 
     \int _{\partial _x u < 0 } 
     \left|\, \partial _x u \, \right|^{p+1} 
     \, \mathrm{d}x \, 
     \right)^{\frac{1}{2p+r+1}}
 \quad \bigl( t \ge 0 \bigr). 
 \end{aligned}
 \end{align*}
}

\medskip

{\bf Proof of Proposition 6.1.}\ 
By using (5.34), 
we estimate the each terms on the right-hand side of (6.2) as 
\begin{align}
\begin{aligned}
&C_{\alpha,p,r} \int ^t_0 
         (1+\tau )
         ^{\alpha -{\frac{2p+r+1}{3p+1}}} 
         \left( \, 
         \int _{-\infty}^{\infty} 
         \left|\, \partial _x u \, \right|^{p+1} 
         \, \mathrm{d}x \, 
         \right)^{\frac{p+2r+1}{3p+1}}
         \, \mathrm{d}\tau \\[5pt]
& \leq \left\{\begin{array} {ll} 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
      \displaystyle{ \int ^t_0 
      (1+\tau )
      ^{\alpha - \frac{2p+r+1}{3p+1} -{\frac{p(p+2r+1)}{(p+1)(3p+1)}}} 
      \, \mathrm{d}\tau } \\[15pt] 
      \, \: \: \; \; \quad \quad \qquad \qquad \qquad \, 
      \left( \, 1 < p \le 
      \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      \, \right),\\[15pt] 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
      \displaystyle{ \int ^t_0 
      (1+\tau )
      ^{\alpha - \frac{2p+r+1}{3p+1} -{\frac{3(p+2r+1)}{2(3p+1)(3p-2)}}
      + \frac{p+2r+1}{3p+1} \epsilon} 
      \, \mathrm{d}\tau } \\[15pt] 
      \, \: \: \; \; \quad \quad \qquad \qquad \qquad \qquad 
      \left( \,  \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      < p \, \right)
      \end{array}
      \right.\, \\[5pt] 
& \leq \left\{\begin{array} {ll} 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t )
      ^{\alpha - \frac{r}{p+1} } \\[15pt] 
      \, \: \: \; \; \quad \quad \qquad \qquad \qquad \, 
      \left( \, 1 < p \le 
      \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      \, \right),\\[15pt] 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t )
      ^{\alpha - \frac{6p(r-p)+7p+2r+3}{2(3p+1)(3p-2)} 
      + \frac{p+2r+1}{3p+1} \epsilon} \\[15pt] 
      \, \: \: \; \; \quad \quad \qquad \qquad \qquad \qquad 
      \left( \,  \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      < p \, \right),
      \end{array}
      \right.\, %\\[5pt] 
%& \leq \left\{\begin{array} {ll} 
%      C( \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
%      (1+t)^{-{p}} \, \: \; \quad \qquad 
%      \left( \, 1 < p \le \displaystyle{\frac{2 + \sqrt{22}}{6}} \, \right),\\[10pt] 
%      C( \, p, \, \phi_0, \, \partial _x u_0 \, ) \, 
%      (1+t)^{-\frac{3}{2(3p-2)}} \, \, \: \quad \quad 
%      \left( \,  \displaystyle{\frac{2 + \sqrt{22}}{6}} < p \, \right).
%      \end{array}
%      \right.\,
\end{aligned}
\end{align}
\begin{align}
\begin{aligned}
&C_{p,r} \int ^t_0 (1+\tau )^\alpha 
    \left( \, 
    \int _{\partial _x u < 0 } 
    \left|\, \partial _x u \, \right|^{p+1} 
    \, \mathrm{d}x \, 
    \right)^{\frac{p+2r+2}{3p}}
    \, \mathrm{d}\tau  \\
& \leq C_{p,r} \int ^t_0 (1+\tau )^\alpha 
       \left( \, 
       \int _{-\infty}^{\infty}
       \left|\, \partial _x \phi \, \right|^{p+1} 
       \, \mathrm{d}x \, 
       \right)
       \| \, \partial _x u (\tau) \, \| _{L^{p+1}}
       ^{\frac{2(p+1)(r-p+1)}{3p}} 
       \, \mathrm{d}\tau \\[5pt]
& \leq \left\{\begin{array} {ll} 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
      \displaystyle{ \int ^t_0 
      (1+\tau )
      ^{\alpha - \frac{2(r-p+1)}{3(p+1)} } 
      \| \, \partial _x \phi (\tau) \, \| _{L^{p+1}}^{p+1}
      \, \mathrm{d}\tau } \\[15pt] 
      \, \: \: \; \; \; \; \; \quad \quad \qquad \qquad \qquad \, 
      \left( \, 1 < p \le 
      \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      \, \right),\\[15pt] 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
      \displaystyle{ \int ^t_0 
      (1+\tau )
      ^{\alpha - \frac{r-p+1}{p(3p-2)} + \frac{2(r-p+1)}{3p} \epsilon} 
      \| \, \partial _x \phi (\tau) \, \| _{L^{p+1}}^{p+1}
      \, \mathrm{d}\tau } \\[15pt] 
      \, \: \: \; \; \; \; \; \quad \quad \qquad \qquad \qquad \qquad 
      \left( \,  \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      < p \, \right)
      \end{array}
      \right.\, 
\end{aligned}
\end{align}
for any $0<\epsilon \ll 1$. \\
\noindent
By using Lemma 5.9 with 
\begin{align*}
\begin{aligned}
\alpha \mapsto \left\{\begin{array} {ll} 
               \alpha-\displaystyle{\frac{2(r-p+1)}{3(p+1)}} 
               \, \, \: \; \quad \quad \qquad \qquad 
               \left( \, 1 < p \le 
               \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
               \, \right),\\[15pt] 
               \alpha
               -\displaystyle{\left( \, 
                \frac{r-p+1}{p(3p-2)} - \frac{2(r-p+1)}{3p} \epsilon 
                \, \right)}
               \quad 
               \left( \,  \displaystyle{\frac{7}{12} + 
               \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
               < p \, \right)
               \end{array}
               \right.\, 
\end{aligned}
\end{align*}
and $q=2$, 
we get 
\begin{align}
\begin{aligned}
&C_{p,r} \int ^t_0 (1+\tau )^\alpha 
    \left( \, 
    \int _{\partial _x u < 0 } 
    \left|\, \partial _x u \, \right|^{p+1} 
    \, \mathrm{d}x \, 
    \right)^{\frac{p+2r+2}{3p}}
    \, \mathrm{d}\tau  \\[5pt]
& \leq \left\{\begin{array} {ll} 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t )
      ^{\alpha - \frac{4p(r-p)+7p+3}{6p} } \\[15pt] 
      \, \: \: \; \; \quad \qquad \qquad \qquad \qquad \, 
      \left( \, 1 < p \le 
      \displaystyle{\frac{7}{12} + 
        \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      \, \right),\\[15pt] 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) \, 
      (1+t )
      ^{\alpha - \frac{p+2r}{2p(3p-2)} + \frac{2(r-p+1)}{3p} \epsilon} \\[15pt] 
      \, \: \: \; \; \quad \qquad \qquad \qquad \qquad \qquad 
      \left( \,  \displaystyle{\frac{7}{12} + 
        \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      < p \, \right)
      \end{array}
      \right.\, 
\end{aligned}
\end{align}
for any $0<\epsilon \ll 1$. \\
\noindent
Substituting (6.3) and (6.5) into (6.2), 
we have 
\begin{align}
\begin{aligned}
 &(1+t)^\alpha  
  \| \, \partial _x u (t)\, \| _{L^{r+1}}^{r+1} \\
 & \quad 
  + \int ^t_0 (1+\tau )^\alpha  \int _{-\infty}^{\infty} 
    \bigl| \, \partial _x u \, \bigr|^{p+r-2} 
    \left( \, \partial _x^2 u \, \right)^2 
    \, \mathrm{d}x \mathrm{d}\tau \\
 & \quad 
  + \int ^t_0 (1+\tau )^\alpha  
    \int _{\partial _x u \geq 0} 
    f''(u) \left|\, \partial _x u \, \right|^{r+2} 
    \, \mathrm{d}x \mathrm{d}\tau  \\
& \leq C_{\alpha,p,r} 
       \| \, \partial _x u_{0} \, \| _{L^{p+1}} \\[5pt]
& \quad + 
      \left\{\begin{array} {ll} 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) 
      (1+t)^{\alpha}\\[5pt]
      \times 
      \left( \, 
       (1+t)
       ^{- \frac{r}{p+1}} 
       + (1+t)
       ^{- \frac{4p(r-p)+7p+3}{6p(p+1)} } 
       \,  \right) \\[15pt] 
      \, \: \: \; \; \quad \qquad \qquad \qquad \, 
      \left( \, 1 < p \le 
      \displaystyle{\frac{7}{12} + 
      \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      \, \right),\\[15pt] 
      C( \, \alpha, \, \epsilon, \, p, \, r, \, \phi_0, \, \partial _x u_0 \, ) 
      (1+t)^{\alpha}\\[5pt]
      \times 
      \left( \, 
       (1+t)
       ^{- \frac{6p(r-p)+7p+2r+3}{2(3p+1)(3p-2)} 
       + \frac{p+2r+1}{3p+1} \epsilon} 
       + (1+t)
       ^{- \frac{p+2r}{2p(3p-2)} 
       + \frac{2(r-p+1)}{3p} \epsilon} 
       \,  \right) \\[15pt] 
      \, \: \: \; \; \quad \qquad \qquad \qquad \qquad 
      \left( \,  \displaystyle{\frac{7}{12} + 
      \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon } }
      < p \, \right)
      \end{array}
      \right.\, \\[5pt] 
\end{aligned}
\end{align}
for any $0<\epsilon \ll 1$.\\
\noindent
Here, we note the following: 
if $1 < p 
    \le \frac{7}{12} + 
        \sqrt{\frac{73}{144} - \frac{(p+1)(3p-2)}{3}\, \epsilon }
    \left( \, < \frac{7+\sqrt{73}}{12} \, \right)$, 
then 
$$
p<\frac{-4p^2+7p+3}{2p}. 
$$
Therefore, it follows that 
$$
(1+t)^{- \frac{r}{p+1} } 
\leq (1+t)^{- \frac{4p(r-p)+7p+3}{6p(p+1)}} \quad 
\left( \, r>\frac{-4p^2+7p+3}{2p}>p \, \right)
$$
and 
$$
(1+t)^{- \frac{4p(r-p)+7p+3}{6p(p+1)}}
\leq (1+t)^{- \frac{r}{p+1} }  \quad 
\left( \, p<r\leq \frac{-4p^2+7p+3}{2p} \, \right).
$$
If $\epsilon 
< \frac{3(r-p)(p-1)(3p+1)}{(3p-2)|\, 9p^2-p-2r-2\, |}$, 
then, for any $p>1$ with $r>p$, 
$$
(1+t)^{- \frac{6p(r-p)+7p+2r+3}{2(3p+1)(3p-2)} 
       + \frac{p+2r+1}{3p+1} \epsilon} 
\leq (1+t)^{- \frac{p+2r}{2p(3p-2)} 
       + \frac{2(r-p+1)}{3p} \epsilon}. 
$$

\medskip

\noindent
Thus, we do complete the proof of Proposition 6.1. 


%
%\bigskip
%
%{\bf Acknowledgement.}\quad
%%The author would like to thank 
%%an anonymous referee for his/her 
%%many significant comments and kind advices 
%%on our manuscript. 
%The auther %also 
%thanks 
%Professor Akitaka Matsumura for 
%his constant help and guidance. 
%%Especially, it might be difficult to be born the ideas, (4.29) and (5.20) 
%%without the discussion with him. 
 
 
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{\it On the motion of suspended particles in a turbulent flow 
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\bibitem{carillo-toscani}
J. A. Carrillo and G. Toscani, 
{\it Asymptotic $L^1$-decay of solutions of the porous medium equation 
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Indiana Univ. Math. J. {\bf 49} (2000), pp. 113-142.

\bibitem{gag} 
E. Gagliardo, 
{\it Ulteriori propriet{\`{a}} di alcune classi 
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Ricerche di Mat. Napoli {\bf 8} (1959), pp. 24-51 (in Italian).

\bibitem{gur-mac} 
M. E. Gurtin and R. C. MacCamy, 
{\it On the diffusion of biological populations}, 
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\bibitem{hashimoto-kawashima-ueda} 
I. Hashimoto, S. Kawashima and Y. Ueda, 
{\it Convergence rate to the nonlinear waves 
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Methods Appl. Anal. {\bf 16} (2009), pp. 389-402.

\bibitem{hashimoto-matsumura} 
I. Hashimoto and A. Matsumura, 
{\it Large time behavior 
of solutions to an initial boundary value problem 
on the half space for scalar viscous conservation law}, 
Methods Appl. Anal. {\bf 14} (2007), pp. 45-59.

\bibitem{hattori-nishihara} 
Y. Hattori and K. Nishihara, 
{\it A note on the stability of rarefaction wave of the 
Burgers equation}, 
Japan J. Indust. Appl. Math. {\bf 8} (1991), pp. 85-96.

\bibitem{huang-pan-wang} 
F. Huang, R. Pan and Z. Wang, 
{\it $L^1$Convergence to the Barenblatt Solution for 
Compressible Euler Equations with Damping}, 
Arch. Rational Mech. Anal. {\bf 200} (2011), pp. 665-689.

\bibitem{ilin-kalashnikov-oleinik} 
A. M. Il'in, A. S. Kala{\v{s}}nikov and O. A. Ole{\u\i}nik,
{\it Second-order linear equations of parabolic type}, 
Uspekhi Math. Nauk SSSR {\bf 17} (1962), pp. 3-146 (in Russian);
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\bibitem{ilin-oleinik} 
A. M. Il'in and O. A. Ole{\u\i}nik, 
{\it Asymptotic behavior of the solutions of the Cauchy problem for 
some quasi-linear equations for large values of the time}, 
Mat. Sb. {\bf 51} (1960), pp. 191-216 (in Russian).

\bibitem{kamin} 
S. Kamin, 
{\it Source-type solutions for equations of nonstationary filtration}, 
J. Math. Anal. Appl. {\bf 64} (1978), pp. 263-276. 

\bibitem{lad-sol-ura} 
O. A. Lady{\v{z}}enskaja, V. A. Solonnikov and N. N. Ural'ceva, 
{\it Linear and quasilinear equations of parabolic type}, 
Transl. Math. Monographs, vol. 23, Amer. Math. Soc., 
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\bibitem{lax} 
P. D. Lax, 
{\it Hyperbolic systems of conservation laws  I\hspace{-.1em}I}, 
Comm. Pure Appl. Math. {\bf 10} (1957), pp. 537-566.

\bibitem{liep-rosh} 
H. W. Liepmann and A. Roshko, 
{\it Elements of Gasdynamics}, 
John Wiley \& Sons, Inc., New York, 1957.

\bibitem{lions} 
J. L. Lions, 
{\it Quelques m{\`{e}}thodes de r{\`{e}}solution 
des probl{\`{e}}mes aux limites non lin{\`{e}}aires}, 
Dunod Gauthier-Villars, Paris, 1969 (in French).

\bibitem{liu-matsumura-nishihara} 
T.-P. Liu, A. Matsumura and K. Nishihara, 
{\it Behaviors of solutions for the Burgers equation 
with boundary corresponding to rarefaction waves}, 
SIAM J. Math. Anal. {\bf 29} (1998), pp. 293-308.

\bibitem{matsu-nishi1} 
A. Matsumura and K. Nishihara,
{\it Asymptotic toward the rarefaction wave of solutions of a
one-dimensional model system for compressible viscous gas}, Japan
J. Appl. Math. {\bf 3} (1986), pp. 1-13.

\bibitem{matsu-nishi2} 
A. Matsumura and K. Nishihara, 
{\it Asymptotics toward the rarefaction wave 
of the solutions of Burgers' equation with 
nonlinear degenerate viscosity
}
Nonlinear Anal. TMA {\bf 23} (1994), pp. 605-614.

\bibitem{matsu-nishi3} 
A. Matsumura and K. Nishihara, 
{\it Asymptotic stability of traveling waves for scalar 
viscous conservation laws with non-convex nonlinearity}, 
Comm. Math. Phys. {\bf 165} (1994), pp. 83-96.

\bibitem{matsumura-yoshida} 
A. Matsumura and N. Yoshida, 
{\it Asymptotic behavior of solutions to the Cauchy problem 
for the scalar viscous conservation law 
with partially linearly degenerate flux}, 
SIAM J. Math. Anal. {\bf 44} (2012), pp. 2526-2544.

\bibitem{nagai-mimura1} 
T. Nagai and M. Mimura, 
{\it Some nonlinear degenerate diffusion equations 
related to population dynamics}, 
J. Math. Soc. Japan {\bf 35} (1983), pp. 539-562.

\bibitem{nagai-mimura2} 
T. Nagai and M. Mimura, 
{\it Asymptotic behavior for 
a nonlinear degenerate diffusion equation in population dynamics}, 
SIAM J. Appl. Math. {\bf 43} (1983), pp. 449-464.

\bibitem{nakamura} 
T. Nakamura, 
{\it Asymptotic decay toward the rarefaction waves 
of solutions for viscous conservation laws 
in a one-dimensional half space}, 
SIAM J. Math. Anal. {\bf 34} (2003), pp. 1308-1317.

\bibitem{nash} 
J. Nash, 
{\it Continuity of solutions of parabolic and elliptic equations}, 
Amer. J. Math. 
{\bf 80} (1958), pp. 931-954.

\bibitem{nir} 
L. Nirenberg, 
{\it On elliptic partial differential equations}, 
Ann. Scuola Norm. Sup. Pisa (Ser. I\hspace{-.1em}I\hspace{-.1em}I) 
{\bf 13} (1959), pp. 115-162.

\bibitem{pattle} 
R. E. Pattle, 
{\it Diffusion from an instantaneous point source with 
a concentration-dependent coefficient}, 
Quart. J. Mech. Appl. Math. {\bf 12} (1959), pp. 407-409.

\bibitem{pro-wei} 
M. H. Protter and H. F. Weinberger, 
{\it Maximum Principles in Differential Equations}, Springer-Verlag, 
New York-Berlin, 1984. 

\bibitem{smoller} 
J. Smoller, 
{\it Shock Waves and Reaction-diffusion Equations}, Springer-Verlag, 
New York-Berlin, 1983. 

\bibitem{yoshida} 
N. Yoshida, 
{\it Decay properties of solutions toward a multiwave pattern 
for the scalar viscous conservation law 
with partially linearly degenerate flux}, 
Nonlinear Anal. TMA {\bf 96} (2014), pp. 189-210.

\bibitem{yoshida'} 
N. Yoshida, 
{\it Asymptotic behavior of solutions toward a multiwave pattern 
for the scalar conservation law 
with degenerate flux and viscosity}, 
submitted. 

\bibitem{yoshida''} 
N. Yoshida, 
{\it Decay properties of solutions 
to the Cauchy problem for the scalar conservation law 
with nonlinearly degenerate viscosity}, 
submitted. 

\bibitem{zel-kom} 
Ya. B. Zel'dovi{\v{c}} and A. S. Kompanceec, 
{\it On the theory of propagation 
of heat with the heat conductivity depending upon the temperature}, 
Collection in honor of the seventieth birthday of academician A. F. Ioffe, 
Izdat. Akad. Nauk SSSR, Moscow (1950), pp. 61-71 (in Russian). 

\end{thebibliography} 

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%% Start line numbering here if you want
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%%%%%%%%%%%%%%
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1502.04750.extracted_bib
{carillo-toscani J. A. Carrillo and G. Toscani, {Asymptotic $L^1$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J. {49 (2000), pp. 113-142.
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{gag E. Gagliardo, {Ulteriori propriet{\`{a di alcune classi di funzioni in pi{\`{u variabili, Ricerche di Mat. Napoli {8 (1959), pp. 24-51 (in Italian).
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{huang-pan-wang F. Huang, R. Pan and Z. Wang, {$L^1$Convergence to the Barenblatt Solution for Compressible Euler Equations with Damping, Arch. Rational Mech. Anal. {200 (2011), pp. 665-689.
1502.04750.extracted_bib
{ilin-kalashnikov-oleinik A. M. Il'in, A. S. Kala{\v{snikov and O. A. Ole{\u\inik, {Second-order linear equations of parabolic type, Uspekhi Math. Nauk SSSR {17 (1962), pp. 3-146 (in Russian); English translation in Russian Math. Surveys {17 (1962), pp. 1-143.
1502.04750.extracted_bib
{ilin-oleinik A. M. Il'in and O. A. Ole{\u\inik, {Asymptotic behavior of the solutions of the Cauchy problem for some quasi-linear equations for large values of the time, Mat. Sb. {51 (1960), pp. 191-216 (in Russian).
1502.04750.extracted_bib
{kamin S. Kamin, {Source-type solutions for equations of nonstationary filtration, J. Math. Anal. Appl. {64 (1978), pp. 263-276.
1502.04750.extracted_bib
{lad-sol-ura O. A. Lady{\v{zenskaja, V. A. Solonnikov and N. N. Ural'ceva, {Linear and quasilinear equations of parabolic type, Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, Rhode. Island, 1968.
1502.04750.extracted_bib
{lax P. D. Lax, {Hyperbolic systems of conservation laws I\hspace{-.1emI, Comm. Pure Appl. Math. {10 (1957), pp. 537-566.
1502.04750.extracted_bib
{liep-rosh H. W. Liepmann and A. Roshko, {Elements of Gasdynamics, John Wiley \& Sons, Inc., New York, 1957.
1502.04750.extracted_bib
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1502.04750.extracted_bib
{liu-matsumura-nishihara T.-P. Liu, A. Matsumura and K. Nishihara, {Behaviors of solutions for the Burgers equation with boundary corresponding to rarefaction waves, SIAM J. Math. Anal. {29 (1998), pp. 293-308.
1502.04750.extracted_bib
{matsu-nishi1 A. Matsumura and K. Nishihara, {Asymptotic toward the rarefaction wave of solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. {3 (1986), pp. 1-13.
1502.04750.extracted_bib
{matsu-nishi2 A. Matsumura and K. Nishihara, {Asymptotics toward the rarefaction wave of the solutions of Burgers' equation with nonlinear degenerate viscosity Nonlinear Anal. TMA {23 (1994), pp. 605-614.
1502.04750.extracted_bib
{matsu-nishi3 A. Matsumura and K. Nishihara, {Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys. {165 (1994), pp. 83-96.
1502.04750.extracted_bib
{matsumura-yoshida A. Matsumura and N. Yoshida, {Asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law with partially linearly degenerate flux, SIAM J. Math. Anal. {44 (2012), pp. 2526-2544.
1502.04750.extracted_bib
{nagai-mimura1 T. Nagai and M. Mimura, {Some nonlinear degenerate diffusion equations related to population dynamics, J. Math. Soc. Japan {35 (1983), pp. 539-562.
1502.04750.extracted_bib
{nagai-mimura2 T. Nagai and M. Mimura, {Asymptotic behavior for a nonlinear degenerate diffusion equation in population dynamics, SIAM J. Appl. Math. {43 (1983), pp. 449-464.
1502.04750.extracted_bib
{nakamura T. Nakamura, {Asymptotic decay toward the rarefaction waves of solutions for viscous conservation laws in a one-dimensional half space, SIAM J. Math. Anal. {34 (2003), pp. 1308-1317.
1502.04750.extracted_bib
{nash J. Nash, {Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. {80 (1958), pp. 931-954.
1502.04750.extracted_bib
{nir L. Nirenberg, {On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (Ser. I\hspace{-.1emI\hspace{-.1emI) {13 (1959), pp. 115-162.
1502.04750.extracted_bib
{pattle R. E. Pattle, {Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. {12 (1959), pp. 407-409.
1502.04750.extracted_bib
{pro-wei M. H. Protter and H. F. Weinberger, {Maximum Principles in Differential Equations, Springer-Verlag, New York-Berlin, 1984.
1502.04750.extracted_bib
{smoller J. Smoller, {Shock Waves and Reaction-diffusion Equations, Springer-Verlag, New York-Berlin, 1983.
1502.04750.extracted_bib
{yoshida N. Yoshida, {Decay properties of solutions toward a multiwave pattern for the scalar viscous conservation law with partially linearly degenerate flux, Nonlinear Anal. TMA {96 (2014), pp. 189-210.
1502.04750.extracted_bib
{yoshida' N. Yoshida, {Asymptotic behavior of solutions toward a multiwave pattern for the scalar conservation law with degenerate flux and viscosity, submitted.
1502.04750.extracted_bib
{yoshida'' N. Yoshida, {Decay properties of solutions to the Cauchy problem for the scalar conservation law with nonlinearly degenerate viscosity, submitted.
1502.04750.extracted_bib
{zel-kom Ya. B. Zel'dovi{\v{c and A. S. Kompanceec, {On the theory of propagation of heat with the heat conductivity depending upon the temperature, Collection in honor of the seventieth birthday of academician A. F. Ioffe, Izdat. Akad. Nauk SSSR, Moscow (1950), pp. 61-71 (in Russian).
1502.04750.extracted_bib
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1502.04750.extracted_bib
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1502.04750.extracted_bib
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