\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 256, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/256\hfil Pointwise estimates for solutions]
{Pointwise estimates for solutions to a system of nonlinear
damped wave equations}

\author[W. Wang \hfil EJDE-2013/256\hfilneg]
{Wenjun Wang}

\address{Wenjun Wang \newline
College of Science, University of Shanghai for
Science and Technology, Shanghai 200093, China}
\email{wwj001373@hotmail.com}

\thanks{Submitted June 24, 2013. Published November 20, 2013.}
\subjclass[2000]{35L15}
\keywords{Damped wave system; global solution; classical solution;
\hfill\break\indent pointwise estimate; Green function}

\begin{abstract}
 In this article, we consider the existence of global solutions
 and pointwise estimates for the Cauchy problem of a nonlinear
 damped wave equation. We obtain the existence by using the 
 approach introduced by Li and Chen in \cite{Li-Chen}
 and some estimates of the solution. The proofs of the estimates
 are based on a detailed analysis of the Green function of the linear
 damped wave equations. Also, we show the $L^p$ convergence rate 
 of the solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

In this paper, we consider the nonlinear damped wave equation
\begin{equation}\label{1.0}
\begin{gathered}
 \partial_t^2u-\Delta u+\partial_tu=F(u),\quad t>0,x\in \mathbb{R}^n,\\
 u(0,x)=a(x),\quad \partial_tu(0,x)=b(x),\quad x\in \mathbb{R}^n,
\end{gathered}
\end{equation}
where $u(t,x)=(u_1(t,x),u_2(t,x),\dots,u_m(t,x)):
(0,T)\times\mathbb{R}^n\to\mathbb{R}^m$ is the unknown
vector valued function and $a(x)=(a_1(x),\dots,a_m(x))$ and
$b(x)=(b_1(x),\dots,b_m(x))$ are given initial data. The nonlinear
smooth vector function $F:\mathbb{R}^m\to\mathbb{R}^m$,
$F(u)=(F_1(u),\dots,F_m(u))$ such that
\begin{equation}
F_j(u) = O\Big(\prod_{k=1}^l u_k^{p_{j,k}}\Big),
\end{equation}
with $p_{j,k}\geq 1$ or $p_{j,k}=0$ for $j,k=1,\dots,m$.

The first aim of this paper is to obtain the existence of
classical global solutions to system \eqref{1.0}. We show the 
existence directly by using the Banach fixed point theorem with a
detailed analysis of the Green function. At the same time, we have
the following decay rates of the solutions
\begin{equation}
\|u_j(t)\|_{L^\infty} \leq C(1+t)^{-n/2}, \quad
\|u_j(t)\|_{L^2} \leq C(1+t)^{-n/4},\quad j=1,\dots,m.
\end{equation}
The second aim is to get the pointwise estimate of the solutions to
system \eqref{1.0}. With the help of the pointwise estimates of the
Green function and using the method of the Green function, we show
the pointwise estimates of the solutions to system \eqref{1.0}. This
estimates represent a clear decaying structure of the solutions.
Furthermore, we get the optimal $L^p$ decay estimates of the
solutions.

There are many authors working in this field. For the single
nonlinear damped wave equation
\begin{equation}\label{1.3}
\begin{gathered}
 \partial_t^2u-\Delta u+\partial_tu=f(u),\quad t>0,x\in \mathbb{R}^n,\\
 u(0,x)=a_1(x),\quad \partial_tu(0,x)=b_1(x),\quad x\in \mathbb{R}^n,
 \end{gathered}
\end{equation}
many results have been published. For the case $f(u)=-|u|^\theta u$,
Kawashima, Nakao and Ono \cite{Kawashima-Nakao-Ono} studied the
decay properties of solutions to \eqref{1.3} by using the energy
method combined with $L^p$-$L^q$ estimates. Ono \cite{Ono-1}
derived sharp decay rates in the subcritical case of solutions in
unbounded domains in $\mathbb{R}^n$. Nakao and Ono in
\cite{Nakao-Ono} proved the existence
and decay of global solutions weak solutions for \eqref{1.3}
 by using the potential well method.
By employing the weighted $L^2$
energy method, Nishihara and Zhao \cite{Nishihara-Zhao} obtained
that the behavior of solutions to \eqref{1.3} as $t\to
\infty$ is expected to be same as that for the corresponding heat
equation. The global asymptotic behaviors were studied by Nishihara
\cite{Nishihara,Nishihara-1} for $n=3,4$ and Ikehata, Nishihara
and Zhao \cite{Ikehata-Nishihara-Zhao} for $n\geq 1$. In
\cite{Liu,Liu-Wang}, the pointwise estimates of classical
solutions to \eqref{1.3} were obtained.


For the case of $f(u)= |u|^\theta u $, Ikehata, Miyaoka and Nakatake
\cite{Ikehata-Miyaoka-Nakatake} obtained the global existence of
weak solutions to \eqref{1.3}. Furthermore, Hosono and Ogawa
\cite{Hosono-Ogawa} obtained the $L^p$-$L^q$ type estimate of the
difference between the solution to \eqref{1.3} and the solutions of
corresponding heat and wave equations in the two-dimensional space.
Meanwhile, when $2\leq n\leq 5$, the same type estimate was studied
by Narazaki in \cite{Narazaki}.

For the general case $f(u)=O(u^{\theta+1})$, Wang and Wang
\cite{Wang-Wang} proved the pointwise estimates of classical
solutions to \eqref{1.1}. There also have been a lot of
investigations for those cases. For detail results, please refer to
\cite{Ikehata,Ikehata-Ohta,Li-Zhou,Ono-2,Ono-3,Todorova-Yordanov,Zhang}.

For the system of the nonlinear damped wave equations
\begin{equation}\label{1.5}
\begin{gathered}
\partial_t^2u_1-\Delta u_1+\partial_tu_1=|u_m|^{p_1},\quad t>0,\;x\in \mathbb{R}^n,\\
\partial_t^2u_2-\Delta u_2+\partial_tu_2=|u_1|^{p_2},\quad t>0,\;x\in \mathbb{R}^n,\\
\dots\\
 \partial_t^2u_m-\Delta u_m+\partial_tu_m=|u_{m-1}|^{p_m},\quad
 t>0,x\in \mathbb{R}^n,\\
 u_j(0,x)=a_j(x),\quad \partial_tu_j(0,x)=b_j(x),\quad x\in \mathbb{R}^n, \;
 (1\leq j\leq m),
\end{gathered}
\end{equation}
Sun and Wang \cite{Sun-Wang} for $m=2$ and Takeda \cite{Takeda} for
$m\geq 2$ obtained global weak solutions to system
\eqref{1.5}.

For the general case \eqref{1.0}, Ogawa and Takeda in
\cite{Ogawa-Takeda} obtained the existence of the global solution
under some conditions, which include the results of \cite{Sun-Wang}
and \cite{Takeda}. Recently, Ogawa and Takeda in
\cite{Ogawa-Takeda-2} proved the asymptotic behavior of solutions to
the problem \eqref{1.0} by using the $L^p$-$L^q$ type decomposition
of the fundamental solution of the linear damped wave equations into
the dissipative part and hyperbolic part.

However, there are few studies concerning the global existence and
decay property of classical solutions to the Cauchy problem of the
nonlinear damped wave system. In this paper, we investigate the
global existence and pointwise estimates of classical solution to
system \eqref{1.0}. First of all, we employ the Green function of
the linear damped wave equation to express the solution of system
\eqref{1.0}. Then, we obtain the global solution directly by using
the method introduced by Li and Chen in \cite{Li-Chen}. Unlike the
usual energy method, this method needn't to prove the local
existence and extend the local solution to the global one in time.
In this process, the decay properties of the Green function play an
important role. We employ $G_1$ and $G_2$ to define the Green
function of linear equation. By a detailed
analysis of the Green function, we obtain the pointwise estimates of
the Green function. Compared with the methods in
\cite{Liu-Wang,Narazaki,Nishihara}, the method of dealing with the
existence theory in this paper is more useful to show a clear
decaying structure of the solution. Secondly, with the obtained
pointwise estimates of the Green function, we give the pointwise
estimates of the solution to \eqref{1.3} by the method of the Green
function. Finally, as a corollary of the pointwise estimates, the
optimal $L^p\ (1\leq p\leq \infty)$ convergence rate can be obtained
easily.

Throughout this paper, we assume that the nonlinear term
$\{F_j(u)\}_{j=1}^m$ satisfies the following conditions, for
$p_{j,k}\in [0,+\infty)\cup\{0\}$, ($j=1,\dots,m$; $k=1,\dots,m$),
\begin{gather}\label{1.6}
|\partial^{\tilde \alpha_j}F_j(u)|
 \leq C_{\tilde \alpha_j,\delta}
 \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde \alpha_j}
 \prod_{k=1}^m|u_k|^{(p_{j,k}-\alpha_{j,k})_+},\quad
 |u_j|\leq \delta,\; 0\leq \tilde \alpha_j \leq \tilde p_j,
\\
\label{1.7}
|\partial^{\tilde \alpha_j}F_j(u)|
 \leq C_{\tilde \alpha_j,\delta},\quad
|u_j|\leq \delta,\; \tilde p_j \leq \tilde \alpha_j \leq l,
\end{gather}
and for $|u_j|\leq \delta,\ |v_j|\leq \delta,\ \tilde \alpha_j\leq
l$,
\begin{equation}\label{1.8}
\begin{split}
 & |\partial^{\tilde \alpha_j}F_j(u)-\partial^{\tilde
 \alpha_j}F_j(v)|\\
&\leq C_{\tilde \alpha_j,\delta}\sum_{\alpha_{j,1}+\dots+\alpha_{j,m}
=\tilde \alpha_j}
 \sum_{l=1}^m
 \Big\{ \prod_{k=1}^{s-1}|u_k|^{(p_{j,k}-\alpha_{j,k})_+}
 \prod_{k=s+1}^m
 |v_k|^{(p_{j,k}-\alpha_{j,k})_+}\\
& \quad\times \Big(
 |u_l|^{(p_{j,s}-\alpha_{j,s}-1)_+}
 + |v_l|^{(p_{j,s}-\alpha_{j,s}-1)_+}\Big)
 |u_s-v_s|
 \Big\},
\end{split}
\end{equation}
where
\[
\tilde p_j=\sum_{k=1}^mp_{j,k}, \quad
\tilde \alpha_j=\sum_{k=1}^m\alpha_{j,k}
\]
 with $\alpha_{j,k}\geq 0$ and $(a)_+=\max\{a,0\}$.


Our main results are the following two theorems.

\begin{theorem}\label{thm1.1}
Assume that $\tilde p_j\geq 2$, $\tilde p_j > 1+\frac{2}{n}$,
$l\geq n+1$ and the initial data
$\{(a_j,b_j)|_{j=1}^m\subset(H^{l+1}(\mathbb{R}^n)\cap
W^{l,1}(\mathbb{R}^n))\times(H^l(\mathbb{R}^n)\cap
W^{l,1}(\mathbb{R}^n))$ and
\begin{equation}
N_0 := \sum_{j=1}^m
 \big(\|a_j\|_{H^{l+1}(\mathbb{R}^n)\cap
W^{l,1}(\mathbb{R}^n)}
 + \|b_j\|_{H^l(\mathbb{R}^n)\cap
W^{l,1}(\mathbb{R}^n)} \big),
\end{equation}
is sufficiently small and the nonlinear coupling $F(u)$ satisfies
the assumptions \eqref{1.6}, \eqref{1.7} and \eqref{1.8}. Then there
exists a unique global classical solution $\{u_j(t)\}_{j=1}^m$ of
system \eqref{1.0}.

Moreover, for $j=1,2,\dots,m$,  we have the  decay estimates
\begin{equation}
\|u_j\|_{W^{l-n-1,\infty}(\mathbb{R}^n)}
 \leq C(1+t)^{-n/2},\quad\text{and}\quad
 \|u_j\|_{H^l} \leq C(1+t)^{-n/4}\,.
\end{equation}
\end{theorem}

For the solution  in the above theorem, we have the following pointwise estimates.

\begin{theorem}\label{thm1.2}
Under the assumptions of Theorem \ref{thm1.1}, if for any
multi-index $\alpha$, $|\alpha|<l$, there exist some constant
$r>n$ and a small positive constant $\varepsilon_0$, such that
\begin{equation}\label{1.11}
|\partial_x^\alpha a_j|
 + |\partial_x^\alpha b_j|
 \leq \varepsilon_0 (1+|x|^2)^{-r},
\end{equation}
then for $|\alpha|<l-n$ the solution to \eqref{1.0} satisfies:
\begin{equation}
|D_x^\alpha u_j(t)|
 \leq C(1+t)^{-\frac{n+|\alpha|}{2}}B_{\frac{n}{2}}(|x|,t),
\end{equation}
where $B_N(|x|,t)=\big(1+\frac{|x|^2}{1+t}\big)^{-N}$. And for
$p\in [1,\infty]$, we have
\begin{equation}
\|D_x^\alpha u_j(t)\|_{L^p(\mathbb{R}^n)}
 \leq C(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\alpha|}{2}}.
\end{equation}
\end{theorem}


\subsection*{Notation}
Various positive constants will be denoted by $C$.
 $W^{m,p}(\mathbb{R}^n)$, with $ m\in \mathbb{Z}_+$ and $p\in[1,\infty]$,
denotes the usual Sobolev space with the norm
\[
\|f\|_{ W^{m,p}(\mathbb{R}^n)}:=\sum_{k=0}^m\|\partial_x^kf\|_{L^p(\mathbb{R}^n)}.
\]
In particular, we use $W^{m,2}(\mathbb{R}^n)= H^m(\mathbb{R}^n)$,
$\|\cdot\|=\|\cdot\|_{ L^2(\mathbb{R}^n)}$,
$\|\cdot\|_{m,p}=\|\cdot\|_{W^{m,p}(\mathbb{R}^n)}$
and $\|\cdot\|_m=\|\cdot\|_{H^m(\mathbb{R}^n)}$.

The rest of this article is organized as follows. In the next section, we
show the pointwise estimates of the Green function. Then the
existence of global solutions is proved in Section 3. Furthermore, the
pointwise estimates of solutions for nonlinear equations are
obtained in Section 4.


\section{Green Function}

To obtain the global existence and pointwise estimates of
the solutions, we should first derive representation formulas of the
solutions through the Green function. The single linearized equation
of \eqref{1.0} is
\begin{equation}\label{1.1}
\begin{gathered}
\partial_t^2u_j-\Delta u_j+\partial_t u_j=0,\\
u_j|_{t=0}=a_j,\quad u_{jt}|_{t=0}=b_j.
\end{gathered}
\end{equation}
Then, the Green function of \eqref{1.1} can be defined as follows:
\begin{gather*}
\partial_t^2G_1-\Delta G_1+\partial_t G_1=0,\\
G_1|_{t=0}=\delta(x),\quad G_{1t}|_{t=0}=0,
\end{gather*}
 and
\begin{gather*}
\partial_t^2G_2-\Delta G_2+\partial_t G_2=0,\\
G_2|_{t=0}=0,\quad
G_{2t}|_{t=0}=\delta(x).
\end{gather*}

Now, we show the formulas of the solutions in the following theorem.
The proof of the theorem is similar to that of
\cite[Theorem 2.5]{Xu-Wang}.
We show the proof here for the convenience of the readers.

\begin{theorem} \label{thm2.1}
The solution of \eqref{1.1} is
\begin{equation}
u_j(x,t)
 = G_1\ast a_j + G_2\ast b_j +
 \int_0^t G_2(t-s)\ast F_j(u)(s){\rm d}s.
\end{equation}
\end{theorem}


\begin{proof}
It is obvious that
\begin{equation}
u_j(x,0)
 = G_1(x,0)\ast a_j + G_2(x,0)\ast b_j
 = \delta(x)\ast a_j = a_j.
\end{equation}
\begin{equation}
\begin{aligned}
u_{jt}(x,t)
&= G_{1t}(x,0)\ast a_j + G_{2t}(x,0)\ast b_j + G_2(t-t)\ast F_j(u)(t)\\
&\quad + \int_0^t G_{2t}(t-s)\ast F_j(u)(s){\rm d}s.
\end{aligned}
\end{equation}
Then, we have
\begin{gather}
u_{jt}(x,0)
 = \delta(x)\ast b_j = b_j, \\
\Delta u_j
 = \Delta G_1\ast a_j + \Delta G_2\ast b_j
 + \int_0^t \Delta G_2(t-s)\ast F_j(u)(s){\rm d}s,\\
\partial_t u_j
 = \partial_t G_1\ast a_j + \partial_t G_2\ast b_j
 + G_2(t-t)\ast F_j(u)(t) + \int_0^t
 \partial_t G_2(t-s)\ast F_j(u)(s){\rm d}s,
\\
\begin{aligned}
\partial_t^2 u_j
&=  \partial_t^2 G_1\ast a_j + \partial_t^2 G_2\ast b_j
 + G_{2t}(t-t)\ast F_j(u)(t) \\
&\quad + \int_0^t \partial_t^2 G_2(t-s)\ast F_j(u)(s){\rm d}s\\
&= \partial_t^2 G_1\ast a_j + \partial_t^2 G_2\ast b_j
 + F_j(u)(t) + \int_0^t \partial_t^2 G_2(t-s)\ast F_j(u)(s){\rm d}s.
\end{aligned}
\end{gather}
Then, by the definition of the Green function, we obtain
\begin{equation}
\begin{split}
&\partial_t^2 u_j - \Delta u_j + \partial_t u_j\\
 &= (\partial_t^2 G_1 - \Delta G_1 + \partial_t G_1) \ast a_j
  +  (\partial_t^2 G_2 - \Delta G_2 + \partial_t G_2) \ast b_j\\
 & \quad + F_j(u)(t) + \int_0^t(\partial_t^2 G_2 - \Delta G_2
  + \partial_t G_2)(t-s) \ast F_j(u)(s){\rm d}s\\
 &=  F_j(u)(t).
\end{split}
\end{equation}
The proof is complete.
\end{proof}

By the Fourier transform and a direct calculation, we have
\begin{gather}
\hat G_1(\xi,t)  = -\frac{\tau_-}{\tau_+-\tau_-}{\rm e}^{\tau_+ t}
 + \frac{\tau_+}{\tau_+-\tau_-}{\rm e}^{\tau_- t},
\\
\hat G_2(\xi,t)  = \frac{1}{\tau_+-\tau_-}{\rm e}^{\tau_+ t}
 -  \frac{1}{\tau_+-\tau_-}{\rm e}^{\tau_- t},
\end{gather}
where $\hat G_i(\xi,t)\ (i=1,2)$ are the Fourier transform
corresponding to $G_i(x,t)$ and
\begin{equation}
\tau_\pm  = \frac{-1\pm\sqrt{1-4|\xi|^2}}{2}.
\end{equation}
In what follows, we show the asymptotic behavior of $u$ by using the
pointwise estimates of $G_1$, $G_2$. First, we divide $|\xi|$ into
three cases: $|\xi|$ is small, bounded and big enough. Here, we set
\begin{equation}
\chi_1(\xi) =  \begin{cases}
 1, &\text{if }\xi <\epsilon,\\
 0, &\text{if }\xi >2\epsilon,
 \end{cases}
 \quad\text{and}\quad
 \chi_3(\xi)  =  \begin{cases}
 1, & \text{if }\xi >R,\\
 0, & \text{if }\xi <R-1,
 \end{cases}
\end{equation}
where $\epsilon$ is small enough, $R$ is large enough, and
$\chi_1, \chi_3$ are smooth functions. 
We denote $\chi_2(\xi)=1-\chi_1(\xi)-\chi_3(\xi)$. Then, we set 
$\hat G_{i,j}(\xi,t)=\chi_j(\xi)G_i(\xi,t)$ where $i=1,2;j=1,2,3$.

To deal with the coupling of Green function, we need the
following two lemmas. The following lemma corresponds to 
\cite[Lemma 3.2]{Wang-Yang}. We omit its proof here.

\begin{lemma}\label{lem2.1}
Assume that $\operatorname{supp}\hat f\subset \{\xi: |\xi|>R\}$ with 
$R$ large enough, $|\partial_\xi^\beta \hat f|\leq C|\xi|^{-|\beta|-1}$,
then there exist distributions $f_1(x),f_2(x)$, such that
$f(x)=f_1(x)+f_2(x)$. And $|\partial_x^\alpha f_1(x)|\leq
 C(1+|x|^2)^{-N}$ for positive number $2N>n+|\alpha|$,
 $\|f_2\|_{L^1}\leq C$, $\operatorname{supp} f_2(x)\subset
 \{x,|x|<2\varepsilon\}$ with $\varepsilon$ being sufficiently
 small.
\end{lemma}

 The proof of the following lemma can be founded in
\cite{Liu-Wang-0}. We omit it here.

\begin{lemma}\label{lem2.3}
If the functions $H(x,t)$ and $S(x,t)$  satisfy
\begin{gather*}
|\partial_x^\alpha H(x,t)|
 \leq  C(1+t)^{-(n+|\alpha|)/2}B_{N}(|x|,t), \\
|\partial_x^\alpha S(x,t)|
 \leq  C(1+t)^{-(2n+|\alpha|)/2}B_{n }(|x|,t),
\end{gather*}
then
\[
\Big|\partial_x^\alpha\int_0^t(H(t-\tau)\ast S(\tau)){\rm d}\tau\Big|
 \leq  C(1+t)^{-(n+|\alpha|)/2}B_{\frac{n}{2}}(|x|,t).
\]
\end{lemma}

Next we estimate $G_{i,j}(x,t)$, ($i=1,2;j=1,2,3$)
which are the inverse Fourier transform corresponding to $\hat
G_{i,j}(\xi,t)$. First of all, for $G_{i,j}$, ($i=1,2;j=1,2$), we
can use the following results from \cite{Liu-Wang}.

\begin{proposition}\label{prop2.1}
For any positive number $N$, we have
\[
|\partial_x^\alpha G_{i,1}|
 \leq  C(1+t)^{-\frac{n+|\alpha|}{2}}B_N(|x|,t), \quad i=1,2.
\]
\end{proposition}

\begin{proposition}\label{prop2.2}
There exists a positive number $c_0$, such that
\[
|\partial_x^\alpha G_{i,2}|
 \leq
 C {\rm e}^{-c_0t}B_N(|x|,t), \quad  i=1,2.
\]
\end{proposition}

For $G_{i,3}$, ($i=1,2$), we show a subtle analysis as follows.
When $|\xi|$ is large enough, using the Taylor expansion, we have
\begin{gather}
\sqrt{1-4|\xi|^2}
 = |\xi|\sqrt{|\xi|^{-2}-4}
 = 2\sqrt{-1}|\xi| -\frac{\sqrt{-1}}{4}|\xi|^{-1}
 + O(|\xi|^{-3}), \\
\label{2.16}
\frac{1}{\sqrt{1-4|\xi|^2}}
 = |\xi|^{-1}\frac{1}{\sqrt{|\xi|^{-2}-4}}
 = -\frac{\sqrt{-1}}{2}|\xi|^{-1}
 + \frac{\sqrt{-1}}{16}|\xi|^{-3}
 + O(|\xi|^{-5}),
\end{gather}
By using the Taylor expansion, we have
\begin{equation}\label{2.17}
\begin{split}
{\rm e}^{\tau_\pm t}
&= {\rm e}^{(-\frac{1}{2}\pm \sqrt{-1}|\xi|)t}
 \Big(
 1+\big(\sum_{j=1}^{k-1}(\pm
 a_j)|\xi|^{1-2j}\big)t
 + \dots\\
&\quad  +  \frac{1}{k!}\big(\sum_{j=1}^{k-1}(\pm
 a_j)|\xi|^{1-2j}\big)^kt^k
 +R^\pm(\xi,t) \Big),
\end{split}
\end{equation}
where $R^\pm (\xi,t) \leq (1+t)^{k+1}(1+|\xi|)^{1-2k}$.

Then, by using \eqref{2.16} and \eqref{2.17}, we have
\begin{gather}
 \frac{1}{\tau_+-\tau_-}{\rm e}^{\tau_+ t}
 = {\rm e}^{(-\frac{1}{2}+\sqrt{-1}|\xi|)t}
 \Big( \sum_{j=1}^{2k-2}p_j^+(t)|\xi|^{-j}+p_{2k-1}^+(t)O(|\xi|^{1-2k}) \Big),
\\
 \frac{1}{\tau_+-\tau_-}{\rm e}^{\tau_- t}
 = {\rm e}^{(-\frac{1}{2}-\sqrt{-1}|\xi|)t}
 \Big( \sum_{j=1}^{2k-2}p_j^-(t)|\xi|^{-j}+p_{2k-1}^-(t)O(|\xi|^{1-2k}) \Big),
\\
 \frac{\tau_+}{\tau_+-\tau_-}{\rm e}^{\tau_- t}
 = {\rm e}^{(-\frac{1}{2}-\sqrt{-1}|\xi|)t}
 \Big( \sum_{j=0}^{2k-2}q_j^-(t)|\xi|^{-j}+q_{2k-1}^-(t)O(|\xi|^{1-2k}) \Big),
\\
 \frac{\tau_-}{\tau_+-\tau_-}{\rm e}^{\tau_+t}
 = {\rm e}^{(-\frac{1}{2}+\sqrt{-1}|\xi|)t}
 \Big( \sum_{j=0}^{2k-2}q_j^+(t)|\xi|^{-j}+q_{2k-1}^+(t)O(|\xi|^{1-2k}) \Big).
\end{gather}
Here, $p_j^\pm,\ q_j^\pm$ are polynomials in $t$ with degree no more
then $j$.

Since $|\xi|>R$, it is observed that
\begin{equation}\label{2.23}
|\hat{G}_{1,3}(\xi,t)| +|\hat{G}_{2,3}(\xi,t)|
 \leq  C {\rm e}^{-t/4}.
 \end{equation}
Now, we take
\begin{equation}
\begin{aligned}
\hat F_{1\alpha}
&= -\chi_3(\xi)
 {\rm e}^{(-\frac{1}{2}+\sqrt{-1}|\xi|)t}
 \sum_{j=0}^{|\alpha|+n+1}q_j^+(t)|\xi|^{-j}\\
&\quad +  \chi_3(\xi)
 {\rm e}^{(-\frac{1}{2}-\sqrt{-1}|\xi|)t}
 \sum_{j=0}^{|\alpha|+n+1}q_j^-(t)|\xi|^{-j},
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
\hat F_{2\alpha}
&= \chi_3(\xi)
 {\rm e}^{(-\frac{1}{2}+\sqrt{-1}|\xi|)t}
 \sum_{j=1}^{|\alpha|+n+1}p_j^+(t)|\xi|^{-j}\\
&\quad - \chi_3(\xi)
 {\rm e}^{(-\frac{1}{2}-\sqrt{-1}|\xi|)t}
 \sum_{j=1}^{|\alpha|+n+1}p_j^-(t)|\xi|^{-j}.
\end{aligned}
\end{equation}
Then, for the high frequency part, we have the following result.

\begin{proposition}\label{prop2.3}
There exists a positive number $c_1$, such that
\begin{equation}
|\partial_x^\alpha(G_{i,3}-F_{i\alpha})|
 \leq
 C {\rm e}^{-c_1t}B_N(|x|,t),\quad i=1,2.
\end{equation}
\end{proposition}

\begin{proof}
It is obvious that
\begin{equation}
\begin{split}
|x^{ \beta}(\partial_x^\alpha(G_{i,3}-F_{i\alpha}))|
&\leq \int |\partial_\xi^{ \beta}(\xi^\alpha(\hat G_{i,3}-\hat
 F_{i\alpha}))|{\rm d}\xi\\
&\leq C {\rm e}^{-c_1t} \int |\xi|^{|\alpha|- |\beta|-(|\alpha|+n+1)-1}{\rm d}\xi\\
&\leq C {\rm e}^{-c_1t}.
\end{split}
\end{equation}
Take $|\beta|=0$ or $|\beta|=2N$, we obtain the the statement 
 of Proposition \ref{prop2.3}.
\end{proof}

\section{Global classical solutions}

The solution can be constructed in the complete metric space
\begin{equation}
X = \{u(t)=(u_1(t),u_2(t),\dots,u_m(t))|  \|u \|_X\leq E\},
\end{equation}
where $E$ is a positive constant and
\begin{align*}
\|u \|_X
&=\sup_{t\geq 0}\sum_{j=1}^m(1+t)^{\frac{n}{2}}\|u_j(\cdot,t)
 \|_{W^{l-n-1,\infty}(\mathbb{R}^n)} 
 +\sup_{t\geq  0}\sum_{j=1}^m(1+t)^{\frac{n}{4}}\|u_j(\cdot,t)
 \|_{H^{l}(\mathbb{R}^n)}.
\end{align*}
Then $(X,\|\cdot\|_X)$ is a Banach space.
Let
\begin{equation}
T[u](t):=(T_1[u](t),T_2[u](t),\dots,T_m[u](t)),
\end{equation}
where
\begin{equation}
T_j[u](t) = G_1\ast a_j + G_2\ast b_j
 + \int_0^t G_2(t-s)\ast F_j(u(s))(x){\rm d}s, \quad (1\leq j\leq m).
\end{equation}
In the following lemma, we show that $T$ is a map from $X$ to
itself.

\begin{lemma}\label{lem3.1}
If $E$ and $N_0$ are sufficiently small with $N_0\ll E$, then $T$ is
a map from $X$ to $X$.
\end{lemma}

\begin{proof}
Firstly, we note that
 \begin{align*}
 \|T_j[u](t)\|_{l-n-1,\infty}
 & \leq
 \|(G_1-F_{1\alpha})(t)\ast a_j\|_{l-n-1,\infty}
 +\|(G_2-F_{2\alpha})(t)\ast b_j\|_{l-n-1,\infty}\\
 & \quad  +\|F_{1\alpha}(t)\ast a_j\|_{l-n-1,\infty}
 +\|F_{2\alpha}(t)\ast b_j\|_{l-n-1,\infty}\\
 & \quad  +
 \int_0^t \|(G_2-F_{2\alpha})(t-\tau)\ast
 F_j(u)(\tau)\|_{l-n-1,\infty}{\rm d}\tau\\
 & \quad  +
 \int_0^t\|F_{2\alpha}(t-\tau)\ast
 F_j(u)(\tau)\|_{l-n-1,\infty}{\rm d}\tau\\
 & :=  \sum_{i=1}^6 I_i.
 \end{align*}
For $I_1$, it follows from the Young inequality and Propositions
\ref{prop2.1}--\ref{prop2.3} that
\begin{equation}\label{(3.8)}
I_1
\leq  \|(G_1-F_{1\alpha})(t)\|_{L^\infty} \|a_j\|_{l-n-1,1}
\leq  C (1+t)^{-n/2}\|a_j\|_{l-n-1,1}.
\end{equation}
Similarly to the estimates of $I_1$, we obtain
\begin{equation}\label{(3.9)}
I_2 \leq  C (1+t)^{-n/2}\|b_j\|_{l-n-1,1}.
 \end{equation}
By noticing $|\alpha|\leq l-n-1$ and the definition of
$F_{1\alpha}$, for some positive number $c_2$, we have
\begin{equation}
\begin{split}
|\partial_x^\alpha F_{1\alpha}\ast a_j|
 &  \leq \int|\xi^\alpha\hat F_{1\alpha}\hat a_j|{\rm d}\xi\\
 & \leq  C \|a_j\|_{ {l,1}} {\rm e}^{-c_2t}\int\chi_3(\xi)|\xi|^{|\alpha|-l}{\rm
 d}\xi\\
 & \leq C\|a_j\|_{l,1}{\rm e}^{-c_2t}.
\end{split}
\end{equation}
Similarly, we have
\begin{equation}
|\partial_x^\alpha F_{2\alpha}\ast b_j|
 \leq  C\|b_j\|_{l-1,1}{\rm e}^{-c_2t}.
\end{equation}
Then, we obtain
\begin{equation}\label{(3.10)}
I_3  \leq  C(1+t)^{-n/2}\|a_j\|_{l,1} \quad\text{and}\quad
I_4  \leq C(1+t)^{-n/2}\|b_j\|_{l-1,1}.
 \end{equation}

To estimate $I_5$ and $I_6$, we give the estimates of
$F_j(u)(t)$ as follows:
\begin{equation}\label{3.10}
\begin{split}
|\partial_x^\alpha F_j(u)(t)|
&\leq |\partial_u^1F_j(u)(t)|\sum_{i=1}^m|\partial_x^\alpha
 u_i|\\
&\quad + |\partial_u^2F_j(u)(t)|\sum_{1\leq k_1,k_2\leq m; \alpha_{1}
+\alpha_{2}=\alpha}|\partial_x^{\alpha_1} u_{k_1} \partial_x^{\alpha_2}
 u_{k_2}|
 + \dots \\
&\quad +
 |\partial_u^{\alpha}F_j(u)(t)|\sum_{1\leq k_1,\dots,k_m\leq m,\ \alpha_{1}
+\dots+\alpha_{m}=\alpha}
 |\partial_x^{\alpha_1} u_{k_1} \dots \partial_x^{\alpha_m}
 u_{k_m}|,
\end{split}
\end{equation}
where $0\leq \alpha_k\leq \alpha$, ($k=1,\dots,m$).

Then, by using \eqref{1.6}, \eqref{1.7}, \eqref{3.10}, the
 H\"older inequality $(\|f g\|_{L^1}\leq
\|f\|_{L^2}\|g\|_{L^2})$ and the assumption $\|u\|_X\leq E$, we have
\begin{gather}\label{3.11}
\| F_j(u)(t)\|_{l-n-1,1}
 \leq  C(1+t)^{-\frac{n}{2}(\tilde p_j-1)}E^{\tilde p_j}, \\
\label{3.12}
\|F_j(u)(t)\|_{l }
 \leq  C(1+t)^{-\frac{n}{2}(\tilde p_j-1)-\frac{n}{4}}E^{\tilde p_j}.
\end{gather}

Using the Young inequality, \eqref{3.11} and Proposition
\ref{prop2.1} and noticing $\tilde p_j>1+\frac{2}{n}$, for
$I_5$, we have
 \begin{align*}
I_5 & \leq
 \int_0^t\|(G_1-F_{1\alpha})(t-\tau)\|_{L^\infty} 
\|F_j(u)(\tau)\|_{l-n-1,1}{\rm d}\tau\\
& \leq
 C \int_0^t(1+t-\tau)^{-n/2}\|F_j(u)(\tau)\|_{l-n-1,1}{\rm d}\tau\\
& \leq
 C E^{\tilde p_j}
 \int_0^t(1+t-\tau)^{-n/2}
 (1+\tau)^{-\frac{n}{2}(\tilde p_j-1)}{\rm d}\tau\\
& \leq
 C(1+t)^{-n/2} E^{\tilde p_j}.
 \end{align*}

For $I_6$, it follows from Lemma \ref{lem2.1}, \eqref{3.12} and
the Sobolev inequality that
\begin{equation}\label{(3.12)}
 \begin{split}
I_6
 & \leq
 C\int_0^t {\rm e}^{-(t-\tau)/4}
 \|(f_1+f_2)\ast F_j(u)(\tau)\|_{l-n-1,1}{\rm d}\tau\\
 & \leq
 C\int_0^t {\rm e}^{-(t-\tau)/4}
 (\|f_1\|_{L^1}+\|f_2\|_{L^1}) \|F_j(u)(\tau)\|_{l-n-1,\infty}{\rm d}\tau\\
 & \leq
 C\int_0^t {\rm e}^{-(t-\tau)/4}
 \|F_j(u)(\tau)\|_{l-n-1,\infty}{\rm d}\tau\\
 & \leq
 C\int_0^t {\rm e}^{-(t-\tau)/4}
 \|F_j(u)(\tau)\|_{l-[\frac{n}{2}]}{\rm d}\tau\\
 & \leq
 CE^{\tilde p_j}
 \int_0^t {\rm e}^{-(t-\tau)/4}
 (1+\tau)^{-\frac{n}{2}(\tilde p_j-1)-\frac{n}{4}}{\rm d}\tau\\
 &  \leq
 C (1+t)^{-n/2} E^{\tilde p_j}.
\end{split}
\end{equation}
Thus, the combination of \eqref{(3.8)}-\eqref{(3.12)} gives
\begin{equation}\label{(3.13)}
 \begin{split}
&\|T_j[u](t)\|_{l-n-1,\infty}\\
& \leq  C(1+t)^{-n/2}
\big( \|a_j\|_{l,1}+\|b_j\|_{l-1,1}
 +\|a_j\|_{l-n-1,1}+\|b_j\|_{l-n-1,1}
 +E^{\tilde p_j}\big).
 \end{split}
 \end{equation}
To estimate $H^l$ norm of $T_j[u](t)$, we consider
 \begin{align*}
 \| T_j[u](t)\|_l
 &  \leq  \|(G_1-G_{1,3})(t)\ast a_j\|_l
 +\|\partial_t(G_2-G_{2,3})(t)\ast b_j\|_l\\
 &  \quad  +\|G_{1,3}(t)\ast a_j\|_l
 +\|G_{2,3}(t)\ast b_j\|_l\\
 &  \quad  +
 \int_0^t \|(G_2-G_{2,3})(t-\tau)\ast
 F_j(u)(\tau)\|_l{\rm d}\tau\\
&\quad + \int_0^t \|G_{2,3}(t-\tau)\ast
 F_j(u)(\tau)\|_l{\rm d}\tau\\
 &: =  \sum_{i=1}^6 J_i.
 \end{align*}
By using Propositions \ref{prop2.1}-\ref{prop2.2}
and the Young inequality, for $J_1$, we obtain
\begin{equation}\label{(3.14)}
J_1  \leq  \|(G_1-G_{1,3})(t)\| \|a_j\|_{l,1 }
 \leq  C (1+t)^{-n/4}\|a_j\|_{l,1 }.
 \end{equation}
For $J_3$, it follows from the Plancherel theorem and \eqref{2.23}
that
\begin{equation}\label{(3.15)}
J_3   \leq
 \sum_{|\alpha|=0}^l\|G_{1,3}(t)\ast \partial_x^\alpha a_j\|
 =  \sum_{|\alpha|=0}^l\|\hat{G}_{1,3}(t)
 \widehat{\partial_x^\alpha a_j}\|
  \leq  C {\rm e}^{-t/4} \|a_j\|_l.
\end{equation}
Similar to the estimates of $J_1$ and $J_3$, we obtain
\begin{equation}\label{(3.16)}
J_2  \leq  C (1+t)^{-n/4}\|b_j\|_{l,1 },\quad\text{and}\quad
J_4  \leq  C {\rm e}^{-t/4} \|b_j\|_l.
 \end{equation}
Using Propositions \ref{prop2.1}-\ref{prop2.2}, the
Young inequality and noticing $\tilde p_j>1+\frac{2}{n}$, we have
\begin{equation}\label{(3.17)}
 \begin{split}
J_5
 & \leq
 C \int_0^t\|(G_2-G_{2,3})(t-\tau)\| \|F_j(u)(\tau)\|_{l }{\rm d}\tau\\
 & \leq
 C E^{\tilde p_j}\int_0^t (1+t-\tau)^{-n/4}(1+t)^{-\frac{n}{2}(\tilde p_j-1)
-\frac{n}{4}}{\rm d}\tau\\
 & \leq
 C (1+t)^{-n/4} E^{\tilde p_j}.
\end{split}
\end{equation}

For $J_6$, by using the Plancherel theorem and \eqref{2.23}, we get
\begin{equation}\label{(3.18)}
J_6
 \leq  C \int_0^t {\rm e}^{-(t-\tau)/4} \|F_j(u)(\tau)\|_l{\rm d}\tau
 \leq  C (1+t)^{-n/4} E^{\tilde p_j}.
\end{equation}
Thus, we obtain
\begin{equation}\label{(3.19)}
\|T_j[u](t)\|_l
 \leq  C(1+t)^{-n/4}\left(\|a_j\|_l+\|b_j\|_l
 +\|a_j\|_{l,1}+\|b_j\|_{l,1}+E^{\tilde p_j}\right).
\end{equation}
By using \eqref{(3.13)}, \eqref{(3.19)}, $\tilde p_j>1+\frac{2}{n}$
and the small assumptions of $E$ and $N_0$ with $N_0\ll E$, we get
$\|T[u](t)\|_X\leq E$. Thus, the proof of Lemma \ref{lem3.1} is
complete.
\end{proof}

Next, we proof that this map $T$ is a contraction mapping.

\begin{lemma}\label{lem3.2}
Assume $u,v\in X$ and $E>0$ is sufficiently small, then there exists
a constant $\gamma$ with $0<\gamma<1$, such that
\[
\|T[u]-T[v]\|_X  \leq  \gamma \|u-v\|_X.
\]
\end{lemma}

\begin{proof}
By the Duhamel principle and the triangle inequality, we have
\begin{equation}\label{(3.22)}
 \begin{split}
&\|T[u]-T[v]\|_{l-n-1,\infty}\\ & \leq
 \int_0^t\left\|(G_2-F_{2\alpha})(t-\tau)\ast
 (F_j(u)-F_j(v))(\tau)\right\|_{l-n-1,\infty}{\rm d}\tau\\
 & \quad + \int_0^t \left\|F_{2\alpha}(t-\tau)\ast
 (F_j(u)-F_j(v))(\tau)\right\|_{l-n-1,\infty}{\rm d}\tau\\
 &  :=H_1+H_2.
\end{split}
\end{equation}
By a directly calculation, we have
\begin{equation}\label{3.23}
\begin{split}
 & |\partial_x^\alpha F(u)-\partial_x^\alpha F(v)|\\
 &\leq \Big|\sum_{k=1}^m\partial_{u_k}^1 F(u)\partial_x^\alpha  u_k
 - \sum_{s=1}^m\partial_{v_s}^1 F(v)\partial_x^\alpha v_s \Big|\\
&\quad +  \Big|
 \sum_{1\leq k_1,k_2\leq m;\alpha_{k_1}+\alpha_{k_2}=\alpha}
 \partial_{u_{k_1}u_{k_2}}^2 F(u)\partial_x^{\alpha_{k_1}} u_{k_1}
 \partial_x^{\alpha_{k_2}}
 u_{k_2} \\
&  \quad - \sum_{1\leq s_1,s_2\leq m;\alpha_{s_1}+\alpha_{s_2}=\alpha}
 \partial_{v_{s_1}v_{s_2}}^2 F(v)\partial_x^{\alpha_{s_1}} v_{s_1}
\partial_x^{\alpha_{s_2}}
 v_{s_2}\Big|+ \dots \\
&\quad   + \Big|  \sum_{1\leq k_i\leq m,i=1,\dots,\alpha;\alpha_{k_1}
 +\dots+\alpha_{k_\alpha}=\alpha}
 \partial_{u_{k_1}u_{k_2}\dots u_{k_\alpha}}^{\alpha} F(u)
\partial_x^{\alpha_{k_1}} u_{k_1}
 \dots  \partial_x^{\alpha_{k_\alpha}}
 u_{k_\alpha} \\
&\quad  -  \sum_{1\leq s_i\leq m,i=1,\dots,\alpha;\alpha_{s_1}
 +\dots+\alpha_{s_\alpha}=\alpha}
 \partial_{v_{s_1}v_{s_2}\dots v_{s_\alpha}}^{\alpha} 
 F(u)\partial_x^{\alpha_{s_1}}  v_{s_1}
 \dots
 \partial_x^{\alpha_{s_\alpha}} v_{s_\alpha}\Big|\\
&\leq \sum_{i=1}^m|\partial_u^1 F(u)\partial_x^\alpha u_i
 - \partial_v^1 F(v)\partial_x^\alpha v_i |\\
&\quad  +  \sum_{1\leq k,s\leq m;\alpha_k+\alpha_s=\alpha}
 |\partial_u^2 F(u)\partial_x^{\alpha_k} u_k\partial_x^{\alpha_s}
 u_s
 - \partial_v^2 F(v)\partial_x^{\alpha_k} v_k\partial_x^{\alpha_s} v_s| + \dots\\
&\quad  + \sum_{1\leq k_i\leq m,i=1,\dots,\alpha;\alpha_{k_1}+\dots
 +\alpha_{k_\alpha}=\alpha}
 |\partial_{u}^{\alpha} F(u)\partial_x^{\alpha_{k_1}} u_{k_1}  \dots
 \partial_x^{\alpha_{k_\alpha}}
 u_{k_\alpha}\\
&\quad - \partial_{v}^{\alpha} F(v)\partial_x^{\alpha_{k_1}}
 v_{k_1} \dots \partial_x^{\alpha_{k_\alpha}}
 v_{k_\alpha}|.
\end{split}
\end{equation}
Using \eqref{3.23} and the assumption \eqref{1.8}, we have
\begin{equation}
\begin{split}
 &\| F_j(u) - F_j(v)\|_{l,1}\\
&\leq C \sum_{s=1}^m
 \sum_{ \tilde \alpha_j =1}^l
 \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde
 \alpha_j}
 \prod_{k=1}^{s-1}\|u_k\|_{L^\infty}^{p_{j,k}-\alpha_{j,k}}\\
&\quad \times \prod_{k=s+1}^m\|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+}
 \big( \|u_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+}
 + \|v_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+} \big)\\
&\quad  \times
 \Big( \|u_s-v_s\|_{ 2}
 \sum_{i=1}^m(\|u_i\|_{l }+\|v_i\|_{l }) \Big) \Big)\\
&\quad  + C \sum_{s=1}^m
 \sum_{ \tilde \alpha_j =1}^l
 \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde
 \alpha_j}
 \prod_{k=1}^{s-1} \|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+}\\
&\quad \times  \prod_{k=s+1}^m \|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+}
 \big(\|v_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+}
 \|v_s\|_{L^2}\big)
 \Big)\\
&\quad\times  \sum_{s=1}^m  \sum_{ \tilde \alpha_j =1}^l
 \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde \alpha_j}
 \prod_{k=1}^{s-1}\|\partial_x^{\alpha_k}u_k\|_{L^\infty}
 \prod_{k=s+1}^m\|\partial_x^{\alpha_k}v_k\|_{L^\infty}
 \|u_s-v_s\|_{ l} \Big).
\end{split}
\end{equation}

For $H_1$, by using the Young inequality and Propositions
\ref{prop2.1}--\ref{prop2.3} and noticing the
definition of $\|\cdot\|_X$ and $\tilde p_j>1+\frac{2}{n}$, we have
\begin{equation}\label{(3.23)}
 \begin{split}
H_1 &  \leq \int_0^t \|(G_2-F_{2\alpha})(t-\tau)\|_{L^\infty}
 \|(F_j(u)-F_j(v))(\tau)\|_{l-n-1,1}{\rm d}\tau\\
 & \leq C \int_0^t (1+t-\tau)^{-n/2}
 \|(F_j(u)-F_j(v))(\tau)\|_{l-n-1,1}{\rm d}\tau\\
 & \leq C E^{\tilde p_j -1} \int_0^t (1+t-\tau)^{-n/2}(1+\tau)^{-n/2(\tilde p_j-2)-n/4}
 \|u-v\|_X{\rm d}\tau\\
 &  \leq C E^{\tilde p_j-1}(1+t)^{-n/2}
 \|u-v\|_X.
\end{split}
\end{equation}
Similarly, by using \eqref{3.23}, we have
\begin{equation}
\begin{split}
&\| F_j(u) - F_j(v)\|_{l}\\
&\leq C \sum_{s=1}^m  \sum_{ \tilde \alpha_j =1}^l
 \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde
 \alpha_j}
 \prod_{k=1}^{s-1}\|u_k\|_{L^\infty}^{p_{j,k}-\alpha_{j,k}} \\
&\quad \times
 \prod_{k=s+1}^m\|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+}
 \big( \|u_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+}
 + \|v_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+} \big)\\
&\quad\times
 \Big( \|u_s-v_s\|_{L^\infty}
 \sum_{i=1}^m(\|u_i\|_{l }+\|v_i\|_{l }) \Big) \Big)\\
&\quad + C \sum_{s=1}^m \sum_{ \tilde \alpha_j =1}^l
 \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde
 \alpha_j}
 \prod_{k=1}^{m} \|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+} \\
&\quad \times
 \sum_{s=1}^m  \sum_{ \tilde \alpha_j =1}^l
 \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde
 \alpha_j}
 \prod_{k=1}^{s-1}\|\partial_x^{\alpha_k}u_k\|_{L^\infty}
 \prod_{k=s+1}^m\|\partial_x^{\alpha_k}v_k\|_{L^\infty}
 \|u_s-v_s\|_{ l} \Big).
\end{split}
\end{equation}

For $H_2$, similar to the estimate of $I_6$, it follows from Lemma
\ref{lem2.1} and the Sobolev inequality that
\begin{align*}
 H_2 &  \leq
 C \int_0^t {\rm e}^{-(t-\tau)/4}
 \|(F_j(u)-F_j(v))(\tau)\|_{l-n-1,\infty}{\rm d}\tau\\
 &
 \leq  C \int_0^t {\rm e}^{-(t-\tau)/4}
 \|(F_j(u)-F_j(v))(\tau)\|_{l-[\frac{n}{2}]}{\rm d}\tau\\
 &
 \leq  CE^{\tilde p_j} \int_0^t {\rm e}^{-(t-\tau)/4} (1+\tau)^{-n/2(\tilde p_j-1)-n/4}
 \|u-v\|_X{\rm d}\tau\\
 &
 \leq  C E^{\tilde p_j} (1+t)^{-n/2} \|u-v\|_X,
\end{align*}
where we used $\tilde p_j>1+\frac{2}{n}$.
Then, we obtain
\begin{equation}\label{(3.24)}
\|T[u]-T[v]\|_{l-n-1,\infty}
 \leq  C E^{\tilde p_j} (1+t)^{-n/2}  \|u-v\|_X.
\end{equation}

On the other hand, by using the Young inequality, the Plancherel
theorem, \eqref{2.23} and Propositions
 \ref{prop2.1}-\ref{prop2.2}, we have
\begin{align*}
\|T[u]-T[v]\|_l
 &  \leq
 \int_0^t \|(G_2-G_{2,3})(t-\tau)\ast
 (F_j(u)-F_j(v))(\tau)\|_l{\rm d}\tau\\
 & \quad  +  \int_0^t \left\|G_{2,3}(t-\tau)(t-\tau)\ast
 (F_j(u)-F_j(v))(\tau)\right\|_l{\rm d}\tau\\
 &  \leq  C \int_0^t (1+t-\tau)^{-n/4}
 \left(\|F_j(u)-F_j(v)\|_{l,1}+\|F_j(u)-F_j(v)\|_l\right){\rm d}\tau\\
 &  \leq C E^{\tilde p_j} (1+t)^{-n/4}  \|u-v\|_X.
\end{align*}
Then
\begin{equation}\label{(3.25)}
\|T[u]-T[v]\|_l \leq  C E^{\tilde p_j} (1+t)^{-n/4} \|u-v\|_X.
\end{equation}
Combining \eqref{(3.24)} with \eqref{(3.25)}, we obtain
$\|Tu-Tv\|_X\leq CE^{\tilde p_j} \|u -v\|_X$. Since the smallness
assumption of $E$ and $\tilde p_j >1+\frac{2}{n}$, we complete the
proof of Lemma \ref{lem3.2}.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.1}] 
Lemmas \ref{lem3.1} and \ref{lem3.2} show that for sufficiently
small initial data 
\[
(a_j,b_j)\in (W^{l,1}(\mathbb{R}^n)\cap
H^{l+1}(\mathbb{R}^n))\times (W^{l,1}(\mathbb{R}^n)\cap
H^{l}(\mathbb{R}^n))
\]
for $j=1,2,\dots,m$, $T: X\to X$ is a contraction mapping.
By the Banach fixed point theorem, there exists a fixed point
 $u\in X$. Here, we obtain the solution $\{u_j(t)\}_{j=1}^m$ to system
\eqref{1.0} satisfies $\|u\|_X\leq E$. Then, the proof is complete.
\end{proof}

\section{Pointwise estimates}

In this section, we show the pointwise estimates of the solutions to
system \eqref{1.0}. First of all, we recall
\begin{equation}\label{4.1}
\begin{split}
&\partial_x^\alpha u_j(x,t)\\
& = \partial_x^\alpha G_1\ast a_j  + \partial_x^\alpha G_2\ast b_j
 +  \partial_x^\alpha \int_0^t  G_2(t-s)\ast F_j(u)(s){\rm d}s\\
& = (\partial_x^\alpha (G_1-F_{1\alpha})(t))\ast a_j
 + (\partial_x^\alpha (G_2-F_{2\alpha})(t))\ast b_j
 + \partial_x^\alpha F_{1\alpha}(t)\ast a_j
 + \partial_x^\alpha F_{2\alpha}(t)\ast b_j\\
 & \quad  + \int_0^t (\partial_x^\alpha (G_2-F_{2\alpha})(t-\tau))\ast
 F_j(u)(\tau) {\rm d}\tau  + \int_0^t \partial_x^\alpha F_{2\alpha}(t-\tau)\ast
 F_j(u)(\tau) {\rm d}\tau .
\end{split}
\end{equation}

From Propositions \ref{prop2.1}--\ref{prop2.2} and
the assumption \eqref{1.11}, by using Lemma \ref{lem2.3}, we have
\begin{equation}
\begin{split}
&|(\partial_x^\alpha (G_1-F_{1\alpha})(t))\ast a_j
+ (\partial_x^\alpha (G_2-F_{2\alpha})(t))\ast b_j|\\
&\leq C\varepsilon_0(1+t)^{-\frac{n+|\alpha|}{2}}B_r(|x|,t).
\end{split}
\end{equation}
For some positive number $b$, by noticing the definition of
$F_{1\alpha}$, $|\alpha|<l$ and assumption \eqref{1.11} with $r>n$,
we have
\begin{equation}
|x^\beta \partial_x^\alpha F_{1\alpha}\ast a_j|
\leq \int |\partial_\xi^\beta\xi^\alpha
 \hat F_{1\alpha}\hat a_j| {\rm d}\xi
\leq  C \varepsilon_0 {\rm e}^{-bt}.
\end{equation}
Take $|\beta|=0$ or $|\beta|=n$, we obtain
\begin{equation}
|\partial_x^\alpha F_{1\alpha}\ast a_j|
\leq  C \varepsilon_0 {\rm e}^{-\frac{b}{2}t}
 B_\frac{n}{2}(|x|,t).
\end{equation}
Similarly, we have
\begin{equation}
|\partial_x^\alpha F_{2\alpha}\ast b_j|
 \leq  C \varepsilon_0 {\rm e}^{-bt/2} B_r(|x|,t).
\end{equation}
To estimate the other parts in \eqref{4.1}, we set
\begin{equation}
\varphi_\alpha (x,t)
 = (1+t)^{\frac{n+|\alpha|}{2}}
 (B_\frac{n}{2}(|x|,t))^{-1},
\end{equation}
and
\begin{equation}
M(t) = \sup_{{0\leq s,\tau\leq t,}\
 {|\alpha|\leq l}}
 \sum_{j=1}^m|\partial_x^\alpha
 u_j(x,\tau)|\varphi_\alpha(x,s).
\end{equation}
When $|\alpha|\leq l-1$, from the assumptions \eqref{1.6},
\eqref{1.7} and the definition of $M$, we have
\begin{equation}\label{4.8}
|\partial_x^\alpha F_j(u)(x,t)|
\leq  M^2(t)(1+t)^{-n-\frac{|\alpha|}{2}}B_n(|x|,t).
\end{equation}
When $|\alpha|= l$, from the definition of $M$, by using Theorem
\ref{thm1.1} and Lemma \ref{lem3.1}, we have
\begin{equation}\label{4.9}
|\partial_x^\alpha F_j(u)(x,s)|
 \leq  M^2(t)(1+t)^{-n-\frac{|\alpha|}{2}}B_n(|x|,t)
 + EM(t)(1+t)^{-n-\frac{|\alpha|}{2}}B_{\frac{n}{2}} (|x|,t).
\end{equation}
Set 
\[
R^\alpha=\big|\int_0^tF_{2\alpha}(t-s)\ast
\partial_x^\alpha F_j(u)(s){\rm d}s\big|.
\]
 From Lemma \ref{lem2.1} and \eqref{4.8} and \eqref{4.9}, we obtain
\begin{equation}
\begin{split}
R^\alpha 
&\leq  \big| \int_0^t {\rm e}^{-b(t-s)}
 (f_1+f_2)\ast \partial_x^\alpha F_j(u)(s){\rm  d}s\big|\\
&\leq \big| \int_0^t {\rm e}^{-b(t-s)}
 f_1 \ast \partial_x^\alpha F_j(u)(s){\rm  d}s\big|
+\big| \int_0^t {\rm e}^{-b(t-s)}  f_2 \ast \partial_x^\alpha F_j(u)(s){\rm
 d}s\big|\\
&:= R_1^\alpha + R_2^\alpha.
\end{split}
\end{equation}
The right-hand side of the above inequality can be estimated as
follows.
\begin{align*}
&R_2^\alpha\\
&\leq \big|\int_0^t {\rm e}^{-b(t-s)} \int_{\mathbb{R}^n} f_2(x-y)
 \partial_x^\alpha F_j(u)(y,s)|_{|y-x|<\varepsilon}{\rm d}y
 {\rm d}s\big|\\
&\leq \int_0^t {\rm e}^{-b(t-s)}
 \int_{\mathbb{R}^n}  |f_2(x-y)|
 (M^2(t)+EM(t))(1+t)^{-n-\frac{|\alpha|}{2}}
 B_{\frac{n}{2}} (|y|,t)_{|y-x|<\varepsilon}{\rm d}y{\rm d}s\\
&\leq \int_0^t {\rm e}^{-b(t-s)} \|f_2\|_{L^1}
 (M^2(t)+EM(t))(1+t)^{-n-\frac{|\alpha|}{2}}
 B_{\frac{n}{2}} (|x|,t) {\rm d}s\\
&\leq C(1+t)^{-\frac{n+|\alpha|}{2}} (M^2(t)+EM(t))
 B_{\frac{n}{2}}(|x|,t).
\end{align*}
Since 
\[
|\partial_x^\alpha ({\rm e}^{-bt}f_1(x))|
 \leq C (1+t)^{-\frac{n+|\alpha|+1}{2}}B_N(|x|,t),
\]
 we have
\[
R_1^\alpha
= \big| \int_0^t {\rm e}^{-b(t-s)} f_1\ast F_j(u)(s){\rm d}s\big|
\leq  C(1+t)^{-\frac{n+|\alpha|}{2}} (M^2(t)+M(t)E)B_{\frac{n}{2}}(|x|,t).
\]
From the definition of $M$, we have
\begin{equation}
|\partial_x^\alpha F_j(u)(x,s)|
\leq  CM^2(t) (1+s)^{-n-\frac{|\alpha|}{2}} B_n(|x|,s).
\end{equation}
From Proposition \ref{prop2.3} and Lemma \ref{lem2.3}, we
obtain
\begin{equation}
\begin{aligned}
&\big|  \int_0^t  (\partial_x^\alpha(G_2(t-s)-F_{2\alpha}))
 \ast F_j(u)(s){\rm d}s \big|\\
&\leq  C(M^2(t)+M(t)E)(1+t)^{-\frac{n+|\alpha|}{2}}
 B_{\frac{n}{2}}(|x|,t).
\end{aligned}
\end{equation}
From the above inequalities and the definition of $M$, we obtain
\begin{equation}
M(t) \leq C\big(M^2(t) + E M(t)+\varepsilon_0 \big).
\end{equation}
Since $E$ and $\varepsilon_0$ are small enough, we have 
$M(t)\leq C$. It yields that
\begin{equation}
|\partial_x^\alpha u_j(t)|
\leq  C (1+t)^{-\frac{n+|\alpha|}{2}} B_{\frac{n}{2}}(|x|,t).
\end{equation}
Thus, we can easily obtain the optimal $L^p$, $1\leq p\leq \infty$,
convergence rate as follows.

\begin{corollary}
Under the assumptions of Theorem \ref{thm1.1}, for
$p\in [1,\infty]$, $|\alpha|\leq l$, we have
\begin{equation}
\|\partial_x^\alpha u_j(\cdot,t)\|_{L^p}
 \leq  C(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\alpha|}{2}},\quad
 j=1,\dots,m.
\end{equation}
\end{corollary}

Thus, we have complete the proof of Theorem \ref{thm1.2}.

\subsection*{Acknowledgments}
This work was supported by NSFC (No. 11201300), Shanghai university
young teacher training program (No. slg11032)
and in part by NSFC (No.11171220, No.11171212).


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\end{document}