\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 $$\label{1.0} \begin{gathered} \partial_t^2u-\Delta u+\partial_tu=F(u),\quad t>0,x\in \mathbb{R}^n,\\ u(0,x)=a(x),\quad \partial_tu(0,x)=b(x),\quad x\in \mathbb{R}^n, \end{gathered}$$ 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 $$F_j(u) = O\Big(\prod_{k=1}^l u_k^{p_{j,k}}\Big),$$ 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 $$\|u_j(t)\|_{L^\infty} \leq C(1+t)^{-n/2}, \quad \|u_j(t)\|_{L^2} \leq C(1+t)^{-n/4},\quad j=1,\dots,m.$$ 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 $$\label{1.3} \begin{gathered} \partial_t^2u-\Delta u+\partial_tu=f(u),\quad t>0,x\in \mathbb{R}^n,\\ u(0,x)=a_1(x),\quad \partial_tu(0,x)=b_1(x),\quad x\in \mathbb{R}^n, \end{gathered}$$ 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 $$\label{1.5} \begin{gathered} \partial_t^2u_1-\Delta u_1+\partial_tu_1=|u_m|^{p_1},\quad t>0,\;x\in \mathbb{R}^n,\\ \partial_t^2u_2-\Delta u_2+\partial_tu_2=|u_1|^{p_2},\quad t>0,\;x\in \mathbb{R}^n,\\ \dots\\ \partial_t^2u_m-\Delta u_m+\partial_tu_m=|u_{m-1}|^{p_m},\quad t>0,x\in \mathbb{R}^n,\\ u_j(0,x)=a_j(x),\quad \partial_tu_j(0,x)=b_j(x),\quad x\in \mathbb{R}^n, \; (1\leq j\leq m), \end{gathered}$$ 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 $|u_j|\leq \delta,\ |v_j|\leq \delta,\ \tilde \alpha_j\leq l$, $$\label{1.8} \begin{split} & |\partial^{\tilde \alpha_j}F_j(u)-\partial^{\tilde \alpha_j}F_j(v)|\\ &\leq C_{\tilde \alpha_j,\delta}\sum_{\alpha_{j,1}+\dots+\alpha_{j,m} =\tilde \alpha_j} \sum_{l=1}^m \Big\{ \prod_{k=1}^{s-1}|u_k|^{(p_{j,k}-\alpha_{j,k})_+} \prod_{k=s+1}^m |v_k|^{(p_{j,k}-\alpha_{j,k})_+}\\ & \quad\times \Big( |u_l|^{(p_{j,s}-\alpha_{j,s}-1)_+} + |v_l|^{(p_{j,s}-\alpha_{j,s}-1)_+}\Big) |u_s-v_s| \Big\}, \end{split}$$ 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 $$N_0 := \sum_{j=1}^m \big(\|a_j\|_{H^{l+1}(\mathbb{R}^n)\cap W^{l,1}(\mathbb{R}^n)} + \|b_j\|_{H^l(\mathbb{R}^n)\cap W^{l,1}(\mathbb{R}^n)} \big),$$ 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 $$\|u_j\|_{W^{l-n-1,\infty}(\mathbb{R}^n)} \leq C(1+t)^{-n/2},\quad\text{and}\quad \|u_j\|_{H^l} \leq C(1+t)^{-n/4}\,.$$ \end{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 $\alpha$, $|\alpha|n$ and a small positive constant $\varepsilon_0$, such that $$\label{1.11} |\partial_x^\alpha a_j| + |\partial_x^\alpha b_j| \leq \varepsilon_0 (1+|x|^2)^{-r},$$ then for $|\alpha|2\epsilon, \end{cases} \quad\text{and}\quad \chi_3(\xi) = \begin{cases} 1, & \text{if }\xi >R,\\ 0, & \text{if }\xi R\}$ 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 $$\label{2.17} \begin{split} {\rm e}^{\tau_\pm t} &= {\rm e}^{(-\frac{1}{2}\pm \sqrt{-1}|\xi|)t} \Big( 1+\big(\sum_{j=1}^{k-1}(\pm a_j)|\xi|^{1-2j}\big)t + \dots\\ &\quad + \frac{1}{k!}\big(\sum_{j=1}^{k-1}(\pm a_j)|\xi|^{1-2j}\big)^kt^k +R^\pm(\xi,t) \Big), \end{split}$$ 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 $$\label{2.23} |\hat{G}_{1,3}(\xi,t)| +|\hat{G}_{2,3}(\xi,t)| \leq C {\rm e}^{-t/4}.$$ Now, we take \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} and \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} 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 $$|\partial_x^\alpha(G_{i,3}-F_{i\alpha})| \leq C {\rm e}^{-c_1t}B_N(|x|,t),\quad i=1,2.$$ \end{proposition} \begin{proof} It is obvious that $$\begin{split} |x^{ \beta}(\partial_x^\alpha(G_{i,3}-F_{i\alpha}))| &\leq \int |\partial_\xi^{ \beta}(\xi^\alpha(\hat G_{i,3}-\hat F_{i\alpha}))|{\rm d}\xi\\ &\leq C {\rm e}^{-c_1t} \int |\xi|^{|\alpha|- |\beta|-(|\alpha|+n+1)-1}{\rm d}\xi\\ &\leq C {\rm e}^{-c_1t}. \end{split}$$ 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 $$X = \{u(t)=(u_1(t),u_2(t),\dots,u_m(t))| \|u \|_X\leq E\},$$ 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 $$T[u](t):=(T_1[u](t),T_2[u](t),\dots,T_m[u](t)),$$ where $$T_j[u](t) = G_1\ast a_j + G_2\ast b_j + \int_0^t G_2(t-s)\ast F_j(u(s))(x){\rm d}s, \quad (1\leq j\leq m).$$ 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 $$\label{(3.8)} I_1 \leq \|(G_1-F_{1\alpha})(t)\|_{L^\infty} \|a_j\|_{l-n-1,1} \leq C (1+t)^{-n/2}\|a_j\|_{l-n-1,1}.$$ Similarly to the estimates of $I_1$, we obtain $$\label{(3.9)} I_2 \leq C (1+t)^{-n/2}\|b_j\|_{l-n-1,1}.$$ By noticing $|\alpha|\leq l-n-1$ and the definition of $F_{1\alpha}$, for some positive number $c_2$, we have $$\begin{split} |\partial_x^\alpha F_{1\alpha}\ast a_j| & \leq \int|\xi^\alpha\hat F_{1\alpha}\hat a_j|{\rm d}\xi\\ & \leq C \|a_j\|_{ {l,1}} {\rm e}^{-c_2t}\int\chi_3(\xi)|\xi|^{|\alpha|-l}{\rm d}\xi\\ & \leq C\|a_j\|_{l,1}{\rm e}^{-c_2t}. \end{split}$$ Similarly, we have $$|\partial_x^\alpha F_{2\alpha}\ast b_j| \leq C\|b_j\|_{l-1,1}{\rm e}^{-c_2t}.$$ Then, we obtain $$\label{(3.10)} I_3 \leq C(1+t)^{-n/2}\|a_j\|_{l,1} \quad\text{and}\quad I_4 \leq C(1+t)^{-n/2}\|b_j\|_{l-1,1}.$$ To estimate $I_5$ and $I_6$, we give the estimates of $F_j(u)(t)$ as follows: $$\label{3.10} \begin{split} |\partial_x^\alpha F_j(u)(t)| &\leq |\partial_u^1F_j(u)(t)|\sum_{i=1}^m|\partial_x^\alpha u_i|\\ &\quad + |\partial_u^2F_j(u)(t)|\sum_{1\leq k_1,k_2\leq m; \alpha_{1} +\alpha_{2}=\alpha}|\partial_x^{\alpha_1} u_{k_1} \partial_x^{\alpha_2} u_{k_2}| + \dots \\ &\quad + |\partial_u^{\alpha}F_j(u)(t)|\sum_{1\leq k_1,\dots,k_m\leq m,\ \alpha_{1} +\dots+\alpha_{m}=\alpha} |\partial_x^{\alpha_1} u_{k_1} \dots \partial_x^{\alpha_m} u_{k_m}|, \end{split}$$ 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 $$\label{(3.12)} \begin{split} I_6 & \leq C\int_0^t {\rm e}^{-(t-\tau)/4} \|(f_1+f_2)\ast F_j(u)(\tau)\|_{l-n-1,1}{\rm d}\tau\\ & \leq C\int_0^t {\rm e}^{-(t-\tau)/4} (\|f_1\|_{L^1}+\|f_2\|_{L^1}) \|F_j(u)(\tau)\|_{l-n-1,\infty}{\rm d}\tau\\ & \leq C\int_0^t {\rm e}^{-(t-\tau)/4} \|F_j(u)(\tau)\|_{l-n-1,\infty}{\rm d}\tau\\ & \leq C\int_0^t {\rm e}^{-(t-\tau)/4} \|F_j(u)(\tau)\|_{l-[\frac{n}{2}]}{\rm d}\tau\\ & \leq CE^{\tilde p_j} \int_0^t {\rm e}^{-(t-\tau)/4} (1+\tau)^{-\frac{n}{2}(\tilde p_j-1)-\frac{n}{4}}{\rm d}\tau\\ & \leq C (1+t)^{-n/2} E^{\tilde p_j}. \end{split}$$ Thus, the combination of \eqref{(3.8)}-\eqref{(3.12)} gives $$\label{(3.13)} \begin{split} &\|T_j[u](t)\|_{l-n-1,\infty}\\ & \leq C(1+t)^{-n/2} \big( \|a_j\|_{l,1}+\|b_j\|_{l-1,1} +\|a_j\|_{l-n-1,1}+\|b_j\|_{l-n-1,1} +E^{\tilde p_j}\big). \end{split}$$ 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 $$\label{(3.14)} J_1 \leq \|(G_1-G_{1,3})(t)\| \|a_j\|_{l,1 } \leq C (1+t)^{-n/4}\|a_j\|_{l,1 }.$$ For $J_3$, it follows from the Plancherel theorem and \eqref{2.23} that $$\label{(3.15)} J_3 \leq \sum_{|\alpha|=0}^l\|G_{1,3}(t)\ast \partial_x^\alpha a_j\| = \sum_{|\alpha|=0}^l\|\hat{G}_{1,3}(t) \widehat{\partial_x^\alpha a_j}\| \leq C {\rm e}^{-t/4} \|a_j\|_l.$$ Similar to the estimates of $J_1$ and $J_3$, we obtain $$\label{(3.16)} J_2 \leq C (1+t)^{-n/4}\|b_j\|_{l,1 },\quad\text{and}\quad J_4 \leq C {\rm e}^{-t/4} \|b_j\|_l.$$ Using Propositions \ref{prop2.1}-\ref{prop2.2}, the Young inequality and noticing $\tilde p_j>1+\frac{2}{n}$, we have $$\label{(3.17)} \begin{split} J_5 & \leq C \int_0^t\|(G_2-G_{2,3})(t-\tau)\| \|F_j(u)(\tau)\|_{l }{\rm d}\tau\\ & \leq C E^{\tilde p_j}\int_0^t (1+t-\tau)^{-n/4}(1+t)^{-\frac{n}{2}(\tilde p_j-1) -\frac{n}{4}}{\rm d}\tau\\ & \leq C (1+t)^{-n/4} E^{\tilde p_j}. \end{split}$$ For $J_6$, by using the Plancherel theorem and \eqref{2.23}, we get $$\label{(3.18)} J_6 \leq C \int_0^t {\rm e}^{-(t-\tau)/4} \|F_j(u)(\tau)\|_l{\rm d}\tau \leq C (1+t)^{-n/4} E^{\tilde p_j}.$$ Thus, we obtain $$\label{(3.19)} \|T_j[u](t)\|_l \leq C(1+t)^{-n/4}\left(\|a_j\|_l+\|b_j\|_l +\|a_j\|_{l,1}+\|b_j\|_{l,1}+E^{\tilde p_j}\right).$$ 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 $$\label{(3.22)} \begin{split} &\|T[u]-T[v]\|_{l-n-1,\infty}\\ & \leq \int_0^t\left\|(G_2-F_{2\alpha})(t-\tau)\ast (F_j(u)-F_j(v))(\tau)\right\|_{l-n-1,\infty}{\rm d}\tau\\ & \quad + \int_0^t \left\|F_{2\alpha}(t-\tau)\ast (F_j(u)-F_j(v))(\tau)\right\|_{l-n-1,\infty}{\rm d}\tau\\ & :=H_1+H_2. \end{split}$$ By a directly calculation, we have $$\label{3.23} \begin{split} & |\partial_x^\alpha F(u)-\partial_x^\alpha F(v)|\\ &\leq \Big|\sum_{k=1}^m\partial_{u_k}^1 F(u)\partial_x^\alpha u_k - \sum_{s=1}^m\partial_{v_s}^1 F(v)\partial_x^\alpha v_s \Big|\\ &\quad + \Big| \sum_{1\leq k_1,k_2\leq m;\alpha_{k_1}+\alpha_{k_2}=\alpha} \partial_{u_{k_1}u_{k_2}}^2 F(u)\partial_x^{\alpha_{k_1}} u_{k_1} \partial_x^{\alpha_{k_2}} u_{k_2} \\ & \quad - \sum_{1\leq s_1,s_2\leq m;\alpha_{s_1}+\alpha_{s_2}=\alpha} \partial_{v_{s_1}v_{s_2}}^2 F(v)\partial_x^{\alpha_{s_1}} v_{s_1} \partial_x^{\alpha_{s_2}} v_{s_2}\Big|+ \dots \\ &\quad + \Big| \sum_{1\leq k_i\leq m,i=1,\dots,\alpha;\alpha_{k_1} +\dots+\alpha_{k_\alpha}=\alpha} \partial_{u_{k_1}u_{k_2}\dots u_{k_\alpha}}^{\alpha} F(u) \partial_x^{\alpha_{k_1}} u_{k_1} \dots \partial_x^{\alpha_{k_\alpha}} u_{k_\alpha} \\ &\quad - \sum_{1\leq s_i\leq m,i=1,\dots,\alpha;\alpha_{s_1} +\dots+\alpha_{s_\alpha}=\alpha} \partial_{v_{s_1}v_{s_2}\dots v_{s_\alpha}}^{\alpha} F(u)\partial_x^{\alpha_{s_1}} v_{s_1} \dots \partial_x^{\alpha_{s_\alpha}} v_{s_\alpha}\Big|\\ &\leq \sum_{i=1}^m|\partial_u^1 F(u)\partial_x^\alpha u_i - \partial_v^1 F(v)\partial_x^\alpha v_i |\\ &\quad + \sum_{1\leq k,s\leq m;\alpha_k+\alpha_s=\alpha} |\partial_u^2 F(u)\partial_x^{\alpha_k} u_k\partial_x^{\alpha_s} u_s - \partial_v^2 F(v)\partial_x^{\alpha_k} v_k\partial_x^{\alpha_s} v_s| + \dots\\ &\quad + \sum_{1\leq k_i\leq m,i=1,\dots,\alpha;\alpha_{k_1}+\dots +\alpha_{k_\alpha}=\alpha} |\partial_{u}^{\alpha} F(u)\partial_x^{\alpha_{k_1}} u_{k_1} \dots \partial_x^{\alpha_{k_\alpha}} u_{k_\alpha}\\ &\quad - \partial_{v}^{\alpha} F(v)\partial_x^{\alpha_{k_1}} v_{k_1} \dots \partial_x^{\alpha_{k_\alpha}} v_{k_\alpha}|. \end{split}$$ Using \eqref{3.23} and the assumption \eqref{1.8}, we have $$\begin{split} &\| F_j(u) - F_j(v)\|_{l,1}\\ &\leq C \sum_{s=1}^m \sum_{ \tilde \alpha_j =1}^l \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde \alpha_j} \prod_{k=1}^{s-1}\|u_k\|_{L^\infty}^{p_{j,k}-\alpha_{j,k}}\\ &\quad \times \prod_{k=s+1}^m\|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+} \big( \|u_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+} + \|v_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+} \big)\\ &\quad \times \Big( \|u_s-v_s\|_{ 2} \sum_{i=1}^m(\|u_i\|_{l }+\|v_i\|_{l }) \Big) \Big)\\ &\quad + C \sum_{s=1}^m \sum_{ \tilde \alpha_j =1}^l \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde \alpha_j} \prod_{k=1}^{s-1} \|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+}\\ &\quad \times \prod_{k=s+1}^m \|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+} \big(\|v_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+} \|v_s\|_{L^2}\big) \Big)\\ &\quad\times \sum_{s=1}^m \sum_{ \tilde \alpha_j =1}^l \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde \alpha_j} \prod_{k=1}^{s-1}\|\partial_x^{\alpha_k}u_k\|_{L^\infty} \prod_{k=s+1}^m\|\partial_x^{\alpha_k}v_k\|_{L^\infty} \|u_s-v_s\|_{ l} \Big). \end{split}$$ 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 $$\label{(3.23)} \begin{split} H_1 & \leq \int_0^t \|(G_2-F_{2\alpha})(t-\tau)\|_{L^\infty} \|(F_j(u)-F_j(v))(\tau)\|_{l-n-1,1}{\rm d}\tau\\ & \leq C \int_0^t (1+t-\tau)^{-n/2} \|(F_j(u)-F_j(v))(\tau)\|_{l-n-1,1}{\rm d}\tau\\ & \leq C E^{\tilde p_j -1} \int_0^t (1+t-\tau)^{-n/2}(1+\tau)^{-n/2(\tilde p_j-2)-n/4} \|u-v\|_X{\rm d}\tau\\ & \leq C E^{\tilde p_j-1}(1+t)^{-n/2} \|u-v\|_X. \end{split}$$ Similarly, by using \eqref{3.23}, we have $$\begin{split} &\| F_j(u) - F_j(v)\|_{l}\\ &\leq C \sum_{s=1}^m \sum_{ \tilde \alpha_j =1}^l \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde \alpha_j} \prod_{k=1}^{s-1}\|u_k\|_{L^\infty}^{p_{j,k}-\alpha_{j,k}} \\ &\quad \times \prod_{k=s+1}^m\|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+} \big( \|u_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+} + \|v_s\|_{L^\infty}^{(p_{j,s}-\alpha_{j,s}-1)_+} \big)\\ &\quad\times \Big( \|u_s-v_s\|_{L^\infty} \sum_{i=1}^m(\|u_i\|_{l }+\|v_i\|_{l }) \Big) \Big)\\ &\quad + C \sum_{s=1}^m \sum_{ \tilde \alpha_j =1}^l \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde \alpha_j} \prod_{k=1}^{m} \|v_k\|_{L^\infty}^{(p_{j,k}-\alpha_{j,k})_+} \\ &\quad \times \sum_{s=1}^m \sum_{ \tilde \alpha_j =1}^l \Big( \sum_{\alpha_{j,1}+\dots+\alpha_{j,m}=\tilde \alpha_j} \prod_{k=1}^{s-1}\|\partial_x^{\alpha_k}u_k\|_{L^\infty} \prod_{k=s+1}^m\|\partial_x^{\alpha_k}v_k\|_{L^\infty} \|u_s-v_s\|_{ l} \Big). \end{split}$$ 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 $$\label{(3.24)} \|T[u]-T[v]\|_{l-n-1,\infty} \leq C E^{\tilde p_j} (1+t)^{-n/2} \|u-v\|_X.$$ 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 $$\label{(3.25)} \|T[u]-T[v]\|_l \leq C E^{\tilde p_j} (1+t)^{-n/4} \|u-v\|_X.$$ 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 $$\label{4.1} \begin{split} &\partial_x^\alpha u_j(x,t)\\ & = \partial_x^\alpha G_1\ast a_j + \partial_x^\alpha G_2\ast b_j + \partial_x^\alpha \int_0^t G_2(t-s)\ast F_j(u)(s){\rm d}s\\ & = (\partial_x^\alpha (G_1-F_{1\alpha})(t))\ast a_j + (\partial_x^\alpha (G_2-F_{2\alpha})(t))\ast b_j + \partial_x^\alpha F_{1\alpha}(t)\ast a_j + \partial_x^\alpha F_{2\alpha}(t)\ast b_j\\ & \quad + \int_0^t (\partial_x^\alpha (G_2-F_{2\alpha})(t-\tau))\ast F_j(u)(\tau) {\rm d}\tau + \int_0^t \partial_x^\alpha F_{2\alpha}(t-\tau)\ast F_j(u)(\tau) {\rm d}\tau . \end{split}$$ From Propositions \ref{prop2.1}--\ref{prop2.2} and the assumption \eqref{1.11}, by using Lemma \ref{lem2.3}, we have $$\begin{split} &|(\partial_x^\alpha (G_1-F_{1\alpha})(t))\ast a_j + (\partial_x^\alpha (G_2-F_{2\alpha})(t))\ast b_j|\\ &\leq C\varepsilon_0(1+t)^{-\frac{n+|\alpha|}{2}}B_r(|x|,t). \end{split}$$ For some positive number $b$, by noticing the definition of $F_{1\alpha}$, $|\alpha|n$, we have $$|x^\beta \partial_x^\alpha F_{1\alpha}\ast a_j| \leq \int |\partial_\xi^\beta\xi^\alpha \hat F_{1\alpha}\hat a_j| {\rm d}\xi \leq C \varepsilon_0 {\rm e}^{-bt}.$$ Take $|\beta|=0$ or $|\beta|=n$, we obtain $$|\partial_x^\alpha F_{1\alpha}\ast a_j| \leq C \varepsilon_0 {\rm e}^{-\frac{b}{2}t} B_\frac{n}{2}(|x|,t).$$ Similarly, we have $$|\partial_x^\alpha F_{2\alpha}\ast b_j| \leq C \varepsilon_0 {\rm e}^{-bt/2} B_r(|x|,t).$$ To estimate the other parts in \eqref{4.1}, we set $$\varphi_\alpha (x,t) = (1+t)^{\frac{n+|\alpha|}{2}} (B_\frac{n}{2}(|x|,t))^{-1},$$ and $$M(t) = \sup_{{0\leq s,\tau\leq t,}\ {|\alpha|\leq l}} \sum_{j=1}^m|\partial_x^\alpha u_j(x,\tau)|\varphi_\alpha(x,s).$$ When $|\alpha|\leq l-1$, from the assumptions \eqref{1.6}, \eqref{1.7} and the definition of $M$, we have $$\label{4.8} |\partial_x^\alpha F_j(u)(x,t)| \leq M^2(t)(1+t)^{-n-\frac{|\alpha|}{2}}B_n(|x|,t).$$ When $|\alpha|= l$, from the definition of $M$, by using Theorem \ref{thm1.1} and Lemma \ref{lem3.1}, we have $$\label{4.9} |\partial_x^\alpha F_j(u)(x,s)| \leq M^2(t)(1+t)^{-n-\frac{|\alpha|}{2}}B_n(|x|,t) + EM(t)(1+t)^{-n-\frac{|\alpha|}{2}}B_{\frac{n}{2}} (|x|,t).$$ 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 $$\begin{split} R^\alpha &\leq \big| \int_0^t {\rm e}^{-b(t-s)} (f_1+f_2)\ast \partial_x^\alpha F_j(u)(s){\rm d}s\big|\\ &\leq \big| \int_0^t {\rm e}^{-b(t-s)} f_1 \ast \partial_x^\alpha F_j(u)(s){\rm d}s\big| +\big| \int_0^t {\rm e}^{-b(t-s)} f_2 \ast \partial_x^\alpha F_j(u)(s){\rm d}s\big|\\ &:= R_1^\alpha + R_2^\alpha. \end{split}$$ 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 $$|\partial_x^\alpha F_j(u)(x,s)| \leq CM^2(t) (1+s)^{-n-\frac{|\alpha|}{2}} B_n(|x|,s).$$ From Proposition \ref{prop2.3} and Lemma \ref{lem2.3}, we obtain \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} From the above inequalities and the definition of $M$, we obtain $$M(t) \leq C\big(M^2(t) + E M(t)+\varepsilon_0 \big).$$ Since $E$ and $\varepsilon_0$ are small enough, we have $M(t)\leq C$. It yields that $$|\partial_x^\alpha u_j(t)| \leq C (1+t)^{-\frac{n+|\alpha|}{2}} B_{\frac{n}{2}}(|x|,t).$$ 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 $$\|\partial_x^\alpha u_j(\cdot,t)\|_{L^p} \leq C(1+t)^{-\frac{n}{2}(1-\frac{1}{p})-\frac{|\alpha|}{2}},\quad j=1,\dots,m.$$ \end{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). \begin{thebibliography}{99} \bibitem{Hosono-Ogawa} T. Hosono, T. Ogawa; \emph{Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations}, J. Differential Equations 203 (2004) 82-118. \bibitem{Ikehata} R. 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