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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 106, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/106\hfil Pointwise estimates for solutions]
{Pointwise estimates for solutions to perturbed Hasegawa-Mima equations}

\author[L. Wang \hfil EJDE-2015/106\hfilneg]
{Lijuan Wang}

\address{Lijuan Wang \newline
Business Information Management School,
Shanghai Institute of Foreign Trade,
201620, China}
\email{ljwang66@suibe.edu.cn}


\thanks{Submitted February 24, 2015. Published April 21, 2015.}
\subjclass[2010]{35B65, 35K55, 76C20}
\keywords{Pointwise estimates; perturbed Hasegawa-Mima equation;
\hfill\break\indent  Green function}

\begin{abstract}
 In this article, we study  pointwise estimates for solutions to the
 two-dimensional perturbed Hasegawa-Mima equation based on the analysis
 of the Green functions for the linearized system.
 The pointwise estimates not only exhibit the Huygen's principle
 but also give additional insight on the description of the evolution
 behavior of the solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

The simplest nonlinear model describing the time evolution of drift waves 
was derived by Hasegawa and Mima \cite{h2},
\begin{equation}\label{eq:hmsta}
 \partial_t(1-\Delta)u=-k\partial_{x_2}u+\{u, \Delta u\}+\nu\Delta^3u,\quad
 x\in \mathbb{R}^3,
\end{equation}
where $u$ is the electrostatic potential fluctuation and $n$ is the density 
fluctuation by  assuming a Boltzmann relation $n\sim u$,
$k=\partial_x\ln n_0$ with $n_0$ being the background particle density and 
$\nu$ is a positive number. This equation also arises in the context of Rossby 
waves in the atmospheres of rotating planets \cite{l5}. The beautiful
structure behind the Hasegawa-Mima equation initiates a lot of mathematical 
investigations, see  \cite{c1,g1,h1,z1},  and the references therein.

In this article, the two-dimensional perturbed Hasegawa-Mima equation derived
 by Liang et al \cite{l1} is studied as a prototype,
\begin{equation}\label{eq:ori}
  \partial_t(u-\Delta u)+k\partial_{x_2} u-\lambda (u-\Delta u)
=-\{u, \Delta u\}, \quad (x_1, x_2)\in \mathbb{R}^2,\quad t>0,
\end{equation}
with initial data
\begin{equation*}
  u(x_1, x_2, 0)=u_0(x_1, x_2),
\end{equation*}
 where $k$ and $\lambda$ are constants and $0<\lambda<1$, $\Delta$ 
is the 2D Laplacian, and $\{\cdot, \cdot\}$ denotes the Poisson bracket
\begin{equation*}
  \{h, g\}=(\partial_{x_1}h)(\partial_{x_2}g)-(\partial_{x_2} h)(\partial_{x_1} g).
\end{equation*}
The term $k\partial_{x_2} u$  describe a drift and $-\lambda (u-\Delta u)$ 
is the perturbation.

Equation \eqref{eq:ori} is the the simplest model for a  two-dimensional 
turbulent system. It describes the time evolution of drift wave in plasma
as mentioned above and  the temporal evolution of geostrophic motion. 
Additionally, it can be reduced to the Euler equation for the incompressible 
homogeneous fluids if we eliminate the term $u_t$ in \eqref{eq:ori}. 
It is well known that Euler equation is a very hot topic in recent years and
numerous papers were devoted to its study, such as 
\cite{d1,l4,t1,y1} and references therein.

In recent years, the  perturbed Hasegawa-Mima equation has also received much 
attention, Many authors have contributed to the study of this equation. 
Guo and  Han \cite{g3} have obtained the global solutions to the Cauchy problem.
  In 1998, Grauer obtained the energy estimate for the perturbed Hasegawa-Mima
equation\cite{g2}. In \cite{z2}, R. F. Zhang and her collaborator got the
existence and the uniqueness of the global solution for the generalized 
Hasegawa-Mima equation. However, there is little information of the
 pointwise estimates for this equations to the best of our knowledge. 
Thus, the principle purpose of this paper is to present the pointwise 
estimates of solutions to \eqref{eq:ori}. In fact, pointwise estimates 
 play a crucial role in the description of the evolution of the solution, 
as it gives explicit expressions of the time asymptotic behavior of solutions 
and helps us get the global existence and the $L^p$ estimates of solutions.

The main approach in this article is using the Green function method.
We can present solutions of the Cauchy problem for the linearized system 
by the fundamental solution (i.e. Green function), which can also be
 applied to write a nonlinear system into an integral system.
Actually, using the Green function to study the pointwise estimate for 
hyperbolic-parabolic systems becomes a very active field of research in recent 
years. Based on the pointwise estimates, we can not only obtain the decay 
rate of the solution due to the parabolicity of the system, but also find 
out the movement of the main part of the solution  caused by the hyperbolicity.  
This method was first introduced by Liu and Zeng in \cite{l3} to get the
pointwise estimates of solutions to the one dimensional quasilinear 
hyperbolic-parabolic systems of conservation laws. 
Later,  Hoff and Zumbru (\cite{h3} and \cite{h4})employed this method
to study the Navier-Stokes equation with viscosity.

Liu and Wang \cite{l2} also use this method to obtain the pointwise estimates
of the solutions to the isentropic Navier-Stokes equations in odd dimensions. 
The classical Green  function method  is  decomposing the Green  function
into three parts: lower frequency part, middle  part and higher frequency part. 
Then by Taylor expansion, Fourier analysis and Kirchhoff formulae, we can obtain 
the pointwise estimates of these three parts, respectively.
The pointwise estimates of the Green function shows that the large time behavior
of Green  function is dominated by the lower frequency waves
while the higher frequency waves play a much more significant role in short time. 
Through the  delicate analysis of the Green  function, we obtain the explicit 
expression of the time-asymptotic behavior of the solutions to the linearized system.


Besides, the detailed analysis of the nonlinear terms is also an indispensable 
part for us because the pointwise estimates of the solutions to the nonlinear 
system are obtained by combining the analysis of the
Green function and the nonlinear terms together. Since there are the third order  
derivatives in the nonlinear terms, how to deal with the nonlinear term of 
equation \eqref{eq:ori} becomes a key point in this paper.
Usually, if there is only the first order derivative in the nonlinear term, 
according to the Duhamel principle the solution can be expressed as 
\[
 u=G(\cdot, t)\ast u_0-\int_0^t G(t-\tau)\ast N(\tau)\,\mathrm{d} \tau,
\]
Then we can transfer the derivative from the nonlinear term to the Green 
function by using the integration by parts.
Finally, using of pointwise estimates of the Green function we can get 
the decay estimates of the solutions.
However, the order of the derivative of the nonlinear term in our equation 
is too  high that the usual way fails.
To overcome this difficulty, a lot of  computations and analysis for the 
nonlinear term will be essential
besides the transfer of the derivatives.
The main result in this article is stated as follows.

\begin{theorem}\label{th:mainhm}
Suppose  $u_0\in H^{s+l}(\mathbb{R}^2)$, $s=2$, $l\leq 9$,  
$E=\max\{\|u_0\|_{H^{s+l}},\|u_0\|_{L^1}\}$ and
\begin{equation}\label{eq:condition}
 \|u_0\|_{H^{s+l}\cap L^1}\leq E,\quad 
|D^\alpha u_0|\leq E(1+|x|)^{-m'}, m'>2, \quad |\alpha|\leq l,
\end{equation}
with $E$  sufficiently small, then when $t$ is large enough,
there exists positive constant $C$ such that the solution $u(x, t)$ 
to \eqref{eq:ori} satisfies
\begin{equation*}
  |D^\alpha_xu(x,t)|\leq CE(1+t)^{-\frac{2+\nu(|\alpha|)}{2}}B_1(x-bt, t).
\end{equation*}
where $b=(0, k)$,
\begin{gather*}
  B_N(x, t)=(1+\frac{|x|^2}{1+t})^{-N}, \\
  \nu(|\alpha|)=\begin{cases}
  |\alpha|, & |\alpha|\leq l-6,\\
  0,& l-6<|\alpha|\leq l-3.\\
  \end{cases}
\end{gather*}
\end{theorem}

The rest of this article is arranged as follows.  
 The energy estimates and the existence of the global solution
to \eqref{eq:ori} will be established in section \ref{sec:energy eatimates}.
In section \ref{sec:pointwise green}, we will
get the analysis of the Green function $G(x, t)$ and the pointwise estimates 
of solutions to \eqref{eq:ori} will follow in section \ref{sec:pointwise}.
For  convenience, we list some important lemmas which are the vital tools 
for obtaining the pointwise estimates of the Green function as a appendix 
at the end of this paper.


\section{Energy estimates}\label{sec:energy eatimates}

In this section, we obtain the existence global solutions and the energy estimates
for \eqref{eq:ori}. We assume that the initial data satisfies
 $\|u_0\|^2_{H^{s+l}} \leq E$, $s\geq 2$, where  $0<E\ll 1 $.


\begin{theorem}\label{thm:prio}
 Suppose the conditions of Theorem \ref{th:mainhm} are satisfied, then for any
positive constants  $m$, we have
  \begin{equation}\label{eq:hmenergy-4}
   \|u\|_{H^m}+C_0\int_0^t\|\nabla u\|_{H^m}\,\mathrm{d} \tau\leq C\|u_0\|_{H^m}.
   \end{equation}
where $C_0$, $C$ are positive constants and independent of $t$.
\end{theorem}

\begin{proof}
In \eqref{eq:ori}, taking the inner product in $L^2(\mathbb{R}^2)$ with $2u$, 
one obtains
\begin{equation*}
  (\partial_t(u-\Delta u), 2u)+(k\partial_{x_2}u, 2u)
-(\lambda \Delta(u-\Delta u), 2u)=(-\{u, \Delta u\}, 2u),
\end{equation*}
since
$(\{u, \Delta u\}, 2u)=0$,
one has
\begin{equation*}
  \frac{\mathrm{d} }{\mathrm{d} t}(\|u\|^2_{L^2}
+\|\nabla u\|^2_{L^2})+2\lambda(\|\nabla u\|^2_{L^2}+\|\Delta u\|_{L^2}^2)=0.
\end{equation*}
By integration, we obtain
\begin{equation}\label{eq:hmenergy-1}
  \|u\|^2_{L^2}+\|\nabla u\|^2_{L^2}
+2\lambda\int_0^t(\|\nabla u\|^2_{L^2}+\|\Delta u\|^2_{L^2})\,\mathrm{d} \tau
=\|u_0\|^2_{L^2}+\|\nabla u_0\|^2_{L^2}.
\end{equation}
Similarly, multiplying \eqref{eq:ori}  by $-2\Delta u$ and integrating the 
result over $\mathbb{R}^2$ for the space variable $(x_1, x_2)$, we have
\begin{equation}\label{eq:hmenergy-2}
  \|\nabla u\|^2_{L^2}+\|\Delta u\|^2_{L^2}
+2\lambda \int_0^t(\|\Delta u\|^2_{L^2}
+\|\nabla\Delta u\|^2_{L^2})\,\mathrm{d} \tau
=\|\nabla u_0\|^2_{L^2}+\|\Delta u_0\|^2_{L^2},
\end{equation}
by noticing that
\begin{equation*}
  \int_{\mathbb{R}^2} \{u, \Delta u\}\Delta u\,\mathrm{d} x=0, \ \ \ t\geq0,
\end{equation*}
\eqref{eq:hmenergy-1} and \eqref{eq:hmenergy-2} imply
\begin{equation*}
  \|u\|^2_{H^2}+\int_0^tC_0\|\nabla u\|_{H^2}^2\,\mathrm{d} t\leq C \|u_0\|_{H^2}^2.
\end{equation*}

Now we aim to obtain an estimates for $\|u\|_{H^3}$. 
Differentiating \eqref{eq:ori}  with respect to $(x_1, x_2)$, one obtains
\begin{equation}\label{eq:hmd-1}
  \partial_t \nabla (u-\Delta u)+k\partial_{x_2} \nabla u
+\{u, \nabla\Delta u\}+\{\nabla u, \Delta u\}-\lambda \nabla \Delta(u-\Delta u)=0.
\end{equation}
Taking the inner product of $-2\nabla\Delta u$ and \eqref{eq:hmd-1}
 in $L^2(\mathbb{R}^2)$, we have
\begin{equation}\label{eq:h3-1}
\frac{\mathrm{d} }{\mathrm{d} t}(\|\Delta u\|^2_{L^2}
+\|\nabla \Delta u\|^2_{L^2})+2\lambda(\|\nabla\Delta u\|^2_{L^2}
+\|\Delta^2 u\|^2_{L^2})
=2\int_\mathbb{R}^2 \nabla\Delta u \cdot \{\nabla u, \Delta u\}\,\mathrm{d} x,
\end{equation}
where we have used the following facts
\begin{equation*}
  \int \nabla\Delta u \cdot \partial_{x_2} \nabla u\,\mathrm{d} x=0, \quad
2\int \nabla\Delta u\cdot\{u, \nabla\Delta u\}\,\mathrm{d} x=0.
\end{equation*}
For the term $2\int \{\nabla u, \Delta u\}\cdot \nabla\Delta u\,\mathrm{d} x$, 
we have
\begin{equation*}
 2\int \{\nabla u, \Delta u\} \cdot \nabla\Delta u\,\mathrm{d} x
\leq C \|D^2 u\|_{L^\infty}\|\nabla\Delta u\|^2_{L^2},
\end{equation*}
where $D^2=\partial_{x_1}^{\alpha_1}\partial_{x_2}^{\alpha_2}$,
$|\alpha_1|+|\alpha_2|=2$.
Thanks to the H$\ddot{o}$lder inequality and the Gagliardo-Nirenberg inequality,
 one has
\begin{equation*}
  \|D^2 u\|_{L^\infty} \leq C \|\Delta u\|^{1/2}_{L^2}\|\Delta^2 u\|^{1/2}_{L^2},
\quad  \|\nabla\Delta u\|_{L^2} \leq C \|\Delta u\|_{L^2}^{1/2}
\|\Delta^2 u\|^{1/2}_{L^2}.
\end{equation*}
The above two inequalities and Young's inequality tell us that
\begin{equation}\label{eq:hmenergy-3}
   \|D^2 u\|_{L^\infty}\|\nabla\Delta u\|^2_{L^2}
\leq C\|\Delta u\|_{L^2}^{\frac{3}{2}}\|\Delta^2 u\|^{\frac{3}{2}}_{L^2} 
\leq \lambda \|\Delta^2 u\|^2_{L^2}+C\|\Delta u\|_{L^2}^6.
\end{equation}
Hence, \eqref{eq:h3-1} and \eqref{eq:hmenergy-3} imply
\begin{equation}\label{eq:hmenergy-6}
  \frac{\mathrm{d} }{\mathrm{d} t}(\|\Delta u\|^2_{L^2}
+\|\nabla \Delta u\|^2_{L^2})+2\lambda \|\nabla\Delta u\|^2_{L^2}
+ \lambda\|\Delta^2 u\|^2_{L^2}
    \leq C\|\Delta u\|^6_{L^2}.
\end{equation}
According to  \eqref{eq:hmenergy-2}, we obtain
\begin{gather}\label{eq:jia1}
  \|\Delta u\|^2_{L^2}\leq \|u_0\|_{H^2}^2\leq E, \quad E\ll 1, \\
\label{eq:energeofdeltau}
  \int_0^t\|\Delta u(\tau)\|^2_{L^2}\,\mathrm{d} \tau
\leq C(\|\nabla u_0\|^2_{L^2}+\|\Delta u_0\|^2_{L^2}).
\end{gather}
The conclusion follows when we combine \eqref{eq:hmenergy-6}, \eqref{eq:jia1}
 and \eqref{eq:energeofdeltau}; i.e.,
\begin{equation*}
  \|\Delta u\|^2_{L^2}+\|\nabla \Delta u\|^2_{L^2}
+\lambda \int_0^t(\|\nabla\Delta u\|^2_{L^2}
+\|\Delta^2 u\|^2_{L^2})\,\mathrm{d} \tau \leq C\|u_0\|_{H^3}.
\end{equation*}
Consequently, 
\begin{equation*}
  \|u\|_{H^m}+C_0\int_0^t\|\nabla u(\tau)\|_{H^m}\,\mathrm{d} \tau
\leq C\|u_0\|_{H^m},\quad 1\leq m\leq 3.
\end{equation*}

Now we will use mathematical induction to prove  \eqref{eq:hmenergy-4}
for any positive $m$.
Assume that \eqref{eq:hmenergy-4} holds for the case $m=M>3$, now we consider
$m=M+1$, 
Differentiating  \eqref{eq:ori} with respect to $(x_1, x_2)$,
\begin{equation}\label{eq:hmenergy-5}
  \partial_tD^\alpha(u-\Delta u)+k\partial_{x_2}D^\alpha u
-\lambda D^\alpha \Delta (u-\Delta u)+D^\alpha\{u, \Delta u\}=0,
\end{equation}
where $|\alpha|=M-1$.



Taking the scalar product of  $-2D^\alpha \Delta u$ and \eqref{eq:hmenergy-5},
 and then integrating the result over $\mathbb{R}^2$ for the space variable 
$(x_1, x_2)$, we obtain
\begin{align*}
&\frac{\mathrm{d}}{\mathrm{d} t}(\|D^\alpha \nabla u\|^2_{L^2}
 +\|D^\alpha \Delta u\|^2_{L^2})+2\lambda(\|D^\alpha \Delta u\|^2_{L^2}
 +\|D^\alpha \nabla\Delta u\|^2_{L^2})   \\
&\leq 2\int_{\mathbb{R}^2} \Big(\sum_{\substack{\beta_1+\beta_2=\alpha\\
    1\leq|\beta_1|\leq M-2,}} 
\{D^{\beta_1}u, D^{\beta_2}\Delta u\}
  +\{u, D^\alpha \Delta u\}+\{D^\alpha u, \Delta u\}\Big)
\cdot D^\alpha \Delta u\,\mathrm{d} x,
\end{align*}
noticing that
\begin{equation*}
  \int_{\mathbb{R}^2}\{u, D^\alpha \Delta u\} \cdot D^\alpha \Delta u\,\mathrm{d} 
x=0, \quad \forall t\geq 0,
\end{equation*}
we have
\begin{align*}
&\int_{\mathbb{R}^2} \Big(
    \sum_{\substack {\beta_1+\beta_2=\alpha\\
    1\leq|\beta_1|\leq M-2,}}
    \{D^{\beta_1}u, D^{\beta_2}\Delta u\}
  +\{u, D^\alpha \Delta u\}+\{D^\alpha, \Delta u\}\Big) 
\cdot D^\alpha \Delta u\,\mathrm{d} x   \\
& \leq 2 \sum_{\substack {\beta_1+\beta_2=\alpha\\
    1\leq|\beta_1|\leq M-2,}}
  \|\nabla D^{\beta_1} u\|_{L^\infty}\|\nabla D^{\beta_2}
 \Delta u\|_{L^2}\|D^\alpha \Delta u\|_{L^2}\\
&\quad +2\|\nabla D^\alpha u\|_{L^4}\|\nabla\Delta u\|_{L^4}
 \|D^\alpha \Delta u\|_{L^2}.
  \end{align*}

By the Gagliardo-Nirenberg inequality and the Sobolev inequality, we have
\begin{gather*}
  \|\nabla D^{\beta_1} u\|_{L^\infty}\leq C \|\nabla D^{\beta_1} u\|_{H^2},\\
 \|\nabla D^\alpha u\|_{L^4}\leq C\|\nabla D^\alpha u\|_{L^2}^{1/2}
 \|\nabla D^\alpha u\|^{1/2}_{H^1}, \\
 \|\nabla\Delta u\|_{L^4}\leq C\|\nabla\Delta u\|_{L^2}^{1-\frac{1}{2(M-2)}}
\|\nabla\Delta u\|_{H^{M-2}}^{\frac{1}{2(M-2)}}.
\end{gather*}
According to the induction assumption and Young's inequality, we conclude
that
\begin{equation*}
  \|u\|_{H^{M+1}}+\int_0^t\|\nabla u(\tau)\|_{H^{M+1}}\,\mathrm{d} \tau
\leq C\|u_0\|_{H^{M+1}}.
\end{equation*}
Thus, by mathematical induction, \eqref{eq:hmenergy-4} is true for any positive $m$.
\end{proof}

Based on the above discussion, we can extend the local solution and get the
 global solution by using the energy
estimates  \eqref{eq:hmenergy-4}. Therefore, we have the following result.

\begin{theorem} \label{thm:cauchyhm}
 Suppose $u_0 \in H^{s+l}$, $s=2$, $l$ is a positive integer, 
 and $\|u_0\|_{H^{s+l}}$ is sufficient small, then there exists a 
global solution $u(x, t)$  to  \eqref{eq:ori}.  Furthermore, when 
$M=s+l-1$, $u(x, t)$  satisfies \eqref{eq:hmenergy-4}, i.e.
 \begin{equation*}
   \|u\|_{H^m}+C_0\int_0^t\|\nabla u\|_{H^m}\,\mathrm{d} \tau
\leq C\|u_0\|_{H^m}.
 \end{equation*}
where $C_0$, $C$ are positive constants and independent of $t$.
\end{theorem}


\section{Pointwise estimates of Green function}\label{sec:pointwise green}

The linearized system of \eqref{eq:ori} about the constant state $u^*$,
 taken to be $0$ without loss of generality, is
\begin{equation}\label{eq:orilinear}
 \partial_t(u-\Delta u)+k\partial_{x_2} u-\lambda (u-\Delta u)=0.
 \end{equation}
The Green function $G(x,t)$ satisfies
\begin{gather*}
     \partial_t(G-\Delta G)+k\partial_{x_2} G-\lambda (G-\Delta G)=0,\\
     G(x, 0)=\delta(x),
\end{gather*}
where $\delta(x)$ is the Dirac function.
By the Fourier transform, we obtain
\begin{equation*}
{\hat G(\xi, t)}=\mathrm{e}^{\eta t},\quad\text{where }
\eta=-\lambda|\xi|^2-\frac{\sqrt{-1}k\xi_2}{1+|\xi|^2}.
\end{equation*}

Our goal in this section is to derive the pointwise estimates for the 
Green function $G(x, t)$ defined above. In order to get the estimates of 
$G(x, t)$, we divide $G$ into the lower frequency part $G_1$, the middle 
frequency part $G_2$ and the higher frequency part $G_3$,
where 
\begin{gather*}
\hat{G}_{i}=\chi_i(\xi)\hat{G}(\xi, t), \quad (i=1,2,3);\\
 \chi_1(\xi)=\begin{cases}
    1, & |\xi|<\epsilon,\\
    0, & |\xi|>2\epsilon;
  \end{cases} \\
\chi_3(\xi)= \begin{cases}
1, & |\xi|>2R,\\
0, & |\xi|<R;
\end{cases}\\
\chi_2(\xi)=1-\chi_1(\xi)-\chi_3(\xi).
\end{gather*}
which are smooth cut-off functions for the fixed constants $0<\epsilon<1$ 
and $R>2$.

First of all, we estimate the lower frequency part $G_1$.

\begin{lemma}\label{lem:g1}
 For any positive integer $N$, if $\epsilon$ is small enough, there exists 
constant $C>0$, such that
   \begin{equation*}
     |D^{\alpha}_x G_1|\leq  Ct^{-\frac{2+|\alpha|}{2}}B_N(x-bt, t),
   \end{equation*}
   where $b=(0, k)$ and $x=(x_1, x_2)$.
 \end{lemma}

 \begin{proof}
When $|\xi|$ is sufficiently small by the Taylor expansion, we obtain
\begin{equation*}
  -\frac{\sqrt{-1}k\xi_2}{1+|\xi|^2}=-\sqrt{-1}k\xi_2+O(|\xi|^2).
\end{equation*}
Hence,
\begin{align*}
  \hat{G}_1
&=\chi_1(\xi)\mathrm{e}^{-\lambda|\xi|^2t} 
 \mathrm{e}^{-\sqrt{-1}k\xi_2t}(1+O(|\xi|^2)t) \\
&=\chi_1(\xi)\mathrm{e}^{-\lambda|\xi|^2t}(1+O(|\xi|^2)t)
 \mathrm{e}^{-\sqrt{-1}k\xi_2t}\\
&=\hat{A} \mathrm{e}^{-\sqrt{-1}k\xi_2t},
 \end{align*}
where $\hat{A}=\chi_1(\xi)\mathrm{e}^{-\lambda|\xi|^2t}(1+O(|\xi|^2)t)$.
By properties of fourier transform, we have
\[
  G_1=A\ast\mathcal {F}^{-1}(\mathrm{e}^{-\sqrt{-1}k\xi_2t})=A(x_1, x_2-kt, t),
\]
and
\begin{equation}\label{eq:g1}
  D^\alpha_x G_1=D^\alpha_x A(x_1, x_2-kt, t).
\end{equation}
Moreover, by the expansion of $\hat{A}$, we have
\begin{equation*}
  |\partial_t^lD^\beta_\xi(\xi^\alpha\hat{A}(\xi,t))|
\leq C|\xi|^{|\alpha|-|\beta|+2l}(1+O(|\xi|^2)t)^{|\beta|+1}
\mathrm{e}^{-\lambda|\xi|^2t}.
\end{equation*}
Thanks to Lemma \ref{lem:3f-nf}, 
\begin{equation*}
  |D^\alpha_x A(x, t)|\leq C_Nt^{-\frac{2+|\alpha|}{2}}B_N(x, t).
\end{equation*}
Thus, by \eqref{eq:g1}, we have
\begin{equation*}
  |D^\alpha_x G_1|\leq C_Nt^{-\frac{2+|\alpha|}{2}}B_N(x-bt, t),
\end{equation*}
with $b=(0, k)$, which is our conclusion.
 \end{proof}

About the middle frequency part $\hat{G}_2$, we have the following Lemma.

\begin{lemma}
Suppose $|\xi|\in(\varepsilon, 2R)$, then there exists a positive $b$, such that
  \begin{equation}\label{eq:shuxueguina}
    |D^\beta_\xi\hat{G}_2(\xi, t)|\leq C(1+t)^{|\beta|}\mathrm{e}^{-bt}.
  \end{equation}
  \end{lemma}

\begin{proof}
If $|\xi|\in(\varepsilon, 2R)$, then there exists a positive $b$, such that
\begin{equation}\label{eq:g22}
  |\hat{G}_2(\xi, t)|=|\chi_2(\xi)\hat{G}(\xi, t)|\leq C\mathrm{e}^{-bt}.
\end{equation}

By \eqref{eq:g22}, this lemma holds when $|\beta|=0$. Now we use 
mathematical induction to prove this Lemma.
Suppose we have
  \begin{equation}\label{eq:dgm-1}
    |D^\beta_\xi\hat{G}_2(\xi, t)|\leq C(1+t)^{|\beta|}\mathrm{e}^{-bt}
  \end{equation}
for $|\beta|\leq l-1$.  By the Fourier transform, 
\begin{gather*}
\partial_t(D^\beta_\xi\hat{G}_2)+\frac{k\sqrt{-1}\xi_2}{1+|\xi|^2}D^\beta_{\xi}
\hat{G}_2+\lambda|\xi|^2D^\beta_\xi\hat{G}_2=-F(\xi),\\
 D^\beta_\xi\hat{G}_2(\xi, 0)=D^\beta_\xi \chi_2(\xi),
\end{gather*}
where
\begin{align*}
F(\xi)&=\sum_{ |\beta_1|+|\beta_2|=|\beta|,
 |\beta_1|\neq 0}
 D_\xi^{\beta_1}(\frac{k\sqrt{-1}\xi_2}{1+|\xi|^2})D^{\beta_2}_\xi\hat{G}_2\\
 &\quad +\sum_{|\gamma_1|+|\gamma_2|=|\beta|,|\gamma_1|\neq 0}
D^{\gamma_1}_\xi(\lambda|\xi|^2)D^{\gamma_2}_\xi\hat{G}_2.
\end{align*}
Note that
\begin{equation*}
D^\beta_\xi\hat{G}_2=D^\beta_\xi \chi_2(\xi)\hat{G}(t, \xi)
-\int_0^t\hat{G}(\xi, t-\tau)F(\tau)\,\mathrm{d} \tau,
\end{equation*}
according to \eqref{eq:dgm-1}, we have
\begin{equation*}
|D^\beta\hat{G}_2|\leq C\int_0^t\mathrm{e}^{-b (t-s)}
(1+t)^{|\beta|-1}\mathrm{e}^{-b s}\,\mathrm{d} s
+C\mathrm{e}^{-b t}\leq C\mathrm{e}^{-b t}(1+t)^{|\beta|},
\end{equation*}
which  implies  that \eqref{eq:shuxueguina} is valid for $|\alpha|=l$. 
 By induction, this lemma is proved.
\end{proof}

Additionally, when $1\leq|\beta|\leq l$,
\begin{equation}\label{eq:g222}
  |x^\beta D^\alpha G_2(x,t)|
\leq C\big|\int_{\mathbb{R}^2}
\mathrm{e}^{\sqrt{-1}x\cdot\xi}D^\beta_\xi(\xi^\alpha\hat{G}_2)
\,\mathrm{d} \xi\big|
\leq C(1+t)^{|\beta|}\mathrm{e}^{-b t}.
\end{equation}
When $|x|^2\leq 1+t$, setting $|\beta|=0$, we have
 \begin{equation*}
   |D^\alpha_x G_2(x, t)|\leq C\mathrm{e}^{-b t}
\leq C\frac{2^N}{(1+\frac{|x|^2}{1+t})^{N}}\mathrm{e}^{-b t}
\leq C\mathrm{e}^{-b t}B_N(x, t)
 \end{equation*}
When $|x|^2>1+t$, setting $|\beta|=2N$,  one has
\begin{equation*}
  |D^\alpha_xG_2(x, t)|\leq C\mathrm{e}^{-b t}\frac{1}{|x|^{2N}}
\leq C\frac{2^N}{(1+\frac{|x|^2}{1+t})^N}\mathrm{e}^{-b t}
\leq C\mathrm{e}^{-b t}B_N(x, t)
\end{equation*}
Therefore, the following Remark holds.

 \begin{remark}\label{lem:g2} \rm
  For any fixed $\varepsilon$ and $R$, there exists positive $b$ such 
that for $|\alpha|\geq 0$
   \begin{equation*}
     |D^\alpha_xG_2(x, t)|\leq C\mathrm{e}^{-b t}B_N(x, t).
   \end{equation*}
 \end{remark}

 Now we consider the higher frequency part $G_3$. 
When $|\xi|\geq 2R$ and $\xi$ is large enough, we obtain
 \begin{equation*}
 \eta=-\lambda|\xi|^2-\frac{\sqrt{-1}k\xi_2}{1+|\xi|^2}
=-\lambda|\xi|^2+\sum_{j=1}^{m}a_j|\xi|^{-(2j-1)}+O(|\xi|^{-(2m+1)}),
 \end{equation*}
So,
\begin{equation*}
\hat{G}_3(\xi, t)
=\chi_3 \mathrm{e}^{-\lambda |\xi|^2 t}(1+\sum_{j=1}^{m}p_i(t)q_j(\xi)
+p_{m+1}(t)O(|\xi|^{-(2m+1)})\big),
\end{equation*}
where $p_j$ and  $q_j$ satisfy
 \begin{equation*}
 |p_j(t)|\leq C(1+t)^j,\quad  1\leq j\leq m+1,\quad
   q_j(\xi)=|\xi|^{-(2j-1)}, \quad 1\leq j\leq m\,.
\end{equation*}
Denote
\begin{equation*}
  \hat{F}_\alpha(\xi, t)=\chi_3(\xi)\mathrm{e}^{-\lambda|\xi|^2t}
\sum_{j=1}^{|\alpha|+2}p_j(t)q_j(\xi).
\end{equation*}
We have the following lemma.

\begin{lemma}\label{lem:g-f}
  When $R$ is sufficient large, there exists a positive $a$, such that
\begin{equation*}
  |D^\alpha_x (G_3-F_{\alpha})(x, t)|\leq C \mathrm{e}^{-at} B_N(x, t)
\end{equation*}
\end{lemma}

\begin{proof}
Notice that the dimension we consider is $n=2$. Hence, we have the 
 estimate
  \begin{equation}\label{eq:g-f}
  \begin{split}
    |x^{2\beta}D^\alpha_x(G_3-F_\alpha)|
&\leq \int|\partial^{2\beta}_\xi(\xi^\alpha(\hat{G}_3-\hat{F}_\alpha))|
\,\mathrm{d} \xi  \\
&\leq \int |\xi|^{|\alpha|-2|\beta|-2(({{|\alpha|}+2})+1)}
 \mathrm{e}^{-\lambda|\xi|^2t}\,\mathrm{d} \xi
\leq C\mathrm{e}^{-at},
\end{split}
\end{equation}
where we have used the following fact:
If $|x|^2\leq t+1$,  let $|\beta|=0$ and let $|\beta|=N$ when $|x|^2>t+1$, 
then 
\begin{equation*}
  |D^\alpha_x(G_3-F_{\alpha})(x, t)|\leq C\mathrm{e}^{-at}
\min\{1, ((1+t)/|x|^2)^N\}.
\end{equation*}

Additionally, we have
\begin{equation*}
  1+\frac{|x|^2}{1+t}\leq 2\begin{cases}
 1, & |x|^2\leq t+1,\\
 \frac{|x|^2}{1+t},& |x|^2>t+1,
  \end{cases}
\end{equation*}
one has
\begin{equation*}
  \min\{1, ((1+t)/|x|^2)^N\}\leq \frac{2^N}{(1+(|x|^2/(1+t)))^N}.
\end{equation*}
By \eqref{eq:g-f} and the above inequality, we have
$$ 
|D^\alpha_x (G_3-F_{\alpha})(x, t)|\leq C \mathrm{e}^{-at} B_N(x, t)
$$
\end{proof}

By Lemma \ref{lem:g1}, Remark \ref{lem:g2}, Lemma \ref{lem:g-f} and Remark
\ref{rem:b-b} we obtain the following result.

 \begin{theorem}\label{thm:dgalpha}
   For any $x\in \mathbb{R}^2$, $t>0$ and  $l\leq 9$ there exists a constant 
$C_\alpha>0$, such that
 \begin{equation*}
|D^\alpha_x(G(x, t)-F_\alpha(x, t))|\leq C_\alpha (1+t)
^{-\frac{2+|\alpha|}{2}}B_N(x-bt, t)\quad |\alpha|\leq l,
 \end{equation*}
where $b=(0, k)$.
\end{theorem}

\section{Pointwise estimates of the nonlinear equations}\label{sec:pointwise}

In this section, we obtain pointwise estimates for the nonlinear equation
 \begin{equation}
 \begin{gathered}
     \partial_t(u-\Delta u)+k\partial_{x_2}u-\lambda\Delta(u-\Delta u)
+\{u, \Delta u\}=0,\\
     u(x,0)=u_0(x).
   \end{gathered}
\end{equation}
 According to the Duhamel principle, 
 \begin{equation*}
   u=G(\cdot, t)\ast u_0-\int_0^t G(t-\tau)\ast \{u, \Delta u\}\,\mathrm{d} \tau;
 \end{equation*}
therefore,
 \begin{equation*}
D^\alpha_x=D^\alpha_x G(\cdot, t)\ast u_0-\int_0^tD^\alpha_xG(t-\tau)
\ast\{u, \Delta u\}\,\mathrm{d} \tau
   ={I}_1+I_2
   \end{equation*}
Next, we obtain estimates for $I_1$ and $I_2$ respectively. 
Suppose that the initial data $u_0$ satisfies the  condition
\begin{equation}\label{eq:initialcondition}
  |D^\alpha_x u_0(x)|\leq CE(1+|x|^2)^{-m'},\quad  m'> 2, \; E\ll 1.
\end{equation}
So we have
\begin{equation*}
  I_1= D^\alpha_x (G(\cdot, t)-F_\alpha(\cdot, t)) \ast u_0+D^\alpha_x F_\alpha(\cdot, t)\ast u_0
  =:I_{11}+I_{12}.
\end{equation*}
By  Theorem  \ref{thm:dgalpha} and Lemma \ref{lem:jisuan},
 we  have the estimate
\begin{align*}
 |I_{11}|
&\leq CE (1+t)^{-\frac{2+|\alpha|}{2}}\int_{\mathbb{R}^2}B_N(|x-bt-y|, t)
 (1+|y|^2)^{-m'}\,\mathrm{d} y   \\
&\leq CE(1+t)^{-\frac{2+|\alpha|}{2}}B_N(x-bt, t).
\end{align*}
Since $\|u_0\|_{L^1}\leq E$, we have 
$\|\hat{u}_0\|_{L^\infty}\leq \|u_0\|_{L^1}\leq E$. 
Hence there exists a positive $p$, such that
\begin{align*}
 |x^\beta D^\alpha_x F_\alpha(\cdot, t)\ast u_0|
&\leq \int |\partial^\beta_\xi \xi^{\alpha} \hat{F}_\alpha \hat{u}_0|\,\mathrm{d}
 \xi\\
&\leq CE\mathrm{e}^{-pt}\int|\xi|^{|\alpha|-|\beta|-(2(|\alpha|+2)-1)}\,\mathrm{d} 
 \xi\\
& \leq CE\mathrm{e}^{-pt}\int|\xi|^{|\alpha|-|\beta|-2|\alpha|-3}\,\mathrm{d} \xi
\leq CE\mathrm{e}^{-pt}.
\end{align*}
Let  $|\beta|=0$ or $|\beta|=2N$, one  gets
\begin{equation*}
 |I_{12}|=|D^\alpha_xF_\alpha (\cdot, t)\ast u_0|
\leq CE\mathrm{e}^{-\frac{pt}{2}}B_N(x, t).
\end{equation*}
By the estimates of  $I_{11}$ and  $I_{12}$, we obtain the estimates of $I_1$.

\begin{lemma}\label{lem:i1}
For $|\alpha|\leq l$ and sufficient large $t$, we have
  \begin{equation*}
  |I_1|\leq CE(1+t)^{-\frac{2+|\alpha|}{2}}B_N(x-bt, t).
  \end{equation*}
\end{lemma}

Now we  establish the estimates of $I_2$:
\begin{align*}
  {I}_2
&=-\int_0^t D^\alpha G(t-\tau)\ast \{u, \Delta u \}\,\mathrm{d} \tau
\\
&=\int_0^t D^\alpha \partial_{x_2} G(t-\tau)\ast h_1(\tau)\,\mathrm{d} \tau
-\int_0^t D^\alpha \partial_{x_1} G(t-\tau)\ast h_2(\tau)\,\mathrm{d} \tau
=:I_{21}+I_{22},
\end{align*}
where $h_1(x, t)=u_{x_1}\Delta u$ and  $h_2(x, t)=u_{x_2}\Delta u$.
For $I_{21}$, we have the following decomposition
\begin{align*}
I_{21} 
&=\int_0^t D^\alpha \partial_{x_2} G(t-\tau)\ast h_1(\tau)\,\mathrm{d} \tau \\
&= \int_0^t D_x^\alpha\partial_{x_2}(G-F_\alpha)(t-\tau)\ast h_1(\tau)\,\mathrm{d}
 \tau+\int_0^t D_x^\alpha\partial_{x_2}F_\alpha(t-\tau) \ast h_1(\tau)\,\mathrm{d}
  \tau  \\
&=:I_{211}+I_{212}.
\end{align*}
Denote
\begin{equation*}
  \varphi_{\alpha}(x, t)=(1+t)^{\frac{2+\gamma(|\alpha|)}{2}}(B_1(x-bt, t))^{-1},
\end{equation*}
where
\begin{equation*}
  \nu(|\alpha|)=\begin{cases}
  |\alpha|, & |\alpha|\leq l-6,\\
  0,& {l-6<|\alpha|\leq l-3}.
  \end{cases}
\end{equation*}
and
\begin{equation*}
  M(t)=\sup_{0\leq \tau\leq t,\ |\alpha|
\leq l-3}\max_{\ x\in \mathbb{R}^2}|D^\alpha u(x,\tau)|\varphi_\alpha(x,\tau).
\end{equation*}
According to the definition of $h_1(x, t)$ and Theorem \ref{thm:cauchyhm},
 we have
\begin{equation*}
|D^\alpha h_1(x, t)|
\leq  \begin{cases}
  M^2(t)(1+t)^{-2-\frac{|\alpha|+3}{2}}B_{2}(x-bt, t),& |\alpha|\leq l-8\\
  M^2(t)(1+t)^{-\frac{5}{2}}B_{2}(x-bt, t),& |\alpha|=l-7, l-6\\
  (M^2(t)+CEM)(1+t)^{-\frac{3}{2}}B_{1}(x-bt, t), & |\alpha|=l-5,\\
  CEM(t)(1+t)^{-\frac{3}{2}}B_{1}(x-bt, t), & l-5<|\alpha|\leq l-3.
  \end{cases}
\end{equation*}
When $|\alpha|\leq l-8$, by Theorem \ref{thm:dgalpha}, we have
\begin{gather*}
|D^\alpha_x(G-F_{\alpha})(x, t)|\leq C (1+t)^{-\frac{2+|\alpha|}{2}}B_1(x-bt, t),\\
  |D^\alpha_xh_1(x)|\leq C M^2(t)(1+t)^{-2-\frac{|\alpha|+3}{2}}B_{2}(x-bt, t).
\end{gather*}
By Lemma $\ref{lem:sandf}$, it is easy to see that
\begin{equation*}
  |I_{211}|\leq  CM^2(t)(1+t)^{-\frac{2+|\alpha|}{2}}B_1(x-bt, t).
\end{equation*}
If $|\alpha|=l-7$, one has
\begin{align*}
 |I_{211}|
&=|\int_0^t D^\alpha \partial_{x_2} (G-F_\alpha)(t-\tau)\ast h_1(\tau)\,\mathrm{d} 
\tau|    \\
&=|\int_0^tD^{\alpha'} (D^\beta\partial_{x_2} G(t-\tau))\ast h_1(\tau)\,\mathrm{d}
 \tau|,
\end{align*}
where $|\alpha'|=l-8$, $|\alpha'|+|\beta|=l-7$.
Since
\begin{align*}
|D^{\alpha'}(D^\beta\partial_{x_2} (G-F_\alpha)(t-\tau))|
&\leq C (1+t)^{-\frac{2+l-6}{2}}B_1(x-bt, t)  \\
& \leq C(1+t)^{-\frac{2+l-7}{2}}B_1(x-bt, t).
\end{align*}
Furthermore,
\begin{align*}
|D^{\alpha'}h_2(x)|
&\leq C M^2(t)(1+t)^{-2-\frac{l-8+3}{2}}B_{2}(x-bt, t) \\
& \leq CM^2(t)(1+t)^{-2-\frac{l-7}{2}}B_{2}(x-bt, t)
 \end{align*}
Using Lemma \ref{lem:sandf}, one gets
\begin{align*}
 &|\int_0^tD^{\alpha'} (D^\beta\partial_{x_2} (G-F_\alpha)(t-\tau))
\ast h_1(\tau)\,\mathrm{d} \tau|  \\
&\leq CM^2(t)(1+t)^{-\frac{2+l-7}{2}}B_{1}(x-bt, t).
\end{align*}
When $|\alpha|=l-7$, we also have
\begin{equation*}
  |I_{211}|\leq CM^2(t)(1+t)^{-\frac{2+l-7}{2}}B_{1}(x-bt, t).
\end{equation*}
When $|\alpha|=l-6$,
\begin{align*}
 |I_{211}|
&=|\int_0^t D^\alpha \partial_{x_2}(G-F_\alpha)(t-\tau)\ast h_1(\tau)\,\mathrm{d} 
\tau| \\
&=|\int_0^tD^{\alpha''} (D^{\beta'}\partial_{x_2}
 (G-F_\alpha)(t-\tau))\ast h_1(\tau)\,\mathrm{d} \tau|,
\end{align*}
where $|\alpha''|=l-8$, $|\alpha''|+|\beta|=l-6$. Besides,
we have
\begin{gather*}
  |D^{\alpha''}(D^{\beta'}\partial_{x_2} G(t-\tau))|
 \leq C (1+t)^{-\frac{2+l-6}{2}}B_{1}(x-bt, t),\\
  |D^{\alpha''} h_1(x)|
 \leq C M^2(t)(1+t)^{-2-\frac{l-6}{2}}B_{2}(x-bt, t).
\end{gather*}
By Lemma \ref{lem:sandf},  we obtain the  estimate
\begin{align*}
  &|\int_0^tD^{\alpha''} (D^{\beta'}\partial_{x_2} 
(G-F_\alpha)(t-\tau))\ast h_1(\tau)\,\mathrm{d} \tau| \\
& \leq CM^2(t)(1+t)^{-\frac{2+l-6}{2}}B_{1}(x-bt, t).
\end{align*}
Therefore, when $|\alpha|=l-6$,
\begin{equation*}
    |I_{211}|\leq CM^2(t)(1+t)^{-\frac{2+l-2}{2}}B_{1}(x-bt, t).
\end{equation*}
When $|\alpha|<l-6$,
\begin{align*}
|I_{211}|
&=|\int_0^t D^\alpha \partial_{x_2} (G-F_\alpha)(t-\tau)\ast h_1(\tau)
 \,\mathrm{d} \tau|    \\
&=|\int_0^tD^{\gamma} (D^{\eta}\partial_{x_2} (G-F_\alpha)(t-\tau))
\ast h_1(\tau)\,\mathrm{d} \tau|,
\end{align*}
where $|\gamma|=l-8$, $|\gamma|+|\eta|=|\alpha|$.

Notice that we also have
\begin{equation*}
  |D^\gamma (D^{\eta}\partial_{x_2}(G-F_\alpha)(t-\tau))|
\leq C(1+t)^{-\frac{2+|\alpha|+1}{2}}B_1(x-bt, t).
\end{equation*}
and
\begin{align*}
 |D^\gamma h_1(x)|
&\leq CM^2(t)(1+t)^{-2-\frac{|\gamma|+3}{2}}B_2(x-bt, t)  \\
& \leq CM^2(t)(1+t)^{-2-\frac{|\alpha|+1}{2}}B_2(x-bt, t)
\end{align*}
By Lemma \ref{lem:sandf}, we have
 \begin{align*}
    |I_{211}|
&=|\int_0^t D^\alpha \partial_{x_2} (G-F_\alpha)(t-\tau)\ast h_1(\tau)\,\mathrm{d}
 \tau|    \\
&=|\int_0^tD^{\gamma} (D^{\eta}\partial_{x_2} (G-F_\alpha)(t-\tau))
 \ast h_1(\tau)\,\mathrm{d} \tau|    \\
& \leq CM^2(t)(1+t)^{-\frac{2+|\alpha|}{2}}B_1(x-bt, t).
  \end{align*}
When $l-6<|\alpha|\leq l-3$, we have $\nu{(|\alpha|)}\leq {2+|\alpha|}$. 
Then we have the following conclusion when $|\alpha|\leq l-3$,
  \begin{equation*}
    |I_{211}|\leq CM^2(t)(1+t)^{-\frac{2+\nu(|\alpha|)}{2}}B_1(x-bt, t).
  \end{equation*}

Now, we aim to deal with  $I_{212}$. Firstly, we can get the following conclusion. 
When $|\xi|$  is large enough, there exist a positive constant $b$, such that
\begin{align*}
\hat{F}_\alpha(\xi, t)
&=\chi_3(\xi)\mathrm{e}^{-\lambda|\xi|^2t}\big(\sum_{j=1}^{m}p_i(t)q_j(\xi)
+p_{m+1}(t)O(|\xi|^{-(2m+1)})\big)  \\
&=\mathrm{e}^{-bt}\sum_{j=1}^{m+1}p_j(t)\chi_3(\xi)
\mathrm{e}^{-\frac{-\lambda|\xi|^2t}{2}}q_j(\xi)  \\
&=\mathrm{e}^{-bt}\sum_{j=1}^{m+1}p_j(t)\hat{f_j}(\xi)
\end{align*}

According to the definition of  $f(\xi)$ and Lemma \ref{lem:3singularfunction}, 
there exists  $f_1$, $f_2$ and $C_0$ such that
\begin{equation*}
  f(x)=f_1(x)+f_2(x)+C_0\delta(x).
\end{equation*}

For any positive $2m>2+|\alpha|$ and $\varepsilon_0$, we have
\begin{gather*}
|D^\alpha_x f_1(x)|\leq C(1+|x|^2)^{-m},\\
\|f_2\|_{L^1}\leq C,  \quad \operatorname{supp} f_2(x) \subset \{x| |x|<2 
\varepsilon_0\}.
\end{gather*}
Therefore,
\begin{align*}
|I_{212}|
&=|\int_0^tF_\alpha(t-\tau)\ast D^\alpha_x\partial_{x_2}
 h_1(\tau)\,\mathrm{d} \tau|  \\
& \leq C \sum_{j=1}^{m+1}|\int_0^tp_j(t)(f_{1j}+f_{2j}+C_0\delta)
\mathrm{e}^{-b(t-\tau)}\ast D^\alpha_x\partial_{x_2}h_1(\tau)\,\mathrm{d} \tau|
\end{align*}
Choose $m=\max\{N', 1+l\}$, then $|D^\alpha_x f_{1j}(x)|\leq (1+|x|)^{-m}$, 
where $N'$ is the constant $N$ from Lemma $\ref{lem:qiyibufen}$.
We have already obtain that when  $l\leq 9$ and $|\alpha|\leq l-3$,
\begin{equation*}
  |D^\alpha_x\partial_{x_2}h_1(\tau)|\leq C(M^2(t)+EM(t))
(1+\tau)^{-1-\nu(|\alpha|)}B_1(x-b\tau, \tau).
\end{equation*}
By Lemma \ref{lem:qiyibufen}, we have
\begin{align*}
&\big|\int_0^tf_{1j}\mathrm{e}^{-b(t-s)}\ast D^\alpha_x\partial_{x_2}h_1(s)
 \,\mathrm{d} s\big|   \\
& \leq C(M^2(t)+EM(t))\Big|\int_0^t\int\mathrm{e}^{-b(t-s)}(1+|x-y|^2)^{-m}
(1+s)^{-1-\nu(|\alpha|)} \\
&\quad\times B_1(x-bs, s)\,\mathrm{d} y\,\mathrm{d} s\Big|\\
& \leq C(M^2(t)+EM(t))(1+t)^{-1-\nu(|\alpha|)}B_1(x-bt, t).
\end{align*}

When $|x-y|\leq 2\varepsilon_0$ and  $\varepsilon_0$ is small enough, 
by Lemma \ref{lem:qiyibufen} we obtain the estimate 
\begin{align*}
  &\big|\int_0^tf_{2j}\mathrm{e}^{-b(t-s)}\ast D^\alpha_x\partial_{x_2}h_{1j}(s)
 \,\mathrm{d} s\big|  \\
& \leq \|f_{2j}\|_{L^1}\|\int_0^tD^\alpha_x\partial_{x_2}h_{1j}(s)(\cdot,s)
 \mathrm{e}^{-b(t-s)}\,\mathrm{d} s\|_{L^\infty}(|\cdot-x|\leq 2\varepsilon_0)\\
& \leq  C(M^2(t)+EM(t))(1+t)^{-1-\nu(|\alpha|)}B_1(x-bt, t).
  \end{align*}
Notice that
\begin{equation*}
  \int_{\mathbb{R}^2}\delta(x-y) D^\alpha_x\partial_{x_2}h_1(t)\,\mathrm{d} y
= D^\alpha_x\partial_{x_2}h_1(x).
\end{equation*}
To conclude,  when $|\alpha|\leq l-3$, $l<9$, and $t$ is large enough, we have
  \begin{equation*}
    |I_{212}|\leq C(M^2(t)+EM(t))(1+t)^{-\frac{2+\nu(|\alpha|)}{2}}B_1(x-bt, t).
  \end{equation*}
By the above two conclusions we obtain the estimates for $I_{21}$, i.e.
when $|\alpha|\leq l-3$ and $l<9$,
  \begin{equation*}
    |I_{21}|\leq C(M^2(t)+EM(t))(1+t)^{-\frac{2+\nu(|\alpha|)}{2}}B_1(x-bt, t).
  \end{equation*}
Similarly, we can get the decay estimates of $I_{22}$. 
Then we obtain the estimates of $I_2$ in the following.

\begin{lemma}\label{lem:i2}
  When  $|\alpha|\leq l-3$, $l<9$, $t$ is large enough, we have
  \begin{equation*}
    |I_2|\leq C(M^2(t)+EM(t))(1+t)^{-\frac{2+\nu(|\alpha|)}{2}}B_1(x-bt, t).
  \end{equation*}
\end{lemma}

Combining Lemma \ref{lem:i1} and Lemma \ref{lem:i2}, we have
\begin{equation*}
  |D^\alpha u(x)|\leq C(E+M^2(t))(1+t)^{-\frac{2+\nu(|\alpha|)}{2}}B_1(x-bt, t).
\end{equation*}
From the definition of  $M(t)$, it is easy to get
\begin{equation*}
  M(t)\leq C(E+M^2(t)).
\end{equation*}
Taking $E$ small enough and using the continuity of $M(t)$ and induction, 
we conclude that
\begin{equation*}
  M(t)\leq CE.
\end{equation*}
Finally, Theorem \ref{th:mainhm} is proved.

\section{Appendix}\label{sec:lemmas}

 We list some known facts and lemmas which used in this paper.
Lemma \ref{lem:3f-nf} can be found in \cite{w2}, which is a vital tool to get
the pointwise estimates of the Green function $G(x, t)$.
Lemma \ref{lem:3singularfunction} gives us a useful tool to deal with
the singular part of the higher frequency part in $G(x, t)$ , which
can be found in \cite{l2}.

\begin{lemma}\label{lem:3f-nf}
 If $\hat{f}(\xi, t)$ satisfies,
 \begin{equation*}
   |D^{2\beta}\partial_t^l(\xi^\alpha\hat{f}(\xi, t))|
\leq C |\xi|^{|\alpha|+k-2|\beta|+2l}(1+(t|\xi|^2))^m\mathrm{e}^{-\nu|\xi|^2t/2},
 \end{equation*}
 for any positive integers $l$ and $m$, and multi-indexes $\alpha, \beta$
with $|\beta|\leq N$, then
 \begin{equation*}
   |D^\alpha f(x,t)|\leq C_N t^{-\frac{n+|\alpha|+k}{2}+l}B_N(x, t),
 \end{equation*}
where $N$ is any fixed integer, and
  $$
B_N(x, t)=(1+\frac{|x|^2}{1+t})^{-N}.
$$
\end{lemma}

\begin{lemma}\label{lem:b-b}
For any positive integer $N$, there exists a constant $C>0$, such that
  \begin{equation}\label{eq:b-b}
    \mathrm{e}^{-t}B_N(x+bt, t)\leq C\mathrm{e}^{-t/2}B_N(x, t),
  \end{equation}
where $x=(x_1, x_2)$ and $b=(b_1, b_2)$.
\end{lemma}

\begin{remark}\label{rem:b-b} \rm
 By a similar proof to that for Lemma $\ref{lem:b-b}$, we can also get
  \begin{equation*}
    \mathrm{e}^{-t}B_N(x, t)\leq C\mathrm{e}^{-t/2}B_N(x-bt, t).
  \end{equation*}
\end{remark}

\begin{lemma}\label{lem:3singularfunction}
If $\operatorname{supp} \hat{f}(\xi)\subset O_{R}=\{\xi:|\xi|>R\}$,
$\hat{f}(\xi)\in L^\infty \cap C^{n+1}(O_R)$, and  $\hat{f}$ satisfies
  \begin{equation*}
  |\hat{f}(\xi)|\leq C_0, \quad
 |D^\beta_{\xi}\hat{f}(\xi)|\leq C_0|\xi|^{-|\beta|-1}, (|\beta|\geq 1),
  \end{equation*}
then there exist distributions  $f_1(x), f_2(x)$ and a constant
$C>0$ depending on $n$, such that
  \begin{equation*}
    f(x)=f_1(x)+f_2(x)+C\delta(x),
  \end{equation*}
  Furthermore, for any positive constant $2m>2+|\alpha|$, we have
  \begin{equation*}
    |D^\alpha_xf_1(x)|\leq C(1+|x|^2)^{-m}, \quad \|f_2\|_{L^1}\leq C,
\quad \operatorname{supp} f_2(x)\subset \{x: |x|<2\varepsilon_0\},
  \end{equation*}
with $\varepsilon_0$ small enough.
\end{lemma}

\begin{lemma}[{\cite[Lemma 5.2]{w2}}] \label{lem:sandf}
 If functions $F(x, t)$ and  $S(x, t)$ satisfy
 \begin{gather*}
   |D^\alpha_xF(x, t)|\leq C(1+t)^{-\frac{n+|\beta|}{2}}B_{n_1}(x-bt, t),\\
   |D^\alpha_xS(x, t)|\leq C(1+t)^{-\frac{2n+|\beta|}{2}}B_{n_2}(x-bt, t),
 \end{gather*}
where $\alpha$ and $\beta$ are multi-indexes.
Then,
 \begin{equation*}
   |D^\alpha_x\Big(\int_0^t F(t-s)\ast S(s)\,\mathrm{d}s\Big)|
\leq C(1+t)^{-\frac{n+|\beta|}{2}}B_{n_3}(x-bt, t),
 \end{equation*}
where $n_1, n_2> n/2$ and $n_3=\min(n_1, n_2)$.
\end{lemma}

\begin{lemma}[{\cite[Proposition 3.2]{w1}}] \label{lem:qiyibufen}
 For any positive integers $\alpha, N'>4$, suppose \ $b>0$, then for
the large $t$, we have
  \begin{align*}
    &\Big|\int_0^t\int_{\mathbb{R}^2}\mathrm{e}^{-b(t-s)}
(1+s)^{-\frac{2+|\alpha|}{2}}(1+|x-y|^2)^{-N'}B_1(y, s)\,\mathrm{d} y\,
\mathrm{d} s\Big|    \\
& \leq C(1+t)^{-\frac{2+|\alpha|}{2}}B_1(x,t).
 \end{align*}
\end{lemma}

\begin{lemma}\label{lem:jisuan}
  Suppose  $n_1, n_2>n/2$, and let $n_3=\min(n_1, n_2)$, then we have
  \begin{equation*}
    \int_{\mathbb{R}^n} (1+\frac{|x-y|^2}{1+t})^{-n_1}(1+|y|^2)^{-n_2}\,\mathrm{d}
y\leq C(1+\frac{|x|^2}{1+t})^{-n_3}.
  \end{equation*}
\end{lemma}

\subsection*{Acknowledgments}
This research is supported by the Science Foundation for The Excellent
Youth Scholars of Ministry of Education of Shanghai, and by the
Shanghai 085 Project.


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