# On the blowup mechanism of smooth solutions to 1D quasilinear strictly
hyperbolic systems with large initial data111Li Jun<EMAIL_ADDRESS>is
supported by NSFC (No.11871030). Xu Gang<EMAIL_ADDRESS><EMAIL_ADDRESS>and Yin Huicheng<EMAIL_ADDRESS><EMAIL_ADDRESS>are
supported by NSFC (No.11731007, No.11971237).

Li Jun  Department of Mathematics, Nanjing University, Nanjing 210093, China
Xu Gang  School of Mathematical Sciences and Institute of Mathematical
Sciences, Nanjing Normal University, Nanjing 210023, China  Yin Huicheng
Department of Mathematics, Nanjing University, Nanjing 210093, China  School
of Mathematical Sciences and Institute of Mathematical Sciences, Nanjing
Normal University, Nanjing 210023, China

###### Abstract

For the first order 1D $n\times n$ quasilinear strictly hyperbolic system
$\partial_{t}u+F(u)\partial_{x}u=0$ with $u(x,0)=\varepsilon u_{0}(x)$, where
$\varepsilon>0$ is small, $u_{0}(x)\not\equiv 0$ and $u_{0}(x)\in
C_{0}^{2}(\mathbb{R})$, when at least one eigenvalue of $F(u)$ is genuinely
nonlinear, it is well-known that on the finite blowup time $T_{\varepsilon}$,
the derivatives $\partial_{t,x}u$ blow up while the solution $u$ keeps to be
small. For the 1D scalar equation or $2\times 2$ strictly hyperbolic system
(corresponding to $n=1,2$), if the smooth solution $u$ blows up in finite
time, then the blowup mechanism can be well understood (_i.e._ , only the
blowup of $\partial_{t,x}u$ happens). In the present paper, for the $n\times
n$ ($n\geq 3$) strictly hyperbolic system with a class of large initial data,
we are concerned with the blowup mechanism of smooth solution $u$ on the
finite blowup time and the detailed singularity behaviours of
$\partial_{t,x}u$ near the blowup point. Our results are based on the
efficient decomposition of $u$ along the different characteristic directions,
the suitable introduction of the modulated coordinates and the global weighted
energy estimates.

Keywords: Blowup mechanism, strictly hyperbolic system, genuinely nonlinear,
geometric blowup, modulated coordinate, global weighted energy estimate.

2010 Mathematics Subject Classification. 35L03, 35L67.

###### Contents

  1. 1 Introduction
  2. 2 Reduction
  3. 3 Reformulation under the modulated coordinates
  4. 4 Bootstrap assumptions
  5. 5 Bootstrap estimates on the bad component of $W$
  6. 6 Bootstrap estimates on good components of $W$
  7. 7 Bootstrap estimates on the modulation variables
  8. 8 Weighted energy estimates
  9. 9 Proof of main theorems
  10. A Appendix

##  1 Introduction

In the paper, we are concerned with the blowup mechanism of smooth solutions
to the following Cauchy problem of 1D $n\times n$ quasilinear strictly
hyperbolic system:

$\displaystyle\partial_{t}u+F(u)\partial_{x}u=0,$ (1.1a) $\displaystyle
u(x,0)=u_{0}(x),$ (1.1b)

where $t\geq 0$, $x\in\mathbb{R}$, $u=(u_{1},\cdots,u_{n})^{\top}$, the
$n\times n$ real matrix $F(u)$ is smooth on its argument $u$, and $u_{0}(x)\in
C^{2}(\mathbb{R})$. The strict hyperbolicity of system (1.1a) means that
$F(u)$ has $n$ distinct real eigenvalues

$\lambda_{1}(u)<\cdots<\lambda_{n}(u),$ (1.2)

meanwhile the corresponding right eigenvectors are denoted by
$\gamma_{1}(u),\cdots,\gamma_{n}(u)$ respectively. One calls system (1.1a) to
be genuinely nonlinear with respect to some eigenvalue $\lambda_{i_{0}}(u)$
($1\leq i_{0}\leq n$) when

$\nabla_{u}\lambda_{i_{0}}(u)\cdot\gamma_{i_{0}}(u)\neq 0.$ (1.3)

Otherwise, (1.1a) is called to be linearly degenerate with respect to the
eigenvalue $\lambda_{i_{0}}(u)$ when

$\nabla_{u}\lambda_{i_{0}}(u)\cdot\gamma_{i_{0}}(u)\equiv 0.$ (1.4)

Our purpose of the paper is to discuss the blowup mechanism of smooth
solutions to problem (1.1) for a class of large smooth initial data $u_{0}(x)$
provided that system (1.1a) is genuinely nonlinear with respect to some
eigenvalue $\lambda_{i_{0}}(u)$ for $i_{0}\in\\{1,\cdots,n\\}$.

###  1.1 Reviews and problems

For the 1D scalar equation

$\begin{cases}\partial_{t}v+f(v)\partial_{x}v=0,\\\
v(x,0)=v_{0}(x),\end{cases}$ (1.5)

where $v_{0}(x)\not\equiv 0$, $v_{0}(x)\in C_{0}^{1}(\mathbb{R})$, $f(v)$ is a
$C^{1}$ smooth function and $f^{\prime}(v)\not=0$ for
$v\in\text{supp}v_{0}(x)$. Set $g(x)=f(v_{0}(x))$, then by the characteristics
method, it is easy to know that the $C^{1}$ solution $v$ will blow up on the
finite positive time $T^{*}=-\frac{1}{\min g^{\prime}(x)}$ due to
$\min\limits_{x\in\mathbb{R}}g^{\prime}(x)<0$. Meanwhile, $v\in
C(\mathbb{R}\times[0,T^{*}])$ and $\lim\limits_{t\nearrow
T^{*}}\|\partial_{t,x}v(\cdot,t)\|_{C(\mathbb{R})}=\infty$ hold. This
illustrates that the blowup of solution $v$ to problem (1.5) corresponds to
the geometric blowup by the terminology in [2].

For the 1D $2\times 2$ strictly hyperbolic system

$\begin{cases}\partial_{t}v+B(v)\partial_{x}v=0,\\\
v(x,0)=v_{0}(x),\end{cases}$ (1.6)

where $v_{0}(x)\not\equiv 0$, $v_{0}(x)\in C_{0}^{1}(\mathbb{R})$, $B(v)\in
C^{1}$ is a $2\times 2$ matrix which admits two distinct real eigenvalues
$\lambda_{1}(v)$ and $\lambda_{2}(v)$, by introducing two Riemann invariants
$w_{1}=w_{1}(v)$ and $w_{2}=w_{2}(v)$, then (1.6) can be decoupled into the
following $2\times 2$ strictly hyperbolic system of $w=(w_{1},w_{2})$:

$\begin{cases}\partial_{t}w_{1}+\lambda_{1}(w)\partial_{x}w_{1}=0,\\\
\partial_{t}w_{2}+\lambda_{2}(w)\partial_{x}w_{2}=0,\\\
w(x,0)=w_{0}(x).\end{cases}$ (1.7)

When the system in (1.6) is genuinely nonlinear with respect to at least one
eigenvalue $\lambda_{i}(v)$ ($i=1,2$), then by (1.7) and [16], one knows that
the smooth solution $v$ will blow up at the maximal finite existence time
$T^{*}$, meanwhile $\|v\|_{L^{\infty}(\mathbb{R}\times[0,T^{*}])}$ is bounded
and $\lim\limits_{t\nearrow
T^{*}}\|\partial_{t,x}v(\cdot,t)\|_{C(\mathbb{R})}=\infty$ holds. This implies
that the blowup of solution $v$ to (1.6) also corresponds to the geometric
blowup.

For the small data solution problem of 1D $n\times n$ quasilinear strictly
hyperbolic system

$\begin{cases}\partial_{t}v+B(v)\partial_{x}v=0,\\\ v(x,0)=\varepsilon
v_{0}(x),\end{cases}$ (1.8)

where $\varepsilon>0$ is small, $v_{0}(x)\not\equiv 0$, $v_{0}(x)\in
C_{0}^{2}(\mathbb{R})$ and $B(v)\in C^{2}$ is a $n\times n$ matrix, when the
system in (1.8) is genuinely nonlinear with respect to at least one eigenvalue
of $B(v)$, it follows from the results in [13] and [14] that the lifespan
$T_{\varepsilon}$ of smooth solution $v$ to (1.8) satisfies

$\lim\limits_{\varepsilon\to 0^{+}}\varepsilon T_{\varepsilon}=\tau_{0}>0.$

Moreover, $\|v\|_{C(\mathbb{R}\times[0,T_{\varepsilon}])}\leq C\varepsilon$
and $\displaystyle\lim_{t\to
T_{\varepsilon}-}\|\partial_{t,x}v(\cdot,t)\|_{C(\mathbb{R})}=\infty$ hold.
This means that the blowup of solution $v$ to (1.8) corresponds to the
geometric blowup.

Compared with the results on problem (1.5) and problem (1.6), two natural
problems arise for the system (1.1a) with $n\geq 3$: when at least one
eigenvalue of $F(u)$ is genuinely nonlinear,

Q1. Can we find a class of large initial data (1.1b) such that the blowup of
solution $u$ corresponds to the geometric blowup as in the small data solution
problem (1.8)?

Q2. Can we find another class of large initial data (1.1b) such that the
solution $u$ itself blows up in finite time?

In the present paper, we focus on the investigation of Q1.

###  1.2 Statement of main results

By Proposition 2.1 in Section 2, (1.1) can be equivalently changed into the
following problem

$\displaystyle\partial_{t}w+A(w)\partial_{x}w=0,$ (1.9a) $\displaystyle
w(x,-\varepsilon)=w_{0}(x),$ (1.9b)

where $w(x,t)=(w_{1},\cdots,w_{n})^{\top}$, $t\geq-\varepsilon$, and
$\varepsilon>0$ is a small constant (for the convenience of expression, here
the initial temporal variable is shifted from $t=0$ to $t=-\varepsilon$). In
addition, the $n$ distinct real eigenvalues of smooth function matrix
$A(w)=\left(a_{ij}(w)\right)_{n\times n}$ are denoted by
$\mu_{1}(w),\cdots,\mu_{n}(w)$. Based on the reduction in Proposition 2.1 and
the strictly hyperbolic condition (1.2), there hold

$\displaystyle\mu_{1}(w)<\cdots<\mu_{i_{0}-1}(w)<\mu_{n}(w)<\mu_{i_{0}}(w)<\cdots<\mu_{n-1}(w),$
(1.10a) $\displaystyle a_{in}(w)=0\ (1\leq i\leq n-1),$ (1.10b) $\displaystyle
a_{nn}(w)=\mu_{n}(w)=\mu_{n}(0)+\partial_{w_{n}}\mu_{n}(0)w_{n}+\sum\limits_{i=1}^{n-1}\partial_{w_{i}}\mu_{n}(0)w_{i}+O(|w|^{2}),$
(1.10c) $\displaystyle A(0)=\text{diag}\\{\mu_{1}(0),\cdots,\mu_{n}(0)\\},$
(1.10d)

where $\partial_{w_{n}}\mu_{n}(0)\neq 0$. This means that the system (1.9a) is
genuinely nonlinear with respect to the eigenvalue $\mu_{n}(w)$. Let
$\ell_{i}(w)$ and $\gamma_{i}(w)$ be the left and right eigenvectors of the
matrix $A(w)$ corresponding to the eigenvalue $\mu_{i}(w)$ ($1\leq i\leq n$),
respectively. Together with (1.10), without loss of generality, one can assume

$\displaystyle\ell_{i}(w)\cdot\gamma_{j}(w)=\delta_{i}^{j}\ (1\leq i,j\leq
n),$ (1.11a) $\displaystyle\gamma_{n}(w)={\bf e}_{n},\ \gamma_{i}(0)={\bf
e}_{i},\ \|\gamma_{i}(w)\|=1\ (1\leq i\leq n-1),$ (1.11b)
$\displaystyle\ell_{i}(0)={\bf e}_{i}^{\top}\ (1\leq i\leq n),$ (1.11c)

where
$\|\gamma_{i}(w)\|=\sqrt{\displaystyle\sum_{k=1}^{n}(\gamma_{i}^{k})^{2}(w)}$
with $\gamma_{i}(w)=(\gamma_{i}^{1}(w),...,\gamma_{i}^{n}(w))^{\top}$.

To study the blowup mechanism of smooth solution to problem (1.9) with a class
of large initial data $w_{0}(x)=(w_{10},\cdots,w_{n0})(x)$, motivated by
[3]-[5], we choose $w_{0}(x)$ as follows:

At first, let $w_{n0}(x)$ satisfy the following generic nondegenerate
condition at $x=0$:

$w_{n0}(0)=\kappa_{0}\varepsilon^{\frac{1}{3}},\
w_{n0}^{\prime}(0)=-\frac{1}{\varepsilon}=\min\limits_{x\in\mathbb{R}}w_{n0}^{\prime}(x),\
w_{n0}^{\prime\prime}(0)=0,\
w_{n0}^{\prime\prime\prime}(0)=\frac{6}{\varepsilon^{4}},$ (1.12)

where $\kappa_{0}$ is a suitable constant.

Secondly, in order to derive $L^{\infty}$ estimates for the lower order
derivatives of $w$ and track the development of possible singularity, we
require such assumptions of $w_{0}(x)$:

$\displaystyle|w_{n0}(x)-w_{n0}(0)|\leq
2\varepsilon^{\frac{1}{2}-\frac{1}{30}},$ (1.13a)
$\displaystyle|\hat{w}(x)|\leq\varepsilon^{\frac{3}{2}}\eta^{\frac{1}{6}}(\varepsilon^{-\frac{3}{2}}x),\
\
|\hat{w}^{\prime}(x)|\leq\eta^{-\frac{1}{3}}(\varepsilon^{-\frac{3}{2}}x)\quad\text{for
$|x|\leq\mathcal{L}\varepsilon^{\frac{3}{2}}$},$ (1.13b)
$\displaystyle|\hat{w}^{(4)}(x)|\leq\varepsilon^{\frac{1}{9}-\frac{9}{2}}\quad\text{for
$|x|\leq\varepsilon^{\frac{3}{2}}$},$ (1.13c)
$\displaystyle\varepsilon|w_{n0}^{\prime}(x)|\leq
2\eta^{-\frac{1}{3}}(\varepsilon^{-\frac{3}{2}}x)\quad\text{for
$\mathcal{L}\varepsilon^{\frac{3}{2}}\leq|x|\leq
2\mathcal{L}\varepsilon^{\frac{3}{2}}$},$ (1.13d)
$\displaystyle\varepsilon|w_{n0}^{\prime}(x)|\leq\eta^{-1}(\varepsilon^{-\frac{3}{2}}x)\quad\text{for
$|x|\geq 2\mathcal{L}\varepsilon^{\frac{3}{2}}$},$ (1.13e)

and for $1\leq j\leq n-1$,

$|w_{j0}(x)|\leq\varepsilon,\
|w_{j0}^{\prime}(x)|\leq\eta^{-\frac{1}{3}}(\varepsilon^{-\frac{3}{2}}x),\
|w_{j0}^{\prime\prime}(x)|\leq\varepsilon^{-\frac{11}{6}}\eta^{-\frac{1}{3}}(\varepsilon^{-\frac{3}{2}}x),$
(1.14)

where $\mathcal{L}=\varepsilon^{-\frac{1}{10}},\eta(x)=1+x^{2}$ and
$\hat{w}(x)=w_{n0}(x)-\kappa_{0}\varepsilon^{\frac{1}{3}}-\overline{w}(x)$
with
$\overline{w}(x)=\varepsilon^{\frac{1}{2}}\overline{W}(\varepsilon^{-\frac{3}{2}}x)$
and
$\overline{W}(y)=(-\frac{y}{2}+(\frac{1}{27}+\frac{y^{2}}{4})^{\frac{1}{2}})^{\frac{1}{3}}-(\frac{y}{2}+(\frac{1}{27}+\frac{y^{2}}{4})^{\frac{1}{2}})^{\frac{1}{3}}$.

Thirdly, in order to derive the $L^{2}-$energy estimates for the
$\mu_{0}-$order derivatives of $w$, we demand that

$\sum\limits_{j=1}^{n-1}\|\partial_{x}^{\mu_{0}}w_{j0}(\cdot)\|_{L^{2}}+\varepsilon\|\partial_{x}^{\mu_{0}}w_{n0}(\cdot)\|_{L^{2}}\lesssim\varepsilon^{\frac{3}{2}(1-\mu_{0})},$
(1.15)

where $\mu_{0}\geq 6$ is a suitably given constant.

Our main results are stated as:

######  Theorem 1.1.

Under the conditions (1.10), and without loss of generality, $\mu_{n}(0)=0$
and $\partial_{w_{n}}\mu_{n}(0)=1$ are assumed, then there exists a positive
constant $\varepsilon_{0}$ such that when $0<\varepsilon<\varepsilon_{0}$ and
$w_{0}(x)$ satisfies (1.12)-(1.15), the problem (1.9) admits a unique local
smooth solution $w$, which will firstly blow up at the point $(x^{*},T^{*})$.
Moreover,

  1. $(1)$

$x^{*}=O(\varepsilon^{2}),\ T^{*}=O(\varepsilon^{\frac{4}{3}}).$

  2. $(2)$

$w$ lies in the following spaces:

$\begin{cases}w\in C([-\varepsilon,T^{*}),H^{\mu_{0}}(\mathbb{R}))\cap
C^{1}([-\varepsilon,T^{*}),H^{\mu_{0}-1}(\mathbb{R})),\\\\[5.69054pt] w_{i}\in
C^{1}([-\varepsilon,T^{*}]\times\mathbb{R})\ (1\leq i\leq n-1),\ w_{n}\in
L^{\infty}([0,T^{*}],C^{\frac{1}{3}}(\mathbb{R})).\end{cases}$ (1.16)

  3. $(3)$

There exist two smooth functions $\xi(t)$ and $\tau(t)$ such that

$\begin{cases}\lim\limits_{t\nearrow T^{*}}(\xi(t),\tau(t))=(x^{*},T^{*}),\
\lim\limits_{t\nearrow
T^{*}}\partial_{x}w_{n}(\xi(t),t)=-\infty,\\\\[2.84526pt]
-2<(T^{*}-t)\partial_{x}w_{n}(\xi(t),t)<-\frac{1}{2}\quad\text{for
$-\varepsilon\leq t<T^{*}$},\\\\[2.84526pt]
\text{$|\tau(t)-T^{*}|\lesssim\varepsilon^{\frac{1}{3}}(T^{*}-t)$ and
$|\xi(t)-x^{*}|\lesssim\varepsilon(T^{*}-t)$ for $-\varepsilon\leq
t<T^{*}$}.\end{cases}$ (1.17)

######  Remark 1.1.

By Theorem 1.1, we know that the solution $w\in C(\mathbb{R}\times[0,T^{*}])$
of (1.9a) blows up at the point $(x^{*},T^{*})$, i.e., $\lim\limits_{t\nearrow
T^{*}}\|\partial_{t,x}w(\cdot,t)\|_{C(\mathbb{R})}=\infty$. This corresponds
to the geometric blowup for the problem (1.9).

######  Remark 1.2.

The assumptions of $\mu_{n}(0)=0$ and $\partial_{w_{n}}\mu_{n}(0)=1$ in
Theorem 1.1 can be realized by the translation $(t,x)\mapsto(t,x+\mu_{n}(0)t)$
and then the spatial scaling $x\mapsto\partial_{w_{n}}\mu_{n}(0)x$.

######  Remark 1.3.

We now give some comments on (1.9)-(1.10). It is not difficult to find that
there are a great number of $w_{0}(x)$ to fulfill the constrains
(1.12)-(1.15). In addition, it follows from Proposition 2.1 that the unknown
$w$ admits the good components $(w_{1},\cdots,w_{n-1})$ and the bad component
$w_{n}$. The conditions (1.13b)-(1.13d) imply that the bad component $w_{n}$
mainly tracks the possible singularity and it can be thought as a suitable
perturbation of the singular function $\overline{W}$. On the other hand, in
order to control the detailed behaviors of $w_{n}$ near the possible blowup
point, we posed the suitable perturbation for the fourth order derivatives of
$\hat{w}$ in (1.13c) when $|x|\leq\varepsilon^{\frac{3}{2}}$. The conditions
(1.13a) and (1.13e) are posed to control the behavior of $w_{n}$ away from the
blowup position. In addition, to avoid the influence of the initial data
$w_{0}(x)$ at infinity, we naturally pose the appropriate decaying condition
(1.14)-(1.15) for large $|x|$.

######  Remark 1.4.

In [3]-[5], through introducing suitable modulated coordinates and taking the
constructive proofs, the authors systematically study the shock formation of
multidimensional compressible Euler equations with a class of smooth initial
data. Motivated by these papers, we study the geometric blowup mechanism of
problem (1.1), whose nonlinear structure is more general than the 1D
compressible Euler equations. Thanks to the new reformulation in the
equivalent problem (1.9) as well as (1.10), we can establish some suitable
exponential-growth controls on the bounds of the characteristics corresponding
to $\mu_{i}(w)\ (1\leq i\leq n-1)$ (see Lemma 6.1 and Lemma 6.2 below) such
that the problem (1.9) can be mainly dominated by the approximate Burgers
equation of $w_{n}$.

######  Remark 1.5.

When (1.9a) admits the structure of conservation laws, there are some
interesting works on the shock construction through the first-in-time blowup
point $(x^{*},T^{*})$ for $t\geq T^{*}$. For instances, under various
nondegenerate conditions with finite orders or infinitely degenerate
conditions for the initial data, the shock construction from the blowup point
is completed for the 1D scalar equation $\partial_{t}u+\partial_{x}(f(u))=0$
in [22]; under the generic nondegenerate condition of initial data, the shock
surface from the blowup curve has been constructed for the multidimensional
scalar equation
$\partial_{t}u+\partial_{1}(f_{1}(u))+\cdot\cdot\cdot+\partial_{n}(f_{n}(u))=0$
in [23]; under the generic nondegenerate conditions of the initial data, for
the 1-D $2\times 2$ $p-$ system of polytropic gases, the authors in [15],[17]
and [7] obtain the formation and construction of the shock wave starting from
the blowup point under some variant conditions; for the 1-D $3\times 3$
strictly hyperbolic conservation laws with the small initial data or the 3-D
full compressible Euler equations with symmetric structure and small perturbed
initial data, the authors in [8], [24] and [10] also get the formation and
construction of the resulting shock waves, respectively. In the near future,
we hope that the shock formation can be constructed from the blowup point
$(x^{*},T^{*})$ in Theorem 1.1 when (1.9a) has the structure of conservation
laws.

######  Remark 1.6.

In the recent years, the formation of shock waves have made much progress for
the multi-dimensional Euler equations and the quasilinear wave equations under
various restrictions on the related initial data. One can see the remarkable
articles [3]-[6], [9], [11]-[12], [18], [20] and [21].

###  1.3 Comments on the proof of Theorem 1.1

Let us give comments on the proof of Theorem 1.1. Motivated by [5], by
introducing the modulated coordinate which is smooth before the singularity
formation, we can convert the finite time singularity formation of (1.9) into
the global well-posedness of smooth solutions to the resulting new system of
$W=(W_{1},\cdot\cdot\cdot,W_{n})^{\top}$ (see (3.4)-(3.5)). To achieve this
aim, we take the following strategies:

$\bullet$ Due to the important form of (1.9a) with (1.10), we divide $W$ as
the $n-1$ good components $(W_{1},\cdots,W_{n-1})^{\top}$ and the bad unknown
$w_{n}$. Inspired by [4], we continue to decompose $w_{n}$ into another bad
part $W_{0}$ and a good part $\kappa(t)$ (see (3.3)). The $L^{\infty}$
estimates for the lower order derivatives of $W_{0}$ are carried out in two
different domains
$\\{(y,s):|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}\\}$ and
$\\{(y,s):|y|\geq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}\\}$ with
$\mathcal{L}=\varepsilon^{-\frac{1}{10}}$. In the interior domain
$\\{(y,s):|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}\\}$,
$W_{0}$ is expected to have the similar behavior as $\overline{W}(y)$, which
is the steady solution of 1D Burgers type equation
$(\partial_{s}-\frac{1}{2})\overline{W}+\left(\frac{3}{2}y+\overline{W}\right)\partial_{y}\overline{W}=0$.
In the exterior domain
$\\{(y,s):|y|\geq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}\\}$, the
treatment of $W_{0}$ is rather delicate since the temporal and spatial decay
estimates of good components $(W_{1},\cdots,W_{n-1})^{\top}$ are required to
be established simultaneously.

$\bullet$ Due to the partially decoupling form of (1.9a), in order to prove
Theorem 1.1, we need to establish the $L^{\infty}$ estimates of the lower
order derivatives and the $L^{2}$ estimates of the highest order derivatives
for $(W_{1},\cdots,W_{n})^{\top}$. To get the related $L^{\infty}$ estimates,
by utilizing the characteristics method and delicate analysis, at first, we
derive the basic exponential controls on the bounds of the characteristics
corresponding to $\mu_{i}(w)\ (1\leq i\leq n-1)$. Subsequently, the spatial
decay rate $(1+y^{2})^{-\frac{1}{3}}$ of $\partial_{y}w_{0}$ and further the
temporal decay of $\partial_{y}W_{0}$ are obtained. From these, the
$L^{\infty}$ estimates of $(W_{1},\cdots,W_{n})^{\top}$ are achieved. On the
other hand, we observe that the coefficients in the equations of $w$ admit the
key $O(e^{\frac{s}{2}})$ scale because of the strict hyperbolicity of (1.9a)
(see (8.38)). This will lead to the expected $L^{2}$ estimates on the highest
order derivatives of $(W_{1},\cdots,W_{n})^{\top}$. Here, we specially point
out that the $L^{\infty}$ estimates of each related quantity depend on the
information of the higher order derivatives of $W$ since the related (1.9a)
only admits the partial decoupling form. This is the main reason to apply the
$L^{2}$ estimates for dealing with the highest order derivatives of $W$.

When these are done, the proof of Theorem 1.1 can be completed successfully.
It is hoped that our analysis methods in the paper will be adopted to study
the singularity formation problem for the general multi-dimensional symmetric
hyperbolic systems with some classes of large initial data, which is a
generalization of the results in [3]-[6] for the multi-dimensional
compressible Euler equations.

The rest of the paper is arranged as follows: In Section 2, we reduce the
problem (1.1) into the equivalent partially decoupling problem (1.9) via
Proposition 2.1. In Section 3, under the modulated coordinate, the problem
(1.9) and the choice of initial data $w_{0}(x)$ are reformulated. Moreover, as
the heuristics of the formation of the expected singularity, the rigorous
derivation on the resulting Burgers-type equation is also given in this
section. The bootstrapping assumptions and their closure of the arguments are
arranged in Section 4-Section 8 respectively: The descriptions of
bootstrapping assumptions on $w$ and the modulated coordinate are made in
Section 4; the $L^{\infty}$ estimates for the bad unknown $W_{n}$ and the good
components $(W_{1},\cdot\cdot\cdot,W_{n-1})^{\top}$ are taken in Section
5-Section 6 respectively. In addition, the closure of bootstrapping
assumptions for the modulation variables is completed in Section 7; the
related energy estimates for the higher order derivatives of $W$ are derived
in Section 8. In Section 9, we establish the main results in Theorem 3.1 and
further Theorem 1.1. Finally, a useful interpolation inequality and its
application for deriving some delicate estimates are given in Appendix A.

##  2 Reduction

In the section, our main aim is to reduce (1.1a) to a partially decoupling
form (1.9a) such that the resulting new unknown functions
$w=(w_{1},\cdot\cdot\cdot,w_{n})^{\top}$ will admit $n-1$ good components and
only one bad component. The good component and the bad component mean that
their regularities are in $C^{1}$ and in $C^{1/3}$ up to the blowup time,
respectively.

######  Proposition 2.1.

Under assumptions (1.2)-(1.3), there exists a constant $\delta_{0}>0$ such
that when $|u|<\delta_{0}$, the system (1.1a) can be equivalently reduced into

$\partial_{t}w+A(w)\partial_{x}w=0,$ (2.1)

where the smooth mapping $u\mapsto w=w(u)$ is invertible and $w(0)=0$. In
addition, the inverse mapping of $w(u)$ is denoted as $u=u(w)$, and the
$n\times n$ matrix

$A(w)=\left(\frac{\partial w}{\partial u}\right)F(u(w))\left(\frac{\partial
w}{\partial u}\right)^{-1}:=\left(a_{ij}(w)\right)_{n\times n}$

satisfies

  1. $(1)$

$A(w)$ has $n$ distinct eigenvalues $\left\\{\mu_{i}(w)\right\\}_{i=1}^{n}$
with

$\mu_{i}(w)=\lambda_{i}(u(w))\ (1\leq i<i_{0});\
\mu_{i}(w)=\lambda_{i+1}(u(w))\ (i_{0}\leq i<n);\
\mu_{n}(w)=\lambda_{i_{0}}(u(w)).$

  2. $(2)$

$a_{in}(w)=0\ (i\neq n),\ a_{nn}(w)=\mu_{n}(w)$ and
$\partial_{w_{n}}\mu_{n}(w)\neq 0$.

  3. $(3)$

$A(0)=\text{diag}\\{\mu_{1}(0),\cdots,\mu_{n}(0)\\}$.

###### Proof.

At first, we claim that when $|u|\leq\delta_{0}$ for some constant
$\delta_{0}>0$, there exist $(n-1)$ linearly independent Riemann invariants
$\alpha_{i}(u)\ (i\neq i_{0})$ corresponding to $\lambda_{i_{0}}(u)$ such that

$\nabla_{u}\alpha_{i}(u)\cdot\gamma_{i_{0}}(u)=0\ (i\neq i_{0},\
|u|<\delta_{0}).$ (2.2)

Indeed, let $\\{\zeta_{i}\\}_{(i\neq i_{0})}$ be $(n-1)$ linearly independent
column vectors orthogonal to $\gamma_{i_{0}}(0)$ and set
$\alpha_{i}(u)=\zeta_{i}^{\top}\cdot u+\bar{\alpha}_{i}(u)$, then it follows
from (2.2) that the unknowns $\\{\bar{\alpha}_{i}(u)\\}_{i\neq i_{0}}$ should
satisfy

$\nabla_{u}\bar{\alpha}_{i}(u)\cdot\gamma_{i_{0}}(u)=-\zeta_{i}^{\top}\cdot(\gamma_{i_{0}}(u)-\gamma_{i_{0}}(0)),\
\bar{\alpha}_{i}(0)=0\ (i\neq i_{0}).$ (2.3)

It is not difficult to find that there exists a constant $\delta_{0}>0$ such
that (2.3) is uniquely solved when $|u|<\delta_{0}$ and
$|\bar{\alpha}_{i}(u)|\lesssim|u|^{2}$. Hence (2.2) is obtained.

By $\alpha_{i}(0)=0\ (i\neq i_{0})$, we define a mapping $u\mapsto
v=v(u)=(v_{1},\cdots,v_{n})^{\top}(u)$ with $v(0)=0$ as

$v_{i}=\alpha_{i}(u)\ (1\leq i<i_{0});\quad v_{i}=\alpha_{i+1}(u)\ (i_{0}\leq
i<n);\quad v_{n}=\gamma_{i_{0}}^{\top}(0)\cdot u.$ (2.4)

Note that $\\{\zeta_{i}\\}_{i\neq i_{0}}$ are $(n-1)$ linearly independent
column vectors which are orthogonal to $\gamma_{i_{0}}(0)$. Then the
transformation $u\mapsto v=v(u)$ is reversible for $|u|<\delta_{0}$ since its
Jacobian matrix $J_{v}(u)$ satisfies $J_{v}(0)=\left(\frac{\partial
v}{\partial
u}\right)|_{u=0}=\left(\zeta_{1},\cdots,\zeta_{i_{0}-1},\zeta_{i_{0}+1},\cdots,\zeta_{n},\gamma_{i_{0}}(0)\right)^{\top}$
and $J_{v}(0)$ is non-singular. We now denote the inverse mapping of $v=v(u)$
as $u=u(v)$.

By (2.2) and (2.4), the system (1.1a) is equivalently converted into

$\partial_{t}v+G(v)\partial_{x}v=0,$ (2.5)

where $G(v)=J_{v}(u)F(u(v))J_{v}^{-1}(u):=\left(g_{ij}(v)\right)_{n\times n}$,
$G(v)$ has $n$ distinct eigenvalues $\\{\lambda_{i}(u(v))\\}_{i=1}^{n}$ and
the corresponding right eigenvectors are
$\\{J_{v}(u)\gamma_{i}(u(v))\\}_{i=1}^{n}$. In addition, (2.2) shows

$J_{v}(u)\gamma_{i_{0}}(u(v))=\gamma_{i_{0}}^{\top}(0)\cdot\gamma_{i_{0}}(u(v)){\bf
e}_{n}\neq{\bf 0}.$ (2.6)

This implies that

$g_{in}(v)=0\ (i\neq n),\quad g_{nn}(v)=\lambda_{i_{0}}(u(v)).$ (2.7)

It follows from (2.5) and (2.7) that the $(n-1)$ order square matrix

$G_{n-1}(v)=\left(g_{ij}(v)\right)_{(n-1)\times(n-1)}$ (2.8)

has $(n-1)$ eigenvalues

$\lambda_{1}(u(v))<\cdots<\lambda_{i_{0}-1}(u(v))<\lambda_{i_{0}+1}(u(v))<\cdots<\lambda_{n}(u(v)).$

Then there exists a unique $(n-1)$ order invertible constant square matrix
$B_{n-1}=\left(b_{ij}\right)_{(n-1)\times(n-1)}$ such that

$\bar{G}_{n-1}(v)=B_{n-1}G_{n-1}(v)B_{n-1}^{-1}:=\left(\bar{g}_{ij}(v)\right)_{(n-1)\times(n-1)},$
(2.9)

where

$\bar{G}_{n-1}(0)=\text{diag}\\{\lambda_{1}(0),\cdots,\lambda_{i_{0}-1}(0),\lambda_{i_{0}+1}(0),\cdots,\lambda_{n}(0)\\}.$

Furthermore, it is derived from (1.3) and (2.6)-(2.7) that

$\nabla_{v}\lambda_{i_{0}}(u(v))J_{v}(u)\gamma_{i_{0}}(u(v))=\nabla_{u}\lambda_{i_{0}}(u)\cdot\gamma_{i_{0}}(u)\neq
0.$ (2.10)

Denote the invertible transformation $v\mapsto
w=w(v):=(w_{1},\cdots,w_{n})^{\top}(v)$ as

$(w_{1},\cdots,w_{n-1})^{\top}=B_{n-1}(v_{1},\cdots,v_{n-1})^{\top},\quad
w_{n}=v_{n}+\sum\limits_{j=1}^{n-1}b_{nj}w_{j},$ (2.11)

where the constants $\\{b_{nj}\\}_{j=1}^{n-1}$ will be determined later. Set
the inverse mapping of $w=w(v)$ as $v=v(w)$ with $v(0)=0$. By (2.5),
(2.8)-(2.11) and a direct computation, we arrive at

$\partial_{t}w+A(w)\partial_{x}w=0,$ (2.12)

where $A(w)=\left(a_{ij}(w)\right)_{n\times n}$ satisfies

$\begin{cases}a_{ij}(w)=\bar{g}_{ij}(v(w))\ (1\leq i,j\leq
n-1),\\\\[5.69054pt] a_{jn}(w)=0\ (1\leq j\leq n-1),\\\\[5.69054pt]
a_{nn}(w)=g_{nn}(v(w))\end{cases}$ (2.13)

and

$\displaystyle(a_{n1},\cdots,a_{nn-1})(w)$ (2.14) $\displaystyle=$
$\displaystyle(g_{n1},\cdots,g_{nn-1})(v(w))B_{n-1}+(b_{n1},\cdots,b_{nn-1})(\bar{G}_{n-1}-g_{nn}I_{n-1})(v(w)).$

Since the $(n-1)$ order square matrix

$\bar{G}_{n-1}(0)-g_{nn}(0)I_{n-1}=\text{diag}\\{\lambda_{1}(0)-\lambda_{i_{0}}(0),\cdots\lambda_{i_{0}-1}(0)-\lambda_{i_{0}}(0),\lambda_{i_{0}+1}(0)-\lambda_{i_{0}}(0),\cdots,\lambda_{n}(0)-\lambda_{i_{0}}(0)\\}$

is invertible due to (1.2) and (2.7), it is derived from (2.14) that there
exists unique $\\{b_{nj}\\}_{1\leq j\leq n-1}$ such that

$(a_{n1},\cdots,a_{n,n-1})(0)=(0,\cdots,0).$ (2.15)

For the constant invertible square matrix $B=\left(b_{ij}\right)_{n\times n}$
with $b_{in}=\delta_{i}^{n}\ (1\leq i\leq n)$ and

$w=Bz,$ (2.16)

one has from (2.11)-(2.14) that

$A(w)=BG(B^{-1}w)B^{-1}:=(a_{ij}(w))_{n\times n},$ (2.17)

where

$A(0)=BG(0)B^{-1}=\text{diag}\\{\lambda_{1}(0),\cdots,\lambda_{i_{0}-1}(0),\lambda_{i_{0}+1}(0),\cdots,\lambda_{n}(0),\lambda_{i_{0}}(0)\\}.$

The expected invertible mapping $u\mapsto w=w(u)$ is just the composition of
two mappings $v=v(u)$ and $w=w(v)$ defined by (2.4) and (2.16) respectively.
Its inverse mapping is denoted as $u=u(w)$. It is easy to know that the matrix
$A(w)$ has $n$ distant eigenvalues $\\{\lambda_{i}(w(u))\\}_{i=1}^{n}$ and the
corresponding right eigenvectors are
$\\{J_{w}(u)\gamma_{i}(u(w))\\}_{i=1}^{n}$. In addition, it is derived from
(2.6), (2.10) and (2.11) that

$J_{w}(u)\gamma_{i_{0}}(u(w))=BJ_{v}(u)\gamma_{i_{0}}(u(v))=\gamma_{i_{0}}^{\top}(0)\cdot\gamma_{i_{0}}(u(v)){\bf
e}_{n}\neq{\bf 0}$ (2.18)

and

$\displaystyle\nabla_{w}\lambda_{i_{0}}(u(w))\cdot
J_{w}(u)\gamma_{i_{0}}(u(w))$ (2.19) $\displaystyle=$
$\displaystyle\nabla_{v}\lambda_{i_{0}}(u(v))\cdot J_{v}(w)\cdot J_{w}(v)\cdot
J_{v}(u)\gamma_{i_{0}}(u(v))$ $\displaystyle=$
$\displaystyle\nabla_{v}\lambda_{i_{0}}(u(v))\cdot
J_{v}(u)\gamma_{i_{0}}(u(v))\neq 0.$

Then the properties (1)-(3) of $A(w)$ come from (2.8)-(2.9), (2.13), (2.15)
and (2.18)-(2.19). ∎

######  Remark 2.1.

We point out that Proposition 2.1 is a generalization of Lemma 2.1 in [8],
where (1.1) with small initial data and $n=3$ is simplified analogously.

##  3 Reformulation under the modulated coordinates

Motivated by [4], to show the geometric blowup mechanism in Theorem 1.1, we
introduce three modulation variables $\tau(t),\ \xi(t)$ and $\kappa(t)$ as

$\begin{cases}\tau(t):\quad\text{tracking the exact blowup time},\\\
\xi(t):\quad\text{tracking the location of the blowup point},\\\
\kappa(t):\quad\text{fixing the speed of the singularity
development}.\end{cases}$ (3.1)

Set the modulated coordinate $(y,s)$ as follows

$s=s(t)=-\log(\tau(t)-t),\quad
y=(x-\xi(t))(\tau(t)-t)^{-\frac{3}{2}}=(x-\xi(t))e^{\frac{3s}{2}}.$ (3.2)

In addition, the new unknowns $W=(W_{1},\cdots,W_{n})^{\top}$ and $W_{0}$ are
defined as

$w_{i}(x,t)=W_{i}(y,s)(i\neq n),\quad
w_{n}(x,t)=W_{n}(y,s)=e^{-\frac{s}{2}}W_{0}(y,s)+\kappa(t),$ (3.3)

where $\kappa(t)=w_{n}(\xi(t),t)$.

By (3.1)-(3.3), when $t<\tau(t)$, the system (1.9a) can be equivalently
rewritten as

$\partial_{s}W+\left(\frac{3}{2}y-e^{\frac{s}{2}}\beta_{\tau}\dot{\xi}(t)\right)\partial_{y}W+e^{\frac{s}{2}}\beta_{\tau}A(w)\partial_{y}W=0,$
(3.4)

where $\beta_{\tau}(t)=\frac{1}{1-\dot{\tau}(t)}$,
$\dot{\xi}(t)=\xi^{\prime}(t)$ and $\dot{\tau}(t)={\tau}^{\prime}(t)$.

In addition, $W_{0}$ is determined by

$(\partial_{s}-\frac{1}{2})W_{0}+\left(\frac{3}{2}y+e^{\frac{s}{2}}\beta_{\tau}(\mu_{n}(w)-\dot{\xi}(t))\right)\partial_{y}W_{0}+\sum\limits_{j\neq
n}e^{s}\beta_{\tau}a_{nj}(w)\partial_{y}W_{j}=-e^{-\frac{s}{2}}\beta_{\tau}\dot{\kappa}(t),$
(3.5)

where $\dot{\kappa}(t)=\kappa^{\prime}(t)$.

###  3.1 Global steady solution for 1D Burgers equation

The simplest case of problem (1.9) is (i.e., $n=1$ and $a_{n}(w)=w_{n}$)

$\begin{cases}\partial_{t}\overline{\omega}+\overline{\omega}\partial_{x}\overline{\omega}=0,\
x\in\mathbb{R},\ t>-\varepsilon,\\\
\overline{\omega}(x,-\varepsilon)=\overline{\omega}_{0}(x),\
x\in\mathbb{R},\end{cases}$ (3.6)

where $\overline{\omega}_{0}(x)\in C^{\infty}(\mathbb{R})$ and
$\overline{\omega}_{0}^{\prime}(x)\leq 0$. It is assumed that
$\overline{\omega}_{0}(x)$ satisfies the generic nondegenerate condition at
$x=0$:

$\overline{\omega}_{0}(0)=0,\
\overline{\omega}_{0}^{\prime}(0)=-\frac{1}{\varepsilon}=\min\limits_{x\in\mathbb{R}}\overline{\omega}_{0}^{\prime}(x),\
\overline{\omega}_{0}^{\prime\prime}(0)=0,\
\overline{\omega}_{0}^{\prime\prime\prime}(0)=\frac{6}{\varepsilon^{4}}.$
(3.7)

By the characteristics method, it is easy to know that under the assumption
(3.7), the smooth solution $\overline{\omega}(x,t)$ of problem (3.6) will
blowup at the first-in-time singularity point $(0,0)$ and the related
characteristics starting from the point $(0,-\varepsilon)$ is $\\{(x,t):x=0,\
-\varepsilon<t<0\\}$. From this and the procedures in (3.1)-(3.5), we define

$\tau_{0}(t)=0,\ \xi_{0}(t)=0,\ \kappa_{0}(t)=0,$ (3.8)

and

$s=s(t)=-\log(\tau_{0}(t)-t),\
y=(x-\xi_{0}(t))(\tau_{0}(t)-t)^{-\frac{3}{2}}=(x-\xi_{0}(t))e^{\frac{3s}{2}}$
(3.9)

and

$\overline{\omega}=e^{-\frac{s}{2}}\overline{W}(y,s)+\kappa_{0}(t).$ (3.10)

Then it follows from (3.6) and (3.8)-(3.10) that $\overline{W}(y,s)$ satisfies

$(\partial_{s}-\frac{1}{2})\overline{W}+\left(\frac{3}{2}y+\overline{W}\right)\partial_{y}\overline{W}=0.$
(3.11)

In Appendix A.1 of [4], it is proved that equation (3.11) has a group of
steady smooth solutions $\overline{W}=\overline{W}(y)$ satisfying such a
generic nondegenerate condition

$\overline{W}(0)=0,\
\overline{W}^{\prime}(0)=\min\limits_{y\in\mathbb{R}}\overline{W}^{\prime}(y)<0,\
\overline{W}^{\prime\prime}(0)=0,\ \overline{W}^{\prime\prime\prime}(0)>0.$
(3.12)

According to the initial data (3.7) and the transformation (3.8)-(3.9), the
solution $\overline{W}(y)$ of (3.11) is introduced in [4]

$\overline{W}(y)=\left(-\frac{y}{2}+\left(\frac{1}{27}+\frac{y^{2}}{4}\right)^{\frac{1}{2}}\right)^{\frac{1}{3}}-\left(\frac{y}{2}+\left(\frac{1}{27}+\frac{y^{2}}{4}\right)^{\frac{1}{2}}\right)^{\frac{1}{3}}.$
(3.13)

In addition, it is easy to obtain

$\displaystyle\overline{W}(0)=0,\ \ \ \overline{W}^{\prime}(0)=-1,\ \ \
\overline{W}^{\prime\prime}(0)=0,\ \ \
\overline{W}^{\prime\prime\prime}(0)=6,$ (3.14a)
$\displaystyle\|\eta^{-\frac{1}{6}}\overline{W}\|_{L^{\infty}}\leq 1,\
-1\leq\eta^{\frac{1}{3}}\overline{W}^{\prime}\leq-\frac{1}{6},\
\|\eta^{\frac{5}{6}}\overline{W}^{\prime\prime}\|_{L^{\infty}}\leq 2,$ (3.14b)
$\displaystyle\|\eta^{\frac{5}{6}}\overline{W}^{(\mu)}\|_{L^{\infty}}\lesssim_{\mu}1\quad\text{for
$\mu\geq 3$}.$ (3.14c)

###  3.2 Evolution for the modulation variables

With the expectation
$\lim\limits_{s\nearrow+\infty}W_{0}(y,s)=\overline{W}(y)$ and by the
properties of $\overline{W}(y)$, we pose

$W_{0}(0,s)=0,\quad\partial_{y}W_{0}(0,s)=-1,\quad\partial_{y}^{2}W_{0}(0,s)=0,\quad\partial_{y}^{3}W_{0}(0,s)=6.$
(3.15)

Next, we derive the equations of the modulation variables in (3.1). For any
nonnegative integer $\mu$, acting $\partial^{\mu}=\partial_{y}^{\mu}$ on both
sides of (3.4) and (3.5) yields

$\displaystyle(\partial_{s}+\frac{3}{2}\mu)\partial^{\mu}W+(\frac{3}{2}y-e^{\frac{s}{2}}\beta_{\tau}\dot{\xi}(t))\partial_{y}\partial^{\mu}W+e^{\frac{s}{2}}\beta_{\tau}A(w)\partial_{y}\partial^{\mu}W=F_{\mu},$
(3.16a)
$\displaystyle(\partial_{s}+\frac{3\mu-1}{2})\partial^{\mu}W_{0}+\left(\frac{3}{2}y+e^{\frac{s}{2}}\beta_{\tau}(\mu_{n}(w)-\dot{\xi}(t))\right)\partial_{y}\partial^{\mu}W_{0}=F_{\mu}^{0},$
(3.16b)

where

$\begin{cases}F_{\mu}=-e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{1\leq\beta\leq\mu}C_{\mu}^{\beta}\partial^{\beta}A(w)\partial_{y}\partial^{\mu-\beta}W,\\\\[5.69054pt]
F_{\mu}^{0}=-e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{1\leq\beta\leq\mu}C_{\mu}^{\beta}\partial^{\beta}\mu_{n}(w)\partial_{y}\partial^{\mu-\beta}W_{0}+e^{s}\beta_{\tau}\sum\limits_{j\neq
n}\partial^{\mu}\left(a_{nj}(w)\partial_{y}W_{j}\right)-e^{-\frac{s}{2}}\beta_{\tau}\dot{\kappa}(t)\delta_{\mu}^{0}.\\\
\end{cases}$

Due to (3.15), it is derived from (3.16b) that for $\mu=0,1,2$, the modulation
variables satisfy the following ordinary differential system:

$\displaystyle\dot{\kappa}(t)$
$\displaystyle=e^{s}\left(\mu_{n}(w^{0})-\dot{\xi}(t)\right)-e^{\frac{3s}{2}}\sum\limits_{j\neq
n}a_{nj}(w^{0})(\partial_{y}W_{j})^{0},$ (3.17a) $\displaystyle\dot{\tau}(t)$
$\displaystyle=1-\partial_{w_{n}}\mu_{n}(w^{0})+e^{\frac{s}{2}}\sum\limits_{j\neq
n}\partial_{w_{j}}\mu_{n}(w^{0})(\partial_{y}W_{j})^{0}$
$\displaystyle\quad-e^{s}\sum\limits_{j\neq
n}\left(\partial_{y}(a_{nj}(w)\partial_{y}W_{j})\right)^{0},$ (3.17b)
$\displaystyle\dot{\xi}(t)$
$\displaystyle=\mu_{n}(w^{0})-\frac{1}{6}(\partial_{y}^{2}\mu_{n}(w))^{0}+\frac{1}{6}\sum\limits_{j\neq
n}e^{\frac{s}{2}}\left(\partial_{y}^{2}(a_{nj}(w)\partial_{y}W_{j})\right)^{0},$
(3.17c)

where the notation $v^{0}$ represents $v(0,s)$ for the function $v(y,s)$.

###  3.3 The equation of $\mathcal{W}=W_{0}-\overline{W}$

Set

$\mathcal{W}=W_{0}-\overline{W}.$ (3.18)

It follows from (3.5) and (3.11) that for any nonnegative integer $\mu$,

$\left(\partial_{s}+\left(\frac{3}{2}y+e^{\frac{s}{2}}\beta_{\tau}(\mu_{n}(w)-\dot{\xi}(t)\right)\partial_{y}\right)\partial^{\mu}\mathcal{W}+\mathcal{D}_{\mu}\partial^{\mu}\mathcal{W}=\mathcal{F}_{\mu},$
(3.19)

where

$\mathcal{D}_{\mu}=\frac{3\mu-1}{2}+\beta_{\tau}\overline{W}^{\prime}+e^{\frac{s}{2}}\beta_{\tau}\mu\partial_{y}\mu_{n}(w)\\\
$

and

$\displaystyle\mathcal{F}_{\mu}=$
$\displaystyle-\sum\limits_{1\leq\beta\leq\mu}C_{\mu}^{\beta}\beta_{\tau}\partial_{y}^{1+\beta}\overline{W}\partial_{y}^{\mu-\beta}\mathcal{W}-\sum\limits_{2\leq\beta\leq\mu}C_{\mu}^{\beta}e^{\frac{s}{2}}\beta_{\tau}\partial_{y}^{\beta}\mu_{n}(w)\partial_{y}^{\mu-\beta+1}\mathcal{W}-e^{-\frac{s}{2}}\beta_{\tau}\dot{\kappa}(t)\delta_{\mu}^{0}$
$\displaystyle-\sum\limits_{j\neq
n}e^{s}\beta_{\tau}\partial_{y}^{\mu}\left(a_{nj}(w)\partial_{y}W_{j}\right)-\beta_{\tau}e^{\frac{s}{2}}\partial_{y}^{\mu}\left(\overline{W}^{\prime}(\mu_{n}(w)-e^{-\frac{s}{2}}W_{0}-\dot{\xi}(t))\right)$
$\displaystyle+(1-\beta_{\tau})\partial_{y}^{\mu}(\overline{W}^{\prime}\overline{W}).$

In addition, for the purpose to establish the weighted estimates of
$\mathcal{W}$, it is derived from (3.19) that for any real number $\nu$,

$\left(\partial_{s}+\left(\frac{3}{2}y+e^{\frac{s}{2}}\beta_{\tau}(\mu_{n}(w)-\dot{\xi}(t))\right)\partial_{y}\right)\left[\eta^{\nu}\partial^{\mu}\mathcal{W}\right]+\mathcal{D}_{\mu,\nu}\left[\eta^{\nu}\partial^{\mu}\mathcal{W}\right]=\eta^{\nu}\mathcal{F}_{\mu},$
(3.20)

where

$\mathcal{D}_{\mu,\nu}=\mathcal{D}_{\mu}-2\nu
y\eta^{-1}\left(\frac{3}{2}y+e^{\frac{s}{2}}\beta_{\tau}(\mu_{n}(w)-\dot{\xi}(t))\right).$

###  3.4 The decomposition on the derivatives of $W$

To deal with the derivatives of $W$, we adopt the method of eigendecomposition
in [16]. Set

$\partial_{y}^{\mu}W=\sum\limits_{m=1}^{n}W_{\mu}^{m}\gamma_{m}(w),$ (3.21)

where $\left\\{\gamma_{m}(w)\right\\}_{m=1}^{n}$ have been defined in (1.11).

Acting each left eigenvector $\ell_{m}(w)$ on both sides of (3.16a) and
substituting the expansion (3.21) into (3.16a) yield

$\left(\partial_{s}+\left(\frac{3}{2}y+e^{\frac{s}{2}}\beta_{\tau}(\mu_{m}(w)-\dot{\xi}(t))\right)\partial_{y}+\frac{3}{2}\mu\right)W_{\mu}^{m}=\mathbb{F}_{\mu}^{m},$
(3.22)

where

$\mathbb{F}_{\mu}^{m}=-\sum\limits_{j=1}^{n}W_{\mu}^{j}\ell_{m}(w)\cdot\left(\partial_{s}\gamma_{j}(w)+((\frac{3}{2}y-e^{\frac{s}{2}}\beta_{\tau}\dot{\xi}(t))I_{n}+e^{\frac{s}{2}}\beta_{\tau}A(w))\partial_{y}\gamma_{j}(w)\right)+\ell_{m}(w)\cdot
F_{\mu}.$

On the other hand, for any real number $\nu$, it is derived from (3.22) that

$\left(\partial_{s}+\left(\frac{3}{2}y+e^{\frac{s}{2}}\beta_{\tau}(\mu_{m}(w)-\dot{\xi}(t))\right)\partial_{y}\right)\left[\eta^{\nu}W_{\mu}^{m}\right]+\mathbb{D}_{\mu,\nu}^{m}\left[\eta^{\nu}W_{\mu}^{m}\right]=\eta^{\nu}\mathbb{F}_{\mu}^{m},$
(3.23)

where

$\mathbb{D}_{\mu,\nu}^{m}=\frac{3\mu}{2}-2\nu
y\eta^{-1}\left(\frac{3}{2}y+e^{\frac{s}{2}}\beta_{\tau}(\mu_{m}(w)-\dot{\xi}(t)\right).$

###  3.5 Initial data and main results under the modulation coordinates

Under the constrains (1.12)-(1.15) and the definition (3.3), the initial data
of $W(y,s)$ on $s=-\log\varepsilon$ can be determined accordingly.

Indeed, due to (1.12) and the definitions (3.1) and (3.3) (also see (3.15)),
the initial data of the modulation variables $\tau(t),\xi(t)$ and $\kappa(t)$
on $t=-\varepsilon$ are

$\tau(-\varepsilon)=0,\ \xi(-\varepsilon)=0,\
\kappa(-\varepsilon)=w_{n}(0,-\varepsilon)=\kappa_{0}\varepsilon^{\frac{1}{3}}.$
(3.24)

In addition, for the bad component $W_{0}(y,s)$, it is derived from
(1.12)-(1.13) and (3.2)-(3.3) that on $s=-\log\varepsilon$,

$W_{0}(0,s)=\partial_{y}^{2}W_{0}(0,s)=0,\
\partial_{y}W_{0}(0,s)=\min\limits_{y\in\mathbb{R}}\partial_{y}W_{0}(y,s)=-1,\
\partial_{y}^{3}W_{0}(0,s)=6$ (3.25)

and

$\displaystyle|W_{0}(y,-\log\varepsilon)|\leq 2\varepsilon^{-\frac{1}{30}},$
(3.26a)
$\displaystyle\text{$|\mathcal{W}(y,-\log\varepsilon)|\leq\varepsilon\eta^{\frac{1}{6}}(y)$
and
$|\partial_{y}\mathcal{W}(y,-\log\varepsilon)|\leq\varepsilon\eta^{-\frac{1}{3}}(y)$
for $|y|\leq\mathcal{L}$},$ (3.26b)
$\displaystyle|\partial_{y}^{4}\mathcal{W}(y,-\log\varepsilon)|\leq\varepsilon^{\frac{1}{9}}\quad\text{for
$|y|\leq 1$},$ (3.26c)
$\displaystyle|\partial_{y}W_{0}(y,-\log\varepsilon)|\leq
2\eta^{-\frac{1}{3}}(y)\boldsymbol{1}_{\\{\mathcal{L}\leq|y|\leq
2\mathcal{L}\\}}+\eta^{-1}(y)\boldsymbol{1}_{\\{|y|\geq
2\mathcal{L}\\}}\quad\text{for $|y|\geq\mathcal{L}$}.$ (3.26d)

For the good components of $W$, it follows from (1.14) and (3.2)-(3.3) that
for $j\neq n$

$\displaystyle|W_{j}(y,-\log\varepsilon)|\leq\varepsilon,$ (3.27a)
$\displaystyle|\partial_{y}W_{j}(y,-\log\varepsilon)|\leq\varepsilon^{\frac{3}{2}-3\nu}\eta^{-\nu}(y)\
(\nu\in[0,\frac{1}{3}]),$ (3.27b)
$\displaystyle|\partial_{y}^{2}W_{j}(y,-\log\varepsilon)|\leq\varepsilon^{\frac{7}{6}}\eta^{-\frac{1}{3}}(y).$
(3.27c)

Following (1.15), the initial energy of $\partial_{y}^{\mu_{0}}W$ on
$s=-\log\varepsilon$ satisfies

$\sum\limits_{j=1}^{n-1}\|\partial_{y}^{\mu_{0}}W_{j}(\cdot,-\log\varepsilon)\|_{L^{2}(\mathbb{R})}+\varepsilon^{\frac{3}{2}}\|\partial_{y}^{\mu_{0}}W_{0}(\cdot,-\log\varepsilon)\|_{L^{2}(\mathbb{R})}\lesssim\varepsilon^{\frac{3}{2}}.$
(3.28)

Under the preparations above, the new version of Theorem 1.1 under the
modulated coordinates can be stated as:

######  Theorem 3.1.

Under the conditions in (1.10) and the notations in (3.1)-(3.3), there exists
a positive constant $\varepsilon_{0}$ such that when
$0<\varepsilon<\varepsilon_{0}$, the system (3.4)-(3.5) and (3.17) with the
initial data satisfying (3.24)-(3.28) has a global-in-time solution $W$ and
$\tau(t),\xi(t),\kappa(t)$, which satisfy

  1. $(1)$

$\lim\limits_{s\to+\infty}\xi(t)=x^{*}=O(\varepsilon^{2}),\lim\limits_{s\to+\infty}\tau(t)=T^{*}=O(\varepsilon^{\frac{4}{3}})$.

  2. $(2)$

$|\dot{\tau}(t)|\lesssim\varepsilon^{\frac{1}{3}},\
|\dot{\xi}(t)|\lesssim\varepsilon,\ |\dot{\kappa}(t)|\lesssim 1$, and
$|\tau(t)|\lesssim\varepsilon^{\frac{4}{3}},\
|\xi(t)|\lesssim\varepsilon^{2},\
|\kappa(t)-\kappa_{0}\varepsilon^{\frac{1}{3}}|\lesssim\varepsilon$.

  3. $(3)$

$|W_{0}(y,s)|\lesssim\varepsilon^{\frac{1}{3}}e^{\frac{s}{2}},|W_{i}(y,s)|\lesssim\varepsilon\
(1\leq i\leq n-1)$.

  4. $(4)$

With respect to $W_{0}$,

$\begin{cases}\text{$|W_{0}(y,s)-\overline{W}(y)|\leq\varepsilon^{\frac{1}{11}}\eta^{\frac{1}{6}}(y)$
and
$|\partial_{y}(W_{0}(y,s)-\overline{W}(y))|\leq\varepsilon^{\frac{1}{12}}\eta^{-\frac{1}{3}}(y)$
for $|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$},\\\
|\partial_{y}W_{0}(y,s)|\leq\frac{7}{6}\eta^{-\frac{1}{3}}(y)\quad\text{for
$|y|\geq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$}.\end{cases}$

  5. $(5)$

For $1\leq j\leq n-1$,

$|\partial_{y}W_{j}(y,s)|\lesssim e^{(3\nu-\frac{3}{2})s}\eta^{-\nu}(y)\
(\nu\in[0,\frac{1}{3}]),|\partial_{y}^{2}W_{j}(y,s)|\lesssim
e^{(\nu^{+}-\frac{7}{6})s}\eta^{-\frac{1}{3}}(y)\ (\nu^{+}>0).$

  6. $(6)$

For $\mu_{0}$ given in (4.5),

$\|\partial_{y}^{\mu_{0}}W_{j}(\cdot,s)\|_{L^{2}(\mathbb{R})}\lesssim
e^{-\frac{3}{2}s}\ (1\leq j\leq n-1),\
\|\partial_{y}^{\mu_{0}}W_{n}(\cdot,s)\|_{L^{2}(\mathbb{R})}\lesssim
e^{-\frac{s}{2}}.$

######  Remark 3.1.

In Theorem 3.1, the spatial decay estimates in $(4)$ and $(5)$ come from the
influences of the initial data $W(y,-\log\varepsilon)$ without compact
support.

##  4 Bootstrap assumptions

Since the local existence of (1.9) was known already (one can see [19] for
instance), we utilize the continuous induction to establish the global-in-time
estimates in Theorem 3.1. According to the initial data in (3.24)-(3.28), we
first make the following induction assumptions. In what follows, $M>0$ is
denoted as a suitably large constant, which is independent of $\varepsilon$.

For the modulation variables in (3.1), suppose that

$\displaystyle|\kappa(t)-\kappa_{0}\varepsilon^{\frac{1}{3}}|\leq
M\varepsilon,\quad|\tau(t)|\leq
M\varepsilon^{\frac{4}{3}},\quad\quad|\xi(t)|\leq M\varepsilon^{2},$ (4.1a)
$\displaystyle|\dot{\kappa}(t)|\leq M,\quad\quad|\dot{\tau}(t)|\leq
M\varepsilon^{\frac{1}{3}},\quad\quad|\dot{\xi}(t)|\leq M\varepsilon.$ (4.1b)

For the bad unknown $W_{0}$ and the related $\mathcal{W}$ in (3.18), the
bootstrap assumptions are

$\displaystyle\|W_{0}(\cdot,s)\|_{L^{\infty}}\leq
M\varepsilon^{\frac{1}{3}}e^{\frac{s}{2}},\ |\partial_{y}^{\mu}W_{0}(y,s)|\leq
M\ (1\leq\mu\leq 4),$ (4.2a)
$\displaystyle|\partial_{y}W_{0}(y,s)|\leq\frac{7}{6}\eta^{-\frac{1}{3}}(y)\quad\text{for
$|y|\geq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$},$ (4.2b)
$\displaystyle\text{$|\mathcal{W}(y,s)|\leq\varepsilon^{\frac{1}{11}}\eta^{\frac{1}{6}}(y)$
and
$|\partial_{y}\mathcal{W}(y,s)|\leq\varepsilon^{\frac{1}{12}}\eta^{-\frac{1}{3}}(y)$
for $|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$}.$ (4.2c)

For the good unknowns $W_{i}(1\leq i\leq n-1)$, it is assumed that

$\displaystyle|W_{i}(y,s)|\leq M\varepsilon,\
|\partial_{y}^{\mu}W_{i}(y,s)|\leq Me^{-\frac{3}{2}s}\ (1\leq\mu\leq 4),$
(4.3a) $\displaystyle|\partial_{y}W_{i}(y,s)|\leq
Me^{(3\nu-\frac{3}{2})s}\eta^{-\nu}(y)\ (\nu\in[0,\frac{1}{3}]),$ (4.3b)
$\displaystyle|\partial_{y}^{2}W_{i}(y,s)|\leq
Me^{(\nu^{+}-\frac{7}{6})s}\eta^{-\frac{1}{3}}(y)\ (\nu^{+}>0).$ (4.3c)

In addition, we make the following auxiliary assumptions with
$\ell=\frac{1}{M^{4}}$,

$|\partial_{y}^{\mu}\mathcal{W}(y,s)|\leq\varepsilon^{\frac{1}{10}}\ell^{4-\mu},\
0\leq\mu\leq 4,\ |y|\leq\ell.$ (4.4)

With respect to the energies of the higher order derivatives of $W$, by fixing
$\mu_{0}$ to be the minimum positive integer such that

$\mu_{0}\geq 6,\
3\mu_{0}-e^{\frac{s}{2}}\beta_{\tau}\partial_{y}\mu_{m}(w)\geq\frac{13}{2}\
(1\leq m\leq n-1),$ (4.5)

we assume

$\sum\limits_{m=1}^{n-1}\|\partial_{y}^{\mu_{0}}W_{m}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq
Me^{-\frac{3}{2}s},\quad\|\partial_{y}^{\mu_{0}}W_{n}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq
Me^{-\frac{s}{2}}.$ (4.6)

##  5 Bootstrap estimates on the bad component of $W$

In the section, we close the bootstrap arguments on $W_{0}$ and $\mathcal{W}$.

###  5.1 The analysis on the characteristics of (3.5)

We now study some properties of the characteristics of (3.5). For any point
$(y_{0},\zeta_{0})$ with $\zeta_{0}\geq-\log\varepsilon$, the characteristics
$y(\zeta)=y(\zeta;y_{0},\zeta_{0})$ of (3.5) starting from $(y_{0},\zeta_{0})$
is defined as

$\begin{cases}\dot{y}(\zeta)=\frac{3}{2}y(\zeta)+e^{\frac{s}{2}}\beta_{\tau}(\mu_{n}(w)-\dot{\xi}(t(\zeta)))(y(\zeta),\zeta),\
\zeta\geq\zeta_{0},\\\ y(\zeta_{0})=y_{0}.\end{cases}$ (5.1)

######  Proposition 5.1.

Under the assumptions (1.10c) and (4.1)-(4.3), when $|y_{0}|\geq\ell$, one has

$|y(\zeta)|\geq|y_{0}|e^{\frac{\zeta-\zeta_{0}}{2}}\ (\zeta\geq\zeta_{0}).$
(5.2)

In addition, if $(y(\zeta),\zeta)$ goes through some point $(y,s)$ with
$|y|\leq\ell$ and $s\geq\zeta_{0}$, then

$|y(\zeta)|\leq\ell\ (\zeta_{0}\leq\zeta\leq s).$ (5.3)

To prove Proposition 5.1 and for later uses, we first establish the following
results on $\mu_{n}(w)-\dot{\xi}(t)$.

######  Lemma 5.2.

One has

$|\mu_{n}(w)-\dot{\xi}(t)|\leq
e^{-\frac{s}{2}}(\frac{7}{6}+\varepsilon^{\frac{1}{20}})|y|+M^{2}e^{-s}$ (5.4)

and

$|\mu_{n}(w)-\dot{\xi}(t)-e^{-\frac{s}{2}}W_{0}|\leq\varepsilon^{\frac{1}{8}}e^{-\frac{s}{2}}|y|^{\frac{1}{2}}+M^{2}e^{-s}.$
(5.5)

###### Proof.

Note that

$(\mu_{n}(w)-\dot{\xi}(t))(y,s)=(\mu_{n}(w)-\mu_{n}(w^{0}))(y,s)+(\mu_{n}(w^{0})-\dot{\xi}(t))(s).$
(5.6)

Then it follows from (3.3), (3.15), (3.17c), (4.2a) and (4.3a) that

$|\mu_{n}(w^{0})-\dot{\xi}(t)|(s)\leq M^{2}e^{-s}.$ (5.7)

In addition, due to $\partial_{w_{n}}\mu_{n}(0)=1$, it is derived from
(1.10c), (3.15), (4.2a)-(4.2b), (4.3a) and (4.3b) with $\nu=\frac{1}{4}$ that

$\displaystyle|\mu_{n}(w)-\mu_{n}(w^{0})-e^{-\frac{s}{2}}W_{0}|(y,s)$ (5.8)
$\displaystyle=$
$\displaystyle|\mu_{n}(w)-\mu_{n}(w^{0})-e^{-\frac{s}{2}}\int_{0}^{y}\partial_{y}W_{0}(z,s)dz|(y,s)$
$\displaystyle\leq$
$\displaystyle\sum\limits_{i=1}^{n}|\int_{0}^{1}\partial_{w_{i}}\mu_{n}(\beta
w+(1-\beta)w^{0})d\beta-\delta_{n}^{i}|\cdot|\int_{0}^{y}\partial_{y}W_{i}(z,s)dz|$
$\displaystyle\leq$ $\displaystyle
M^{2}\varepsilon^{\frac{1}{4}}e^{-\frac{s}{2}}\min\\{|y|^{\frac{1}{2}},|y|\\}.$

By (4.2b)-(4.2c), (3.14b) and (3.15), we arrive at

$|W_{0}(y,s)|=|\int_{0}^{y}\partial_{y}W_{0}(z,s)dz|\leq(\frac{7}{6}+\varepsilon^{\frac{1}{14}})|y|.$
(5.9)

Substituting (5.7)-(5.9) into (5.6) yields (5.4)-(5.5) and then the proof of
Lemma 5.2 is completed.∎

We now start the proof of Proposition 5.1.

###### Proof.

Since (5.3) can be easily derived from (5.2), it suffices to prove (5.2). Due
to $|\beta_{\tau}|\leq 2$ by (3.4) and (4.1b), then it follows from (5.1) and
(5.4) that

$\frac{d}{d\zeta}y^{2}(\zeta)\geq\frac{7}{12}y^{2}(\zeta)-2M^{2}e^{-\zeta}|y(\zeta)|\geq\frac{13}{24}y^{2}(\zeta)-M^{2}e^{-\zeta}.$
(5.10)

We derive from (5.10) and the assumption $|y_{0}|\geq\ell$ that

$e^{-\frac{13}{24}\zeta}y^{2}(\zeta)\geq
e^{-\frac{13}{24}\zeta_{0}}y_{0}^{2}-M^{2}e^{-(1+\frac{13}{24})\zeta_{0}}\geq\frac{1}{4}e^{-\frac{13}{24}\zeta_{0}}\ell^{2}\
(\zeta\geq\zeta_{0})$

and

$|y(\zeta)|\geq\frac{1}{2}\ell\quad\text{when $\zeta\geq\zeta_{0}$}.$

Together with (5.10), this yields

$\frac{d}{d\zeta}y^{2}(\zeta)\geq\frac{1}{2}y^{2}(\zeta).$

Then (5.2) is obtained and the proof of Proposition 5.1 is completed.∎

######  Remark 5.1.

For each point $(y,s)\in\mathbb{R}\times[-\log\varepsilon,+\infty)$, one can
define the following backward characteristics $y=y(\zeta)$ starting from
$(y_{0},\zeta_{0})=(y_{0}(y,s),\zeta_{0})$

$\begin{cases}\dot{y}(\zeta)=\frac{3}{2}y(\zeta)+e^{\frac{s}{2}}\beta_{\tau}(\mu_{n}(w)-\dot{\xi}(t))(y(\zeta),\zeta),\
\zeta_{0}\leq\zeta\leq s,\\\ y(\zeta_{0})=y_{0}.\end{cases}$ (5.11)

According to Proposition 5.1, $(y(\zeta),\zeta)$ can be clarified into one of
the following cases:

  1. Case $1$.

Set $\zeta_{0}=-\log\varepsilon$ and there are no additional constrains on
$y(\zeta)$.

  2. Case $2$.

When $|y|\leq\ell$, set $\zeta_{0}=-\log\varepsilon$, then
$|y(\zeta)|\leq\ell$ holds for $-\log\varepsilon\leq\zeta\leq s$ due to (5.3).

  3. Case $3$.

When $|y|\geq\ell$, set $(y_{0},\zeta_{0})$ with $|y_{0}|>\ell$ and
$\zeta_{0}=-\log\varepsilon$ or $|y_{0}|=\ell$ and
$\zeta_{0}\geq-\log\varepsilon$, then
$|y(\zeta)|\geq|y_{0}|e^{\frac{\zeta-\zeta_{0}}{2}}$ holds for
$\zeta_{0}\leq\zeta\leq s$ due to (5.2).

###  5.2 Bootstrap estimate on $W_{0}$

For each point $(y,s)\in\mathbb{R}\times[-\log\varepsilon,+\infty)$, it
follows from the Case 1 in Remark 5.1 and (3.16b) with $\mu=0$ that

$(\frac{d}{d\zeta}-\frac{1}{2})W_{0}(y(\zeta),\zeta)=F_{0}^{0}(y(\zeta),\zeta).$
(5.12)

By (4.1b), (4.2a) and (4.3a), we have

$\displaystyle|F_{0}^{0}(y(\zeta),\zeta)|\leq$ $\displaystyle
e^{-\frac{\zeta}{2}}(|\beta_{\tau}||\dot{\kappa}(t)|)(\zeta)+\beta_{\tau}e^{\zeta}\sum\limits_{j=1}^{n-1}(|a_{nj}(w)||\partial_{y}W_{j}|)(y(\zeta),\zeta)$
(5.13) $\displaystyle\leq$ $\displaystyle
2Me^{-\frac{\zeta}{2}}+M^{2}e^{-\frac{\zeta}{2}}\leq
2M^{2}e^{-\frac{\zeta}{2}}.$

Combining (5.12) with (5.13) yields

$\displaystyle e^{-\frac{s}{2}}|W_{0}(y,s)|$
$\displaystyle=|e^{\frac{1}{2}\log\varepsilon}W_{0}(y_{0},-\log\varepsilon)+\int_{-\log\varepsilon}^{s}e^{-\frac{\zeta}{2}}F_{0}^{0}(y(\zeta),\zeta)d\zeta|$
(5.14)
$\displaystyle\leq\sqrt{\varepsilon}\|W_{0}(\cdot,-\log\varepsilon)\|_{L^{\infty}}+2M^{2}\varepsilon.$

Then it is derived from (5.14) and (3.26a) that

$\|W_{0}(\cdot,s)\|_{L^{\infty}}\leq\varepsilon^{\frac{1}{3}}e^{\frac{s}{2}}.$
(5.15)

###  5.3 Bootstrap estimate on $\mathcal{W}$ when $|y|\leq\ell$

For each $(y,s)\in\mathbb{R}\times[-\log\varepsilon,+\infty)$, by Case 1 in
Remark 5.1, it follows from (3.19) that

$\displaystyle\partial^{\mu}\mathcal{W}(y,s)=$
$\displaystyle\partial^{\mu}\mathcal{W}(y_{0}(y,s),-\log\varepsilon)\exp\left({-\int_{-\log\varepsilon}^{s}\mathcal{D}_{\mu}(y(\alpha),\alpha)d\alpha}\right)$
(5.16)
$\displaystyle+\int_{-\log\varepsilon}^{s}\mathcal{F}_{\mu}(y(\zeta),\zeta)\exp\left(-{\int_{\zeta}^{s}\mathcal{D}_{\mu}(y(\alpha),\alpha)d\alpha}\right)d\zeta.$

We next estimate $\partial_{y}^{4}\mathcal{W}(y,s)$ when $|y|\leq\ell$. In
this situation, $|y(\alpha)|\leq\ell$ holds for
$-\log\varepsilon\leq\alpha\leq s$ by (5.3) in Proposition 5.1. For $\mu=4$ in
(3.19), one has from (3.3) and (3.18) that

$\displaystyle\mathcal{D}_{4}(y(\alpha),\alpha)$ (5.17) $\displaystyle=$
$\displaystyle\frac{11}{2}+\beta_{\tau}\overline{W}^{\prime}(y(\alpha))+\sum\limits_{m=1}^{n}4e^{\frac{s}{2}}\beta_{\tau}\partial_{w_{m}}\mu_{n}(w)\partial_{y}W_{m}$
$\displaystyle=$
$\displaystyle\frac{11}{2}+\beta_{\tau}\overline{W}^{\prime}(y(\alpha))+4\beta_{\tau}\partial_{y}W_{0}(y(\alpha),\alpha)+\sum\limits_{m=1}^{n}4e^{\frac{s}{2}}\beta_{\tau}\left(\partial_{w_{m}}\mu_{n}(w)-\delta_{n}^{m}\right)\partial_{y}W_{m}$
$\displaystyle=$
$\displaystyle\frac{11}{2}+5\beta_{\tau}\overline{W}^{\prime}(y(\alpha))+4\beta_{\tau}\partial_{y}\mathcal{W}(y(\alpha),\alpha)+\sum\limits_{m=1}^{n}4e^{\frac{s}{2}}\beta_{\tau}\left(\partial_{w_{m}}\mu_{n}(w)-\delta_{n}^{m}\right)\partial_{y}W_{m}.$

Due to $\partial_{w_{n}}\mu_{n}(0)=1$ and $|y(\alpha)|\leq\ell$ for
$-\log\varepsilon\leq\alpha\leq s$, then by (5.17), (1.10), (3.3), (3.14b),
(4.1), (4.2) and (4.3a), we arrive at

$\mathcal{D}_{4}\geq\frac{11}{2}-5-M\varepsilon^{\frac{1}{12}}\geq\frac{1}{3}.$
(5.18)

It follows from (5.16) and (5.18) that

$\displaystyle|\partial_{y}^{4}\mathcal{W}(y,s)|$
$\displaystyle\leq|\partial_{y}^{4}\mathcal{W}(y_{0}(y,s),-\log\varepsilon)|+3\|\mathcal{F}_{4}(y(\zeta),\zeta)\chi_{[-\log\varepsilon,s]}(\zeta)\|_{L^{\infty}}.$
(5.19)

When $|y|\leq\ell$, it is derived from (5.3) in Proposition 5.1 and (3.26c)
that $|y_{0}|=|y_{0}(y,s)|\leq\ell<1$ and

$|\partial_{y}^{4}\mathcal{W}(y_{0}(y,s),-\log\varepsilon)|\leq\varepsilon^{\frac{1}{9}}.$
(5.20)

For $\mathcal{F}_{4}(y(\zeta),\zeta)$ in (5.19), it follows from (3.19),
(1.10), (3.14), (3.17c), (4.1), (4.2a)-(4.2b) and (4.3a)-(4.3b) that

$\displaystyle|\mathcal{F}_{4}(y(\zeta),\zeta)|$ (5.21) $\displaystyle\leq$
$\displaystyle\sum\limits_{\nu=0}^{3}M|\partial_{y}^{\nu}\mathcal{W}(y(\zeta),\zeta)|+M^{2}\varepsilon^{\frac{1}{2}}+Me^{\frac{s}{2}}\sum\limits_{\nu=0}^{4}|\partial_{y}^{\nu}(\mu_{n}(w)-e^{-\frac{s}{2}}W_{0}-\dot{\xi}(t))|(y(\zeta),\zeta)$

and

$\displaystyle\sum\limits_{\nu=0}^{4}|\partial_{y}^{\nu}(\mu_{n}(w)-e^{-\frac{s}{2}}W_{0}-\dot{\xi}(t))|(y(\zeta),\zeta)$
(5.22) $\displaystyle=$
$\displaystyle\sum\limits_{\nu=0}^{4}|\partial_{y}^{\nu}\left(\sum\limits_{m=1}^{n}\int_{0}^{1}(\partial_{w_{m}}\mu_{n}(\beta
w+(1-\beta)w^{0})d\beta-\delta_{n}^{m})\cdot\int_{0}^{y}\partial_{y}W_{m}(z,\zeta)dz\right)|(y(\zeta),\zeta)$
$\displaystyle\leq$ $\displaystyle
M^{2}e^{-\frac{3s}{2}}+M^{2}\varepsilon^{\frac{1}{3}}e^{-\frac{s}{2}}(1+|y(\zeta)|).$

By the definition of $\ell$ in (4.4) and $|y(\zeta)|\leq\ell$ shown in (5.3)
when $|y|\leq\ell$, then it is derived from (5.21)-(5.22) and (4.4) that

$|\mathcal{F}_{4}(y(\zeta),\zeta)|\leq\varepsilon^{\frac{1}{10}}\ell^{\frac{3}{5}}.$

Combining this with (5.19)-(5.20) yields

$|\partial_{y}^{4}\mathcal{W}(y,s)|\leq
2\varepsilon^{\frac{1}{10}}\ell^{\frac{1}{2}}\leq\frac{1}{2}\varepsilon^{\frac{1}{10}}\quad\text{for
$|y|\leq\ell$}.$ (5.23)

In addition, by (3.14a) and (3.15), $\partial_{y}^{\mu}\mathcal{W}(0,s)=0$ for
$\mu=1,2,3$, we have that for $|y|\leq\ell$,

$\displaystyle|\partial_{y}^{\nu}\mathcal{W}(y,s)|=$
$\displaystyle|\int_{0}^{y}\partial_{y}^{\nu+1}\mathcal{W}(z,s)dz|$ (5.24)
$\displaystyle\leq$ $\displaystyle
2\varepsilon^{\frac{1}{10}}\ell^{4-\nu+\frac{1}{2}}\leq\frac{1}{2}\varepsilon^{\frac{1}{10}}\ell^{4-\nu}\quad(0\leq\nu\leq
3).$

###  5.4 Bootstrap estimates on $\mathcal{W}$ when
$|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$

In the region $\\{(y,s):|y|\leq\ell\\}$, it is derived from (5.24) that

$|\eta^{-\frac{1}{6}}(y)\mathcal{W}(y,s)|\leq
2\varepsilon^{\frac{1}{10}}\ell^{4+\frac{1}{2}}\quad\text{for $|y|\leq\ell$}.$
(5.25)

In the region
$\\{(y,s):\ell<|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}\\}$,
by Case 3 in Remark 5.1 and (3.20), one has

$(\frac{d}{d\zeta}+\mathcal{D}_{\mu,\nu}(y(\zeta),\zeta))[\eta^{\nu}\partial^{\mu}\mathcal{W}](y(\zeta)\zeta)=[\eta^{\nu}\mathcal{F}_{\mu}](y(\zeta),\zeta)$

and

$\displaystyle\left[\eta^{\nu}\partial^{\mu}\mathcal{W}\right](y,s)$
$\displaystyle=[\eta^{\nu}\partial^{\mu}\mathcal{W}](y_{0},\zeta_{0})\exp\left(-\int_{\zeta_{0}}^{s}\mathcal{D}_{\mu,\nu}(y(\alpha),\alpha)d\alpha\right)$
(5.26)
$\displaystyle+\int_{\zeta_{0}}^{s}[\eta^{\nu}\mathcal{F}_{\mu}](y(\zeta),\zeta)\exp\left(-\int_{\zeta}^{s}\mathcal{D}_{\mu,\nu}(y(\alpha),\alpha)d\alpha\right)d\zeta.$

For $(\mu,\nu)=(0,-\frac{1}{6})$ in (3.20), it comes from (3.17), (4.2a) and
(4.3a) that

$\mathcal{D}_{0,-\frac{1}{6}}=-\frac{1}{2(1+y^{2})}+\beta_{\tau}\overline{W}^{\prime}+\frac{\beta_{\tau}}{3}\frac{y}{1+y^{2}}e^{\frac{s}{2}}(\mu_{n}(w)-\dot{\xi}(t)).$
(5.27)

In addition, by (5.5), (4.2b) and (3.15), one has

$|\mu_{n}(w)-\dot{\xi}(t)|(y(\alpha),\alpha)\leq
4e^{-\frac{\alpha}{2}}|y(\alpha)|^{\frac{1}{2}}+M^{2}e^{-\alpha}.$ (5.28)

Combining (5.27)-(5.28) with (3.14b) and (4.1b) yields

$|\mathcal{D}_{0,-\frac{1}{6}}(y(\alpha),\alpha)|\leq
10\eta^{-\frac{1}{4}}(y(\alpha))+M^{2}e^{-\frac{\alpha}{2}},$

and then it follows from Proposition 5.1 and Case 3 in Remark 5.1 that

$\int_{\zeta_{0}}^{+\infty}|\mathcal{D}_{0,-\frac{1}{6}}|(y(\alpha),\alpha)d\alpha\leq
2M^{2}\varepsilon^{\frac{1}{2}}+10\int_{\zeta_{0}}^{+\infty}\frac{1}{(1+\ell^{2}e^{\frac{\alpha-\zeta_{0}}{2}})^{\frac{1}{4}}}d\alpha\leq
200\ln\frac{1}{\ell}.$ (5.29)

From (5.26) with $(\mu,\nu)=(0,-\frac{1}{6})$ and (5.29), we obtain

$\displaystyle|\eta^{-\frac{1}{6}}\mathcal{W}|(y,s)$ (5.30)
$\displaystyle\leq$
$\displaystyle\frac{1}{\ell^{200}}\left(\eta^{-\frac{1}{6}}(y(\zeta_{0}))|\mathcal{W}|(y(\zeta_{0}),\zeta_{0})+\int_{\zeta_{0}}^{s}\eta^{-\frac{1}{6}}(y(\zeta))|\mathcal{F}_{0}|(y(\zeta),\zeta)d\zeta\right)\quad\text{for
$|y|\geq\ell$}.$

With (3.14b), (3.25), (4.1), (4.3a) and (5.7)-(5.8) , $\mathcal{F}_{0}$ in
(3.19) satisfies

$\displaystyle|\mathcal{F}_{0}|(y(\zeta),\zeta)$ (5.31) $\displaystyle\leq$
$\displaystyle
2e^{\frac{s}{2}}\eta^{-\frac{1}{3}}(y(\zeta))(|\mu_{n}(w^{0})-\dot{\xi}(t)|+|\mu_{n}(w)-\mu_{n}(w^{0})-e^{-\frac{s}{2}}W_{0}|)$
$\displaystyle+M^{2}e^{-\frac{s}{2}}+2M\varepsilon\eta^{-\frac{1}{6}}(y(\zeta))$
$\displaystyle\leq$ $\displaystyle
2e^{\frac{s}{2}}\eta^{-\frac{1}{3}}(y(\zeta))(M^{2}\varepsilon^{\frac{1}{4}}e^{-\frac{\zeta}{2}}\eta^{\frac{1}{4}}(y(\zeta))+M^{2}e^{-\zeta})+M^{2}e^{-\frac{\zeta}{2}}+2M\varepsilon\eta^{-\frac{1}{6}}(y(\zeta))$
$\displaystyle\leq$ $\displaystyle
M^{3}e^{-\frac{\zeta}{2}}+M^{3}\varepsilon^{\frac{1}{4}}\eta^{-\frac{1}{12}}(y(\zeta)).$

When $|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$ and
$\zeta_{0}=-\log\varepsilon$, one has $|y(\zeta_{0})|\leq\mathcal{L}$ due to
(5.2) in Proposition 5.1. Thus, combining (5.31) with (5.30), (5.25), (3.26b)
and Case 3 in Remark 5.1 shows

$|\eta^{-\frac{1}{6}}(y)\mathcal{W}|(y,s)\leq\frac{1}{2}\varepsilon^{\frac{1}{11}}\quad\text{for
$|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$}.$ (5.32)

###  5.5 Bootstrap estimates on $\partial_{y}\mathcal{W}$ when
$|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$

As in Subsection 5.4, the $L^{\infty}$ estimate of
$\eta^{\frac{1}{3}}\partial_{y}\mathcal{W}$ is still considered in the cases
of $|y|\leq\ell$ and
$\ell<|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$. In the
region
$\\{(y,s):\ell<|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}\\}$,
by Case 3 in Remark 5.1, it follows from (5.2) in Proposition 5.1 that when
$|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$,

$|y(\zeta)|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{\zeta}{4}},\
\zeta_{0}\leq\zeta\leq s.$ (5.33)

For $|y|\leq\ell$, by (5.24), one has that

$\eta^{\frac{1}{3}}(y)|\partial_{y}\mathcal{W}(y,s)|\leq
2\varepsilon^{\frac{1}{10}}\ell^{3+\frac{1}{2}}\ (|y|\leq\ell).$ (5.34)

Next, we estimate $\eta^{\frac{1}{3}}\partial_{y}\mathcal{W}$ when
$\ell\leq|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$. For
$(\mu,\nu)=(1,\frac{1}{3})$ in (3.20), we have

$\displaystyle\mathcal{D}_{1,\frac{1}{3}}=$
$\displaystyle\frac{1}{1+y^{2}}+\beta_{\tau}\overline{W}^{\prime}+\beta_{\tau}e^{\frac{s}{2}}\partial_{y}\mu_{n}(w)-\frac{2y}{3(1+y^{2})}\beta_{\tau}e^{\frac{s}{2}}(\mu_{n}(w)-\dot{\xi}(t))$
$\displaystyle=$
$\displaystyle\frac{1}{1+y^{2}}+\beta_{\tau}\overline{W}^{\prime}+\beta_{\tau}\partial_{w_{n}}\mu_{n}(w)\partial_{y}W_{0}+\beta_{\tau}e^{\frac{s}{2}}\sum\limits_{j\neq
n}\partial_{w_{j}}\mu_{n}(w)\partial_{y}W_{j}$
$\displaystyle-\frac{2y}{3(1+y^{2})}\beta_{\tau}e^{\frac{s}{2}}(\mu_{n}(w)-\dot{\xi}(t)).$

Combining this with (1.10c), (3.14b), (4.1b), (4.2a)-(4.2b), (4.3a) and
(5.28)-(5.29) yields

$|\mathcal{D}_{1,\frac{1}{3}}(y(\alpha),\alpha)|\leq
10\eta^{-\frac{1}{4}}(y(\alpha))+M^{2}e^{-\frac{\alpha}{2}}$

and

$\int_{\zeta_{0}}^{+\infty}|\mathcal{D}_{1,\frac{1}{3}}|(y(\alpha),\alpha)d\alpha\leq
200\ln\frac{1}{\ell}.$ (5.35)

It is derived from (5.26) and (5.35) that for $|y|\geq\ell$,

$\displaystyle|\eta^{\frac{1}{3}}(y)\partial_{y}\mathcal{W}|(y,s)$ (5.36)
$\displaystyle\leq$
$\displaystyle\frac{1}{\ell^{200}}\left(\eta^{\frac{1}{3}}(y(\zeta_{0}))|\partial_{y}\mathcal{W}|(y(\zeta_{0}),\zeta_{0})+\int_{\zeta_{0}}^{s}\eta^{\frac{1}{3}}(y(\zeta))|\mathcal{F}_{1}|(y(\zeta),\zeta)d\zeta\right).$

For $\mathcal{F}_{1}$ in (5.36), one has from (3.19), (1.10d) and (4.1),
(4.2a), (4.3a) that

$\displaystyle|\mathcal{F}_{1}|$ $\displaystyle\leq
2|\overline{W}^{\prime\prime}\mathcal{W}|+Me^{s}|W|\sum\limits_{j=1}^{n-1}|\partial_{y}^{2}W_{j}|+Me^{s}\sum\limits_{j=1}^{n-1}|\partial_{y}W_{j}|^{2}+Me^{\frac{s}{2}}\sum\limits_{j=1}^{n-1}|\partial_{y}W_{0}\partial_{y}W_{j}|$
(5.37)
$\displaystyle+2e^{\frac{s}{2}}|\overline{W}^{\prime\prime}||\mu_{n}(w)-e^{-\frac{s}{2}}W_{0}-\dot{\xi}(t)|+Me^{\frac{s}{2}}|\overline{W}^{\prime}|\sum\limits_{j=1}^{n-1}|\partial_{y}W_{j}|$
$\displaystyle+2|\overline{W}^{\prime}||\partial_{w_{n}}\mu_{n}(w)-1||\partial_{y}W_{0}|+2M\varepsilon(|\overline{W}^{\prime}|^{2}+|\overline{W}^{\prime\prime}\overline{W}|)=\sum\limits_{k=1}^{8}I_{k}.$

For $I_{1}$, it follows from (3.14b), (5.32) and (5.33) that

$\eta^{\frac{1}{3}}(y(\zeta))I_{1}(y(\zeta),\zeta)\leq
2\varepsilon^{\frac{1}{11}}\eta^{-\frac{1}{3}}(y(\zeta)).$ (5.38)

For $I_{2}$, we have from (3.3), (4.1a), (5.15), (4.3a) and (4.3c) with
$\nu^{+}=\frac{1}{24}$ that

$\eta^{\frac{1}{3}}(y(\zeta))I_{2}(y(\zeta),\zeta)\leq
M^{4}\varepsilon^{\frac{1}{3}}e^{-\frac{1}{8}\zeta}.$ (5.39)

For $I_{3}$, (4.3b) with $\nu=\frac{1}{6}$ shows

$\eta^{\frac{1}{3}}(y(\zeta))I_{3}(y(\zeta),\zeta)\leq M^{3}e^{-\zeta}.$
(5.40)

In the similar way, due to $\partial_{w_{n}}\mu_{n}(0)=1$, it is derived from
(1.10), (4.2a), (4.3a) and (3.14b) that

$\eta^{\frac{1}{3}}(y(\zeta))(I_{4}+I_{6}+I_{7}+I_{8})(y(\zeta),\zeta)\leq
M^{3}e^{-\zeta}+M^{2}\varepsilon^{\frac{1}{3}}\eta^{-\frac{1}{3}}(y(\zeta)).$
(5.41)

In addition, it follows from (3.14b) and (5.5) that

$\eta^{\frac{1}{3}}(y(\zeta))I_{5}(y(\zeta),\zeta)\leq
M^{2}\varepsilon^{\frac{1}{8}}\eta^{-\frac{1}{2}}(y(\zeta)).$ (5.42)

Substituting (5.38)-(5.42) into (5.37) yields

$\eta^{\frac{1}{3}}(y(\zeta))|\mathcal{F}_{1}|(y(\zeta),\zeta)\leq
4\varepsilon^{\frac{1}{11}}\eta^{-\frac{1}{3}}(y(\zeta))+\varepsilon^{\frac{1}{11}}e^{-\frac{1}{8}\zeta}.$
(5.43)

Analogously to obtain (5.32), combining (5.43) with (5.36), (5.34) and (3.26b)
derives

$\eta^{\frac{1}{3}}(y)|\partial_{y}\mathcal{W}(y,s)|\leq\frac{1}{2}\varepsilon^{\frac{1}{12}}\quad\text{for
$|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$}.$ (5.44)

###  5.6 More delicate estimates for $W_{0}$

In the Subsection, we mainly estimate the weighted $L^{\infty}$ norms of
$\eta^{-\frac{1}{6}}W_{0}$ and $\eta^{\frac{1}{3}}\partial_{y}W_{0}$ in the
whole spatial space. Since the proof procedures are very similar to the
processes in Subsection 5.4 and Subsection 5.5, we just give the sketch of the
related verifications. For $\mu\in\mathbb{N}_{0}$ and $\nu\in\mathbb{R}$, it
is derived from (3.5) that

$\left(\partial_{s}+(\frac{3}{2}y+\beta_{\tau}e^{\frac{s}{2}}(\mu_{n}(w)-\dot{\xi}(t)))\partial_{y}\right)[\eta^{\nu}\partial^{\mu}W_{0}]+\overline{D}_{\mu,\nu}[\eta^{\nu}\partial^{\mu}W_{0}]=\eta^{\nu}\overline{F}_{\mu}^{0},$
(5.45)

where

$\overline{D}_{\mu,\nu}=\frac{3\mu-1}{2}+\mu\beta_{\tau}e^{\frac{s}{2}}\partial_{y}\mu_{n}(w)+\beta_{\tau}\partial_{y}W_{0}\boldsymbol{1}_{\mu\geq
2}-\frac{2\nu
y}{1+y^{2}}(\frac{3}{2}y+\beta_{\tau}e^{\frac{s}{2}}(\mu_{n}(w)-\dot{\xi}(t)))$

and

$\displaystyle\overline{F}_{\mu}^{0}$
$\displaystyle=-\sum\limits_{2\leq\beta\leq\mu-1}C_{\mu}^{\beta}\beta_{\tau}e^{\frac{s}{2}}\partial_{y}^{\beta}\mu_{n}(w)\partial_{y}^{\mu-\beta+1}W_{0}-\beta_{\tau}e^{\frac{s}{2}}\partial_{y}^{\mu}(\mu_{n}(w)-w_{n})\partial_{y}W_{0}\boldsymbol{1}_{\mu\geq
2}$
$\displaystyle-\sum\limits_{j=1}^{n-1}\beta_{\tau}e^{s}\partial_{y}^{\mu}(a_{nj}(w)\partial_{y}W_{j})-\beta_{\tau}e^{-\frac{s}{2}}\dot{\kappa}(t)\delta_{\mu}^{0}.$

When $|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$, it follows
from (3.14b), (5.32) and (5.44) that for
$|y|\leq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$,

$\eta^{-\frac{1}{6}}(y)|W_{0}(y,s)|\leq
1+\frac{1}{2}\varepsilon^{\frac{1}{11}},\
\eta^{\frac{1}{3}}(y)|\partial_{y}W_{0}(y,s)|\leq
1+\frac{1}{2}\varepsilon^{\frac{1}{12}}.$ (5.46)

When $|y|\geq\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{s}{4}}$, the
backward characteristics $y=y(\zeta)$ is defined by (5.11) with
$(y_{0},\zeta_{0})$ satisfying
$|y_{0}|\geq\mathcal{L},\zeta_{0}=-\log\varepsilon$ or
$|y_{0}|=\mathcal{L}\varepsilon^{\frac{1}{4}}e^{\frac{\zeta_{0}}{4}},\zeta_{0}>-\log\varepsilon$.
In this case, we have $|y(\zeta)|\geq\mathcal{L}e^{\frac{\zeta-\zeta_{0}}{4}}$
for $\zeta\geq\zeta_{0}$ due to (5.2). Moreover, it is derived from (5.45)
that

$\displaystyle\left[\eta^{\nu}\partial^{\mu}W_{0}\right](y,s)$
$\displaystyle=[\eta^{\nu}\partial^{\mu}W_{0}](y_{0},\zeta_{0})\exp\left(-\int_{\zeta_{0}}^{s}\overline{D}_{\mu,\nu}(y(\alpha),\alpha)d\alpha\right)$
(5.47)
$\displaystyle+\int_{\zeta_{0}}^{s}[\eta^{\nu}\overline{F}_{\mu}^{0}](y(\zeta),\zeta)\exp\left(-\int_{\zeta}^{s}\overline{D}_{\mu,\nu}(y(\alpha),\alpha)d\alpha\right)d\zeta.$

In addition, by (5.28) and (4.2a)-(4.2b), (4.3a), we have

$\displaystyle|\overline{D}_{0,-\frac{1}{6}}(y(\alpha),\alpha)|$ (5.48)
$\displaystyle=$
$\displaystyle\left|-\frac{1}{2(1+y(\alpha)^{2})}+\frac{y(\alpha)}{3(1+y(\alpha)^{2})}e^{\frac{\alpha}{2}}\beta_{\tau}(\mu_{n}(w)-\dot{\xi}(t))\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\eta^{-1}(y(\alpha))+e^{\frac{\alpha}{2}}\eta^{-\frac{1}{2}}(y(\alpha))|\mu_{n}(w)-\dot{\xi}(t)|$
$\displaystyle\leq$ $\displaystyle
5\eta^{-\frac{1}{4}}(y(\alpha))+M^{2}e^{-\frac{\alpha}{2}}$

and

$\displaystyle|\overline{D}_{1,\frac{1}{3}}(y(\alpha),\alpha)|$ (5.49)
$\displaystyle=$
$\displaystyle\left|\frac{1}{1+y(\alpha)^{2}}+e^{\frac{\alpha}{2}}\beta_{\tau}\partial_{y}\mu_{n}(w)-\frac{2y(\alpha)}{3(1+y(\alpha)^{2})}e^{\frac{\alpha}{2}}\beta_{\tau}(\mu_{n}(w)-\dot{\xi}(t))\right|$
$\displaystyle\leq$
$\displaystyle\eta^{-1}(y(\alpha))+2e^{\frac{\alpha}{2}}\eta^{-\frac{1}{2}}(y(\alpha))|\mu_{n}(w)-\dot{\xi}(t)|+2e^{\frac{\alpha}{2}}\sum\limits_{j=1}^{n}|\partial_{w_{j}}\mu_{n}(w)||\partial_{y}W_{j}|$
$\displaystyle\leq$ $\displaystyle
20\eta^{-\frac{1}{4}}(y(\alpha))+M^{3}e^{-\frac{\alpha}{2}}.$

Similarly to (5.29), for
$|y(\zeta)|\geq\mathcal{L}e^{\frac{\zeta-\zeta_{0}}{4}}$, one has from
(5.48)-(5.49) that

$\displaystyle\int_{\zeta_{0}}^{s}|\overline{D}_{0,-\frac{1}{6}}|(y(\alpha),\alpha)d\alpha+\int_{\zeta_{0}}^{s}|\overline{D}_{1,\frac{1}{3}}|(y(\alpha),\alpha)d\alpha$
(5.50) $\displaystyle\leq$ $\displaystyle
4M^{3}\varepsilon^{\frac{1}{2}}+\int_{\zeta_{0}}^{s}\frac{25}{(1+\mathcal{L}^{2}e^{\frac{\zeta-\zeta_{0}}{2}})^{\frac{1}{4}}}d\zeta$
$\displaystyle\leq$ $\displaystyle
4M^{3}\varepsilon^{\frac{1}{2}}+600\ln(1+\mathcal{L}^{-1})\leq\varepsilon^{\frac{1}{20}}.$

Based on (5.50), together with (5.47) this yields

$\eta^{-\frac{1}{6}}(y)|W_{0}(y,s)|\leq(1+2\varepsilon^{\frac{1}{20}})\left(\eta^{-\frac{1}{6}}(y_{0})|W_{0}(y_{0},\zeta_{0})|+\int_{\zeta_{0}}^{s}\eta^{-\frac{1}{6}}(y(\zeta))|\overline{F}_{0}^{0}|(y(\zeta),\zeta)d\zeta\right)$
(5.51)

and

$\eta^{\frac{1}{3}}(y)|\partial_{y}W_{0}(y,s)|\leq(1+2\varepsilon^{\frac{1}{20}})\left(\eta^{\frac{1}{3}}(y_{0})|\partial_{y}W_{0}(y_{0},\zeta_{0})|+\int_{\zeta_{0}}^{s}\eta^{\frac{1}{3}}(y(\zeta))|\overline{F}_{1}^{0}|(y(\zeta),\zeta)d\zeta\right).$
(5.52)

By (1.10), (4.1)-(4.3), (4.3b) with $\nu=\frac{1}{3}$ and (4.3c) with
$\nu^{+}=\frac{1}{24}$, then $\overline{F}_{0}^{0}$ and $\overline{F}_{1}^{0}$
in (5.45) satisfy

$|\overline{F}_{0}^{0}(y(\zeta),\zeta)|\leq 4M^{2}e^{-\frac{\zeta}{2}},\
|\overline{F}_{1}^{0}(y(\zeta),\zeta)|\leq
2M^{2}e^{-\frac{\zeta}{9}}\eta^{-\frac{1}{3}}(y(\zeta)).$ (5.53)

Therefore, we derive from (3.26b), (3.26d), (5.46) and (5.51)-(5.53) that

$\eta^{-\frac{1}{6}}(y)|W_{0}(y,s)|\leq 1+\varepsilon^{\frac{1}{21}},\
\eta^{\frac{1}{3}}(y)|\partial_{y}W_{0}(y,s)|\leq
1+\varepsilon^{\frac{1}{21}}.$ (5.54)

For the estimates of $\partial_{y}^{\mu}W_{0}$, together with Lemma A.1 for
$u=\partial_{y}^{\mu}W_{0}$ and $d=1,p,q=\infty,r=2,j=\mu-1,m=\mu_{0}-1$, it
comes from (3.3), (4.6) and (5.54) that for $2\leq\mu\leq 4$ and
$\alpha=\frac{\mu-1}{\mu_{0}-\frac{1}{2}}\in(0,\frac{6}{11}]$,

$\|\partial_{y}^{\mu}W_{0}(\cdot,s)\|_{L^{\infty}}\leq
M^{\frac{1}{20}}\|\partial_{y}^{\mu_{0}}W_{0}(\cdot,s)\|_{L^{2}}^{\alpha}\|\partial_{y}W_{0}(\cdot,s)\|_{L^{\infty}}^{1-\alpha}\leq
2^{1-\alpha}M^{\frac{1}{20}+\alpha}\leq M^{\frac{3}{5}}.$ (5.55)

##  6 Bootstrap estimates on good components of $W$

In the section, we will apply the characteristics method to establish a series
of estimates of $W_{m}\ (m\neq n)$.

###  6.1 Framework for the characteristics method

For $1\leq m\leq n-1$ and any point
$(y_{0},\zeta_{0})\in\mathbb{R}\times[-\log\varepsilon,+\infty)$, we consider
the following forward characteristics $y(\zeta):=y(\zeta;y_{0},\zeta_{0})$ of
(3.22) which starts from $(y_{0},\zeta_{0})$:

$\begin{cases}\dot{y}(\zeta)=\frac{3}{2}y(\zeta)+e^{\frac{\zeta}{2}}\beta_{\tau}(\mu_{m}(w)-\dot{\xi}(t))(y(\zeta),\zeta),\\\
y(\zeta_{0})=y_{0}.\end{cases}$ (6.1)

This yields that for $s\geq\zeta_{0}$,

$y(\zeta)e^{-\frac{3}{2}\zeta}=y_{0}e^{-\frac{3}{2}\zeta_{0}}+\int_{\zeta_{0}}^{\zeta}e^{-\alpha}\left(\beta_{\tau}(\cdot)(\mu_{m}(w)-\dot{\xi}(\cdot))\right)(y(\alpha),\alpha)d\alpha:=G_{m}(\zeta;y_{0},\zeta_{0}).$
(6.2)

Next we discuss the positions of $y(\zeta;y_{0},\zeta_{0})$ for the different
cases of $(y_{0},\zeta_{0})$.

######  Lemma 6.1.

For each $i_{0}\leq m\leq n-1$, $a_{m}:=\mu_{n}(0)>0$ is due to (1.10a) and
the assumption $\mu_{n}(0)=0$ in Remark 1.2. Then for any point
$(y_{0},\zeta_{0})\in\mathbb{R}\times[-\log\varepsilon,+\infty)$,
$y(\zeta):=y(\zeta;y_{0},\zeta_{0})$ can be classified into the following six
cases:

  1. Case $1$.

When $y_{0}<-4a_{m}e^{\frac{\zeta_{0}}{2}}$, $y(\zeta)\leq-
a_{m}e^{\frac{3}{2}\zeta-\zeta_{0}}<0$ holds for $\zeta\geq\zeta_{0}.$

  2. Case $2$.

When $-\frac{a_{m}}{4}e^{\frac{\zeta_{0}}{2}}\leq y_{0}\leq 0$, there exists a
number $\zeta^{*}\geq\zeta_{0}$ such that $G_{m}(\zeta^{*};y_{0},\zeta_{0})=0$
and

$\begin{cases}-\frac{3}{2}a_{m}(e^{-\zeta}-e^{-\zeta^{*}})e^{\frac{3}{2}\zeta}\leq
y(\zeta)\leq-\frac{a_{m}}{2}(e^{-\zeta}-e^{-\zeta^{*}})e^{\frac{3}{2}\zeta}\leq
0\quad\text{for $\zeta_{0}\leq\zeta\leq\zeta^{*}$},\\\\[5.69054pt]
y(\zeta)\geq\frac{a_{m}}{2}(e^{-\zeta^{*}}-e^{-\zeta})e^{\frac{3}{2}\zeta}\geq
0\quad\text{for $\zeta\geq\zeta^{*}$}.\end{cases}$

  3. Case $3$.

When $y_{0}\geq 0$, one has
$y(\zeta)\geq\frac{a_{m}}{2}(e^{-\zeta_{0}}-e^{-\zeta})e^{\frac{3}{2}\zeta}\geq
0$ for $\zeta\geq\zeta_{0}.$

  4. Case $4$.

When $(y_{0},\zeta_{0})\in D^{+}$ and $(y(\zeta),\zeta)$ lies in the domain
$D^{+}$, there holds

$-4a_{m}e^{\frac{\zeta}{2}}\leq
y(\zeta)\leq-\frac{a_{m}}{4}e^{\frac{\zeta}{2}}<0\quad\text{for
$\zeta\geq\zeta_{0}$},$

where $D^{+}=\\{(y,\zeta):-4a_{m}e^{\frac{\zeta}{2}}\leq
y\leq-\frac{a_{m}}{4}e^{\frac{\zeta}{2}},\ \zeta\geq-\log\varepsilon\\}$.

  5. Case $5$.

When $(y_{0},\zeta_{0})\in D^{+}$ and the characteristics $y=y(\zeta)$ goes
through $\partial D^{+}$ at some point $(\hat{y},\hat{\zeta})$ with
$\hat{y}=-4a_{m}e^{\frac{\hat{\zeta}}{2}}$, we have

$\begin{cases}(y(\zeta),\zeta)\in D^{+}\quad\text{for
$\zeta_{0}\leq\zeta\leq\hat{\zeta}$},\\\ y(\zeta)\leq-
a_{m}e^{\frac{3}{2}\zeta-\hat{\zeta}}<0\quad\text{for
$\zeta\geq\hat{\zeta}$.}\end{cases}$

  6. Case $6$.

When $(y_{0},\zeta_{0})\in D^{+}$ and the characteristics $y=y(\zeta)$ goes
through $\partial D^{+}$ at some point $(\hat{y},\hat{\zeta})$ with
$\hat{y}=-\frac{a_{m}}{2}e^{\frac{\hat{\zeta}}{2}}$, there exists
$\tilde{\zeta}>\hat{\zeta}$ such that $G_{m}(\tilde{\zeta};y_{0},\zeta_{0})=0$
and $y=y(\zeta)$ can be divided into the three parts as:

$\begin{cases}(y(\zeta),\zeta)\in D^{+}\quad\text{for
$\zeta_{0}\leq\zeta\leq\hat{\zeta}$},\\\
-2a_{m}(e^{-\zeta}-e^{-\tilde{\zeta}})e^{\frac{3}{2}\zeta}\leq
y(\zeta)\leq-\frac{a_{m}}{2}(e^{-\zeta}-e^{-\tilde{\zeta}})e^{\frac{3}{2}\zeta}\leq
0\quad\text{for $\hat{\zeta}\leq\zeta\leq\tilde{\zeta}$},\\\
y(\zeta)\geq\frac{a_{m}}{2}(e^{-\tilde{\zeta}}-e^{-\zeta})e^{\frac{3}{2}\zeta}\geq
0\quad\text{for $\zeta\geq\tilde{\zeta}$}.\end{cases}$

###### Proof.

Since $a_{m}>0$ for $i_{0}\leq m\leq n-1$, by (3.3), (4.1), (4.2a) and (4.3a),
we then have

$|\beta_{\tau}(\mu_{m}(w)-\dot{\xi}(t))-a_{m}|\leq
M^{3}\varepsilon^{\frac{1}{3}}\leq\frac{a_{m}}{2}.$ (6.3)

When $y_{0}<-4a_{m}e^{\frac{\zeta_{0}}{2}}$, it is derived from (6.2) and
(6.3) that

$y(\zeta)e^{-\frac{3}{2}\zeta}\leq-4a_{m}e^{-\zeta_{0}}+\frac{3}{2}a_{m}(e^{-\zeta_{0}}-e^{-\zeta})\leq-
a_{m}e^{-\zeta_{0}}.$ (6.4)

This shows Case 1.

When $-\frac{a_{m}}{4}e^{\frac{\zeta_{0}}{2}}\leq y_{0}\leq 0$, it follows
from (6.3) that $G_{m}(\zeta;y_{0},\zeta_{0})$ in (6.2) satisfies

$\begin{cases}G_{m}(\zeta_{0};y_{0},\zeta_{0})=y_{0}e^{-\frac{3}{2}\zeta_{0}}\leq
0,\\\ G_{m}(+\infty;y_{0},\zeta_{0})\geq
y_{0}e^{-\frac{3}{2}\zeta_{0}}+\frac{a_{m}}{2}e^{-\zeta_{0}}\geq-\frac{a_{m}}{4}e^{-\zeta_{0}}+\frac{a_{m}}{2}e^{-\zeta_{0}}>0.\end{cases}$
(6.5)

Since $G_{m}(\zeta;y_{0},\zeta_{0})$ is a continuous function with respect to
the variable $\zeta$, (6.5) shows that there exists $\zeta^{*}\geq\zeta_{0}$
such that $G_{m}(\zeta^{*};y_{0},\zeta_{0})=0$. In this situation, we derive
from (6.2) that

$y(\zeta)e^{-\frac{3}{2}\zeta}=G_{m}(\zeta;y_{0},\zeta_{0})-G_{m}(\zeta^{*};y_{0},\zeta_{0})=\int_{\zeta^{*}}^{\zeta}e^{-\alpha}\left(\beta_{\tau}(\cdot)(\mu_{m}(w)-\dot{\xi}(\cdot))\right)(y(\alpha),\alpha)d\alpha.$
(6.6)

Therefore, Case 2 is obtained from (6.3), (6.6) and $a_{m}>0$ for $i_{0}\leq
m\leq n-1$.

Based on the results established in Case 1-Case 2, Case 3-Case 6 in Lemma 6.1
can be carried out in the same way due to the formula (6.2) and the definition
of $D^{+}$, here we omit the details.∎

######  Lemma 6.2.

For each $1\leq m\leq i_{0}-1$, $a_{m}:=\mu_{m}(0)<0$ is due to (1.10a) and
the assumption $\mu_{n}(0)=0$ in Remarks 1.2. Then for any point
$(y_{0},\zeta_{0})\in\mathbb{R}\times[-\log\varepsilon,+\infty)$,
$y(\zeta)=y(\zeta;y_{0},\zeta_{0})$ can be classified into the following six
cases:

  1. Case $1$.

When $y_{0}>-4a_{m}e^{\frac{\zeta_{0}}{2}}$, $y(\zeta)\geq-
a_{m}e^{\frac{3}{2}\zeta-\zeta_{0}}>0.$

  2. Case $2$.

When $0\leq y_{0}\leq-\frac{a_{m}}{4}e^{\frac{\zeta_{0}}{2}}$, there exists a
number $\zeta^{*}\geq\zeta_{0}$ such that $G_{m}(\zeta^{*};y_{0},\zeta_{0})=0$
and

$\begin{cases}0\leq-\frac{a_{m}}{2}(e^{-\zeta}-e^{-\zeta^{*}})e^{\frac{3}{2}\zeta}\leq
y(\zeta)\leq-\frac{3}{2}a_{m}(e^{-\zeta}-e^{-\zeta^{*}})e^{\frac{3}{2}\zeta}\quad\text{for
$\zeta_{0}\leq\zeta\leq\zeta^{*}$},\\\\[5.69054pt]
y(\zeta)\leq\frac{a_{m}}{2}(e^{-\zeta^{*}}-e^{-\zeta})\leq 0\quad\text{for
$\zeta\geq\zeta^{*}$}.\end{cases}$

  3. Case $3$.

When $y_{0}\leq 0$,
$y(\zeta)\leq\frac{a_{m}}{2}(e^{-\zeta_{0}}-e^{-\zeta})e^{\frac{3}{2}\zeta}\leq
0$ for $\zeta\geq\zeta_{0}.$

  4. Case $4$.

When $(y_{0},\zeta_{0})\in D^{-}$ and the characteristics $(y(\zeta),\zeta)$
lies in $D^{-}$, one has

$0<-\frac{a_{m}}{4}e^{\frac{\zeta}{2}}\leq
y(\zeta)\leq-4a_{m}e^{\frac{\zeta}{2}}\quad\text{for $\zeta\geq\zeta_{0}$},$

where $D^{-}=\\{(y,\zeta):-\frac{a_{m}}{4}e^{\frac{\zeta}{2}}\leq
y\leq-4a_{m}e^{\frac{\zeta}{2}},\ \zeta\geq-\log\varepsilon\\}$.

  5. Case $5$.

When $(y_{0},\zeta_{0})\in D^{-}$ and the characteristics $(y(\zeta),\zeta)$
goes through $\partial D^{-}$ at some point $(\hat{y},\hat{\zeta})$ with
$\hat{y}=-4a_{m}e^{\frac{\hat{\zeta}}{2}}$, we have

$\begin{cases}(y(\zeta),\zeta)\in D^{-}\quad\text{for
$\zeta_{0}\leq\zeta\leq\hat{\zeta}$},\\\ y(\zeta)\geq-
a_{m}e^{\frac{3}{2}\zeta-\hat{\zeta}}>0\quad\text{for
$\zeta\geq\hat{\zeta}$}.\end{cases}$

  6. Case $6$.

When $(y_{0},\zeta_{0})\in D^{-}$ and the characteristics $(y(\zeta),\zeta)$
goes through $\partial D^{-}$ at some point $(\hat{y},\hat{\zeta})$ with
$\hat{y}=-\frac{a_{m}}{2}e^{\frac{\hat{\zeta}}{2}}$, there exists
$\tilde{\zeta}>\hat{\zeta}$ such that $y=y(\zeta)$ can be divided into the
three parts as:

$\begin{cases}(y(\zeta),\zeta)\in D^{-}\quad\text{for
$\zeta_{0}\leq\zeta\leq\hat{\zeta}$},\\\
0\leq-\frac{a_{m}}{2}(e^{-\zeta}-e^{-\tilde{\zeta}})e^{\frac{3}{2}\zeta}\leq
y(\zeta)\leq-2a_{m}(e^{-\zeta}-e^{-\tilde{\zeta}})e^{\frac{3}{2}\zeta}\quad\text{for
$\hat{\zeta}\leq\zeta\leq\tilde{\zeta}$},\\\
y(\zeta)\leq\frac{a_{m}}{2}(e^{-\tilde{\zeta}}-e^{-\zeta})e^{\frac{3}{2}\zeta}\leq
0\quad\text{for $\zeta\geq\tilde{\zeta}$}.\end{cases}$

###### Proof.

Since the proof of Lemma 6.2 is just the same as in Lemma 6.1, we omit the
details here. ∎

Based on Lemma 6.1 and Lemma 6.2, we now establish the following results.

######  Lemma 6.3.

For $1\leq m\leq n-1$ and each forward characteristics
$y(\zeta):=y(\zeta;y_{0},\zeta_{0})$ defined by (6.1), when the function
$\mathcal{D}(z,\zeta)$ satisfies that for some positive constant $c_{0}$,

$|\mathcal{D}(z,\zeta)|\leq c_{0}\eta^{-\frac{\kappa}{2}}(z)\ (0<\kappa<1),$
(6.7)

then

$\int_{\zeta_{0}}^{+\infty}|\mathcal{D}(y(\zeta),\zeta)|d\zeta\leq\frac{16c_{0}}{\kappa(1-\kappa)|a_{m}|^{\kappa}}e^{-\frac{\kappa}{2}\zeta_{0}}.$
(6.8)

###### Proof.

We only consider the Case 2 in Lemma 6.1. The estimate (6.8) for other Cases
in Lemma 6.1 and Lemma 6.2 can be done analogously. In the present situation,
we choose $\zeta_{1},\zeta_{2}\in[-\log\varepsilon,+\infty)$ such that

$e^{-\zeta_{1}}=\min\\{2e^{-\zeta^{*}},e^{-\zeta_{0}}\\},\
e^{-\zeta_{2}}=\frac{1}{2}e^{-\zeta^{*}}.$

This implies $\zeta_{0}\leq\zeta_{1}\leq\zeta^{*}<\zeta_{2}$. Then it is
derived from (6.7) and Case 2 in Lemma 6.1 that

$\displaystyle\int_{\zeta_{0}}^{+\infty}|\mathcal{D}(y(\zeta),\zeta)|d\zeta$
$\displaystyle\leq$
$\displaystyle\frac{2c_{0}}{|a_{m}|^{\kappa}}\left(\int_{\zeta_{0}}^{\zeta_{1}}\frac{e^{-\frac{3}{2}\kappa\zeta}}{|e^{-\zeta}-e^{-\zeta^{*}}|^{\kappa}}d\zeta+\int_{\zeta_{1}}^{\zeta_{2}}\frac{e^{-\frac{3}{2}\kappa\zeta}}{|e^{-\zeta}-e^{-\zeta^{*}}|^{\kappa}}d\zeta+\int_{\zeta_{2}}^{+\infty}\frac{e^{-\frac{3}{2}\kappa\zeta}}{|e^{-\zeta}-e^{-\zeta^{*}}|^{\kappa}}ds\right)$
$\displaystyle\leq$
$\displaystyle\frac{2c_{0}}{|a_{m}|^{\kappa}}\left(2\int_{\zeta_{0}}^{\zeta_{1}}e^{-\frac{\kappa}{2}\zeta}d\zeta+2e^{\kappa\zeta^{*}}\int_{\zeta_{2}}^{+\infty}e^{-\frac{3}{2}\kappa\zeta}d\zeta+4e^{-\frac{\kappa}{2}\zeta^{*}}\int_{\frac{1}{2}}^{2}|1-t|^{-\kappa}dt\right)$
$\displaystyle\leq$
$\displaystyle\frac{16c_{0}}{\kappa(1-\kappa)|a_{m}|^{\kappa}}e^{-\frac{\kappa}{2}\zeta_{0}}.$

Therefore, the estimate (6.8) holds for the case $i_{0}\leq m\leq n-1$ in
(6.2) and Case 2 in Lemma 6.1.∎

###  6.2 Auxiliary analysis

As in [16], we will apply the decomposition (3.21) and the reduced system
(3.22)-(3.23) to establish the related estimates for the good components of
$W$. To this end, we first show the relation between $\partial_{y}^{\mu}W$ and
$W_{\mu}^{m}$ ($1\leq m\leq n$).

Due to (1.11), (3.3), (4.1a), (4.2a) and (4.3a), one has $\ell_{m}^{n}(w)=0$
and $\|\ell_{m}(w)\|_{L^{\infty}}\leq 2$ for $1\leq m\leq n-1$. Combining this
with (3.21) yields

$\begin{cases}|W_{\mu}^{m}|=|\ell_{m}(w)\cdot\partial_{y}^{\mu}W|\leq
2\sum\limits_{j=1}^{n-1}|\partial_{y}^{\mu}W_{j}|\ (1\leq m\leq n-1),\\\
|W_{\mu}^{n}|=|\ell_{n}(w)\cdot\partial_{y}^{\mu}W|\leq
2\sum\limits_{j=1}^{n}|\partial_{y}^{\mu}W_{j}|.\end{cases}$ (6.9)

In addition, by $\|\gamma_{j}(w)\|=1$ for $1\leq j\leq n$ and
$\gamma_{k}^{n}(w)=0$ for $1\leq k\leq n-1$, then it follows from (3.21) that

$\begin{cases}|\partial_{y}^{\mu}W_{j}|=|\sum\limits_{m=1}^{n}W_{\mu}^{m}\gamma_{m}^{j}(w)|\leq\sum\limits_{k=1}^{n-1}|W_{\mu}^{k}|\
(1\leq j\leq n-1),\\\
|\partial_{y}^{\mu}W_{n}|=|\sum\limits_{m=1}^{n}W_{\mu}^{m}\gamma_{m}^{n}(w)|\leq|W_{\mu}^{n}|.\end{cases}$
(6.10)

###  6.3 Bootstrap estimates on $W_{j}\ (j\neq n)$

First, by (1.10), (3.14b), (4.1a), (4.2) and (4.3a)-(4.3b) with
$\nu=\frac{1}{3}$, it is derived from (3.22) and (3.3) that

$\displaystyle\partial_{s}\gamma_{j}(w)+((\frac{3}{2}y-e^{\frac{s}{2}}\beta_{\tau}\dot{\xi}(t))I_{n}+e^{\frac{s}{2}}\beta_{\tau}A(w))\partial_{y}\gamma_{j}(w)$
(6.11) $\displaystyle=$ $\displaystyle\frac{\partial\gamma_{j}(w)}{\partial
w}\left(\partial_{s}W+(\frac{3}{2}y-e^{\frac{s}{2}}\beta_{\tau}\dot{\xi}(t))\partial_{y}W\right)+e^{\frac{s}{2}}\beta_{\tau}A(w)\frac{\partial\gamma_{j}(w)}{\partial
w}\partial_{y}W$ $\displaystyle=$ $\displaystyle
e^{\frac{s}{2}}\beta_{\tau}[A(w),\frac{\partial\gamma_{j}(w)}{\partial
w}]\partial_{y}W$

and

$\left|e^{\frac{s}{2}}\beta_{\tau}[A(w),\frac{\partial\gamma_{j}(w)}{\partial
w}]\partial_{y}W\right|\leq M^{2}\eta^{-\frac{1}{3}}(y)(1-\delta_{j}^{n}),$
(6.12)

where $[A,B]=AB-BA$ for two $n\times n$ matrices $A$ and $B$.

For any $(y,s)\in\mathbb{R}\times[-\log\varepsilon,+\infty)$, the backward
characteristics $y(\zeta):=y(\zeta;y,s)$ of (3.22) which starts from
$(y_{0}(y,s),-\log\varepsilon)$, is defined as

$\begin{cases}\dot{y}(\zeta)=\frac{3}{2}y(\zeta)+e^{\frac{\zeta}{2}}\beta_{\tau}(\cdot)(\mu_{m}(w)-\dot{\xi}(\cdot))(y(\zeta),\zeta),\\\
y(-\log\varepsilon)=y_{0}(y,s).\end{cases}$ (6.13)

Then it is derived from (3.22) with $\mu=0$ and (6.13) for $1\leq m\leq n-1$
that

$W_{0}^{m}(y,s)=W_{0}^{m}(y_{0}(y,s),-\log\varepsilon)+\int_{-\log\varepsilon}^{s}\mathbb{F}_{0}^{m}(y(\zeta),\zeta)d\zeta,$
(6.14)

where $\mathbb{F}_{0}^{m}$ is given in (3.22), and it comes from $F_{0}=0$ in
(3.16a) and (6.11)-(6.12) that

$|\mathbb{F}_{0}^{m}(y(\zeta),\zeta)|\leq
M^{3}\eta^{-\frac{1}{3}}(y(\zeta))\sum\limits_{j=1}^{n-1}\|W_{0}^{j}\|_{L^{\infty}}.$
(6.15)

In addition, by Lemma 6.3 with $\kappa=\frac{2}{3}$ and (6.14)-(6.15), we
arrive at

$\displaystyle|W_{0}^{m}(y,s)|$
$\displaystyle\leq\|W_{0}^{m}(\cdot,-\log\varepsilon)\|_{L^{\infty}}+M^{4}\varepsilon^{\frac{1}{3}}\sum\limits_{j=1}^{n-1}\|W_{0}^{j}\|_{L^{\infty}}$
(6.16)
$\displaystyle\leq\|W_{0}^{m}(\cdot,-\log\varepsilon)\|_{L^{\infty}}+\varepsilon^{\frac{1}{4}}\sum\limits_{j=1}^{n-1}\|W_{0}^{j}\|_{L^{\infty}}.$

On the other hand, due to the arbitrariness of $(y,s)$, summing $m$ in both
sides of (6.16) from $1$ to $n-1$ yields

$\sum\limits_{m=1}^{n-1}\|W_{0}^{m}\|_{L^{\infty}}\leq
2\sum\limits_{m=1}^{n-1}\|W_{0}^{m}(\cdot,-\log\varepsilon)\|_{L^{\infty}}.$
(6.17)

Then it follows from (6.17), (6.9)-(6.10) and (3.27a) that

$|W_{j}(y,s)|\leq
4n\sum\limits_{k=1}^{n-1}\|W_{k}(\cdot,-\log\varepsilon)\|_{L^{\infty}}\leq
4n^{2}\varepsilon\ (1\leq j\leq n-1).$ (6.18)

###  6.4 Bootstrap estimates on $\partial_{y}W_{j}\ (j\neq n)$

With the definition (6.13), it is derived from (3.22) for $\mu=1$ and $1\leq
m\leq n-1$ that

$e^{\frac{3}{2}s}W_{1}^{m}(y,s)=\varepsilon^{-\frac{3}{2}}W_{1}^{m}(y_{0}(y,s),-\log\varepsilon)+\int_{-\log\varepsilon}^{s}e^{\frac{3}{2}\zeta}\mathbb{F}_{1}^{m}(y(\zeta),\zeta)d\zeta.$
(6.19)

Since $a_{in}(w)=0$ and $\ell_{i}^{n}(w)=0$ for $1\leq i\leq n-1$ (see (1.10b)
and (1.11)), it follows from (3.22), (6.10)-(6.11), (4.2)-(4.3) and (3.16a)
with $\mu=1$ that

$|\mathbb{F}_{1}^{m}(y(\zeta),\zeta)|\leq
M^{3}\eta^{-\frac{1}{3}}(y(\zeta))\sum\limits_{j=1}^{n-1}|W_{1}^{j}|(y(\zeta),\zeta).$
(6.20)

As in (6.17), combining (6.19)-(6.20) with Lemma 6.3 yields that for $1\leq
m\leq n-1$,

$\displaystyle e^{\frac{3}{2}s}|W_{1}^{m}|(y,s)$
$\displaystyle\leq\varepsilon^{-\frac{3}{2}}\|W_{1}^{m}(\cdot,-\log\varepsilon)\|_{L^{\infty}}+M^{4}\varepsilon^{\frac{1}{3}}\sum\limits_{j=1}^{n-1}\|e^{\frac{3}{2}\varsigma}W_{1}^{j}(z,\varsigma)\|_{L^{\infty}_{z,\varsigma}}$
(6.21)
$\displaystyle\leq\varepsilon^{-\frac{3}{2}}\|W_{1}^{m}(\cdot,-\log\varepsilon)\|_{L^{\infty}}+\varepsilon^{\frac{1}{4}}\sum\limits_{j=1}^{n-1}\|e^{\frac{3}{2}\varsigma}W_{1}^{j}(z,\varsigma)\|_{L^{\infty}_{z,\varsigma}}.$

Similarly to (6.17)-(6.18), it is derived from (6.21) and (3.27b) with $\nu=0$
that

$|\partial_{y}W_{j}(y,s)|\leq 4n^{2}e^{-\frac{3}{2}s}\ (1\leq j\leq n-1).$
(6.22)

In addition, as in (5.55), by (4.6) and (6.22), we have that for $2\leq\mu\leq
4$ and $\alpha=\frac{\mu-1}{\mu_{0}-\frac{1}{2}}\in(0,\frac{6}{11}]$,

$\displaystyle\|\partial_{y}^{\mu}W_{k}(\cdot,s)\|_{L^{\infty}}$
$\displaystyle\leq
M^{\frac{1}{20}}\|\partial_{y}^{\mu_{0}}W_{k}(\cdot,s)\|_{L^{2}}^{\alpha}\|\partial_{y}W_{k}(\cdot,s)\|_{L^{\infty}}^{1-\alpha}$
(6.23)
$\displaystyle\leq(4n^{2})^{1-\alpha}M^{\frac{1}{20}+\alpha}e^{-\frac{3}{2}s}\leq
M^{\frac{3}{5}}e^{-\frac{3}{2}s}\ (1\leq k\leq n-1).$

###  6.5 Weighted bootstrap estimates of the good components

For $a=\max\limits_{1\leq i\leq n-1}\\{5|\mu_{m}(0)|+1\\}$, set the domain
$D^{0}$ as

$D^{0}=\\{(y,s):|y|<4ae^{\frac{s}{2}},-\log\varepsilon\leq s<+\infty\\}.$
(6.24)

Then $D^{\pm}\subset D^{0}$ for the domains $D^{\pm}$ defined in Lemma 6.1 and
Lemma 6.2.

When $(y,s)\in D^{0}$, it is derived from (6.22)-(6.23) that

$\begin{cases}\sum\limits_{j=1}^{n-1}|\partial_{y}W_{j}(y,s)|\leq
4n^{2}e^{-\frac{3}{2}s}\leq
M^{\frac{1}{5}}e^{(\nu-\frac{3}{2})s}\eta^{-\nu}(y)\
(0\leq\nu\leq\frac{1}{3}]),\\\\[11.38109pt]
\sum\limits_{j=1}^{n-1}|\partial_{y}^{2}W_{j}(y,s)|\leq
M^{\frac{2}{5}}e^{-\frac{3}{2}s}\leq
M^{\frac{3}{5}}e^{-\frac{7}{6}s}\eta^{-\frac{1}{3}}(y).\end{cases}$ (6.25)

Next we derive the weighted estimates on the good components of $W$ when
$(y,s)\notin D^{0}$. In this situation, as in (6.13), for each $1\leq m\leq
n-1$, the backward characteristics $y(\zeta):=y(\zeta;y,s)$ of (3.23) which
starts from $(y_{0}(y,s),\zeta_{0})\notin D^{0}$ is defined as

$\begin{cases}\dot{y}(\zeta)=\frac{3}{2}y(\zeta)+e^{\frac{\zeta}{2}}\beta_{\tau}(\mu_{m}(w)-\dot{\xi}(t))(y(\zeta),\zeta),\
\zeta_{0}\leq\zeta\leq s,\\\ y(\zeta_{0})=y_{0}(y,s),\end{cases}$ (6.26)

where either $\zeta_{0}=-\log\varepsilon$ or
$(y_{0}(y,s),\zeta_{0})\in\partial D^{0}$.

Note that $y(\zeta)$ in (6.26) has the following expression

$y(\zeta)e^{-\frac{3}{2}\zeta}=y_{0}(y,s)e^{-\frac{3}{2}\zeta_{0}}+\int_{\zeta_{0}}^{\zeta}e^{-\alpha}\beta_{\tau}(\mu_{m}(w)-\dot{\xi}(t))(y(\alpha),\alpha)d\alpha.$
(6.27)

It follows from (6.3) and (6.27) with $|y_{0}(y,s)|\geq
4ae^{\frac{\zeta_{0}}{2}}$ that for $\zeta_{0}\leq\zeta\leq s$,

$|y(\zeta)|\geq
e^{\frac{3}{2}\zeta}(4ae^{-\zeta_{0}}-2|a_{m}|(e^{-\zeta_{0}}-e^{-\zeta}))\geq
e^{\frac{3}{2}\zeta-\zeta_{0}}.$ (6.28)

Based on the definition of $y(\zeta)=y(\zeta;y,s)$ in (6.26), we derive from
(3.23) that

$\displaystyle\left[\eta^{\nu}W_{\mu}^{m}\right](y,s)$
$\displaystyle=[\eta^{\nu}W_{\mu}^{m}](y_{0},\zeta_{0})\exp(-\int_{\zeta_{0}}^{s}\mathbb{D}_{\mu,\nu}^{m}(y(\alpha),\alpha)d\alpha)$
(6.29)
$\displaystyle+\int_{\zeta_{0}}^{s}[\eta^{\nu}\mathbb{F}_{\mu}^{m}(y(\zeta),\zeta)\exp(-\int_{\zeta}^{s}\mathbb{D}_{\mu,\nu}^{m}(y(\alpha),\alpha)d\alpha))d\zeta$

and

$\displaystyle\mathbb{D}_{\mu,\nu}^{m}(y(\zeta),\zeta)$
$\displaystyle=\frac{3\mu}{2}-3\nu+\frac{3\nu}{1+y^{2}}-\frac{2\nu
y}{1+y^{2}}e^{\frac{s}{2}}\beta_{\tau}(\mu_{m}(w)-\dot{\xi}(t))\bigl{|}_{(y,s)=(y(\zeta),\zeta)}$
(6.30)
$\displaystyle:=\frac{3\mu}{2}-3\nu+\mathbb{D}_{\nu}^{m}(y(\zeta),\zeta).$

It is derived from (6.28), (6.30) and (6.3) that

$\int_{\zeta_{0}}^{+\infty}|\mathbb{D}_{\nu}^{m}|(y(\zeta),\zeta)d\zeta\leq
6|\nu|a\int_{\zeta_{0}}^{+\infty}e^{\zeta_{0}-\zeta}d\zeta\leq 6|\nu|a.$
(6.31)

Due to (6.29) and (6.31), when $\frac{3\mu}{2}-3\nu>0$, we obtain

$\displaystyle e^{(\frac{3\mu}{2}-3\nu)s}\eta^{\nu}(y)|W_{\mu}^{m}(y,s)|\leq
e^{6|\nu|a}\biggl{(}e^{(\frac{3\mu}{2}-3\nu)\zeta_{0}}\eta^{\nu}(y_{0}(y,s))|W_{\mu}^{m}(y_{0}(y,s),\zeta_{0})|$
(6.32)
$\displaystyle\qquad\qquad\qquad+\int_{\zeta_{0}}^{s}e^{(\frac{3\mu}{2}-3\nu)\zeta}\eta^{\nu}(y(\zeta))|\mathbb{F}_{\mu}^{m}|(y(\zeta),\zeta)d\zeta\biggr{)}$

and

$\displaystyle
e^{(\frac{7}{6}-\nu^{+})s}\eta^{\frac{1}{3}}(y)|W_{2}^{m}(y,s)|\leq
e^{2a}\biggl{(}e^{(\frac{7}{6}-\nu^{+})\zeta_{0}}|\eta^{\frac{1}{3}}(y_{0}(y,s))|W_{2}^{m}(y_{0}(y,s),\zeta_{0})|$
(6.33)
$\displaystyle\qquad\qquad\qquad+\int_{\zeta_{0}}^{s}e^{(\frac{7}{6}-\nu^{+})\zeta}\eta^{\frac{1}{3}}(y(\zeta))|\mathbb{F}_{2}^{m}|(y(\zeta),\zeta)dl\biggr{)}\
(0\leq\nu^{+}<\frac{7}{6}).$

Here we point out that the factor $\frac{7}{6}-\nu^{+}$ appeared in (6.33) for
$0\leq\nu^{+}<\frac{7}{6}$ is due to (6.25).

With respect to $\mathbb{F}_{2}^{m}$, similarly to the argument for (6.20), it
is derived from (3.16a), (3.22), (6.9), (6.22) and (4.2)-(4.3) that

$|\mathbb{F}_{2}^{m}(y(\zeta),\zeta)|\leq
M^{3}(e^{-\zeta}+\eta^{-\frac{1}{3}}(y(\zeta)))\sum\limits_{j=1}^{n-1}(|W_{2}^{j}(y(\zeta),\zeta)|+|W_{1}^{j}(y(\zeta),\zeta)|).$
(6.34)

For $\mu=1$, $0\leq\nu\leq\frac{1}{3}$ and $1\leq m\leq n-1$ in (6.32), we
obtain from (3.27b), (6.10), (6.20), (6.25), (6.26) and Lemma 6.3 that

$\displaystyle e^{(\frac{3}{2}-3\nu)s}\eta^{\nu}(y)|W_{1}^{m}(y,s)|$
$\displaystyle\leq
e^{2a}\left(M^{\frac{2}{5}}+M^{3}(\varepsilon+\varepsilon^{\frac{1}{3}})\sum\limits_{j=1}^{n-1}\|W_{1}^{j}(z,\tau)\|_{L^{\infty}(\\{|z|\geq
4ae^{\frac{\tau}{2}}\\})}\right)$ $\displaystyle\leq
M^{\frac{2}{5}}e^{2a}+\varepsilon^{\frac{1}{4}}\sum\limits_{j=1}^{n-1}\|W_{1}^{j}(z,\tau)\|_{L^{\infty}(\\{|z|\geq
4ae^{\frac{\tau}{2}}\\})}.$

Combining this with (6.10) shows that for $1\leq j\leq n-1$,

$|\partial_{y}W_{j}(y,s)|\leq\sum\limits_{m-1}^{n-1}|W_{1}^{m}(y,s)|\leq
M^{\frac{3}{5}}e^{(3\nu-\frac{3}{2})s}\eta^{-\nu}(y)\
(0\leq\nu\leq\frac{1}{3}).$ (6.35)

For the estimates of $\partial_{y}^{2}W_{m}$ with $1\leq m\leq n-1$, it
follows from (3.27c), (6.9), (6.25), (6.33)-(6.35) and Lemma 6.3 that

$\displaystyle
e^{(\frac{7}{6}-\nu^{+})s}\eta^{\frac{1}{3}}(y)|W_{2}^{m}(y,s)|\leq
2e^{2a}M^{\frac{3}{5}}\varepsilon^{\nu^{+}}+M^{5}(\varepsilon+\varepsilon^{\frac{1}{3}})$
$\displaystyle\qquad\qquad+M^{4}(\varepsilon+\varepsilon^{\frac{1}{3}})\sum\limits_{j=1}^{n-1}\|e^{(\frac{7}{6}-\nu^{+})\tau}\eta^{\frac{1}{3}}(z)W_{2}^{j}(z,\tau)\|_{L^{\infty}(\\{|z|\geq
4ae^{\frac{\tau}{2}}\\})}\ (0<\nu^{+}<\frac{7}{6}).$

Together with (6.10), this yields that for $1\leq j\leq n-1$,

$|\partial_{y}^{2}W_{j}(y,s)|\leq\sum\limits_{m=1}^{n-1}|W_{2}^{m}(y,s)|\leq
M^{\frac{4}{5}}e^{(\nu^{+}-\frac{7}{6})s}\eta^{-\frac{1}{3}}(y)\
(0<\nu^{+}<\frac{7}{6}).$ (6.36)

##  7 Bootstrap estimates on the modulation variables

For $\dot{\kappa}(t)$ and $\dot{\tau}(t)$, it follows from (1.10), (3.3),
(3.17a), (3.17c), (3.25) and (4.1)-(4.3) that

$\displaystyle|\dot{\kappa}(t)|$ $\displaystyle\leq
e^{s}|\mu_{n}(w^{0})-\dot{\xi}(t)|+e^{\frac{3s}{2}}\sum\limits_{j\neq
n}|a_{nj}(w^{0})||(\partial_{y}W_{j})^{0}|$ (7.1) $\displaystyle\leq
e^{s}|(\partial_{y}^{2}\mu_{n}(w))^{0}|+e^{\frac{3s}{2}}\sum\limits_{k=0,2}\sum\limits_{j\neq
n}|(\partial_{y}^{k}(a_{nj}(w)\partial_{y}W_{j}))^{0}|$
$\displaystyle\leq|(\partial_{w_{n}w_{n}}(\mu_{n}(w)))^{0}|+M\varepsilon^{\frac{1}{3}}\leq
M^{\frac{1}{4}}\leq M^{\frac{1}{2}}$

and

$\displaystyle|\dot{\tau}(t)|$
$\displaystyle\leq|1-(\partial_{w_{n}}\mu_{n}(w))^{0}|+M^{2}e^{-\frac{s}{2}}\leq
M^{2}\varepsilon^{\frac{1}{2}}\leq\varepsilon^{\frac{1}{3}}\leq
2\varepsilon^{\frac{1}{3}}.$ (7.2)

By the coordinate transformation (3.2), one has $t=t(s)$ and

$\frac{d}{ds}(\kappa,\tau,\xi)(t(s))=(\dot{\kappa}(t),\dot{\tau}(t),\dot{\xi}(t))\frac{e^{-s}}{\beta_{\tau}}.$
(7.3)

Combining (7.3) with (7.1)-(7.2) and (3.24) shows

$|\kappa(t)-\kappa_{0}\varepsilon^{\frac{1}{3}}|=|\int_{-\log\varepsilon}^{s}(\dot{\kappa}(t)\frac{e^{-s}}{\beta_{\tau}})ds|\leq|\kappa_{0}\varepsilon|+2M^{\frac{1}{4}}\varepsilon\leq
M^{\frac{1}{2}}\varepsilon$ (7.4)

and

$|\tau(t)|=|\tau(-\varepsilon)+\int_{-\log\varepsilon}^{s}(\dot{\tau}(t)\frac{e^{-s}}{\beta_{\tau}})ds|\leq
2\varepsilon^{\frac{4}{3}}.$ (7.5)

With respect to $\xi(t)$, by (1.10), (3.3), (3.25), (6.18), (6.22)-(6.23) and
(7.2)-(7.4), then $\xi(t)$ in (3.17c) satisfies

$|\dot{\xi}(t)|\leq|\mu_{n}(w^{0})|+\frac{1}{6}|(\partial^{2}\mu_{n}(w))^{0}|+\frac{1}{6}\sum\limits_{j\neq
n}e^{\frac{s}{2}}|(\partial^{2}(a_{nj}(w)\partial_{y}W_{j}))^{0}|\leq
M^{\frac{3}{4}}\varepsilon\leq 2M^{\frac{3}{4}}\varepsilon$ (7.6)

and

$|\xi(t)|=\left|\xi(-\varepsilon)+\int_{-\log\varepsilon}^{s}(\dot{\xi}(t)\frac{e^{-s}}{\beta_{\tau}})ds\right|\leq
2M^{\frac{3}{4}}\varepsilon^{2}.$ (7.7)

##  8 Weighted energy estimates

In the section, we establish the spatial $L^{2}-$energy estimates of
$\partial_{y}^{\mu_{0}}W$ when $\mu_{0}$ satisfies (4.5).

######  Theorem 8.1.

For $\mu_{0}$ satisfying (4.5), under the assumptions (4.1)-(4.3) and (4.6),
one has

$\sum\limits_{m=1}^{n-1}\|\partial_{y}^{\mu_{0}}W_{m}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq
M^{\frac{1}{2}}e^{-\frac{3}{2}s},\
\|\partial_{y}^{\mu_{0}}W_{n}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq
M^{\frac{1}{2}}e^{-\frac{s}{2}}.$ (8.1)

Based on the expansion (3.21) and the assumption (4.6), we have from (6.9) and
(8.1) that

$\sum\limits_{m=1}^{n-1}\|W_{\mu_{0}}^{m}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq
2nMe^{-\frac{3}{2}s},\ \|W_{\mu_{0}}^{n}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq
2Me^{-\frac{s}{2}}.$ (8.2)

###  8.1 Framework for energy estimates

To prove Theorem 8.1, we first establish the following framework for energy
estimates:

######  Lemma 8.2.

With $\mu_{0}$ satisfying (4.5), under the assumption (8.2) (or see (4.6)),
one has

  1. (1)

When $1\leq m\leq n-1$, for any Lipschitz continuous function $q_{m}(y)$ and
with the notation

$Q_{m}(y,s)=-(\frac{3}{2}y+e^{\frac{s}{2}}\beta_{\tau}(\mu_{m}(w)-\dot{\xi}(t)))q_{m}^{\prime}(y),$
(8.3)

then

$\displaystyle\frac{d}{ds}\int_{\mathbb{R}}e^{q_{m}(y)}|W_{\mu_{0}}^{m}|^{2}(y,s)dy+\int_{\mathbb{R}}(3\mu_{0}-\frac{5}{2}-e^{\frac{s}{2}}\beta_{\tau}\partial_{y}\mu_{m}(w)+Q_{m})e^{q_{m}(y)}|W_{\mu_{0}}^{m}|^{2}(y,s)dy$
(8.4) $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}}e^{q_{m}(y)}|\mathbb{F}_{\mu_{0}}^{m}|^{2}(y,s)dy.$

  2. (2)

For $W_{\mu_{0}}^{n}$,

$\frac{d}{ds}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy+\int_{\mathbb{R}}(2\mu_{0}-\frac{5}{2}-e^{\frac{s}{2}}\beta_{\tau}\partial_{y}\mu_{n}(w))|W_{\mu_{0}}^{n}|^{2}(y,s)dy\leq\frac{1}{\mu_{0}+1}\int_{\mathbb{R}}|\mathbb{F}_{\mu_{0}}^{n}|^{2}(y,s)dy.$
(8.5)

###### Proof.

For $1\leq m\leq n-1$, multiplying both sides of (3.22) with $\mu=\mu_{0}$ by
$2e^{q_{m}(y)}W_{\mu_{0}}^{m}$ and integrating on $\mathbb{R}$ yield

$\displaystyle\frac{d}{ds}\int_{\mathbb{R}}e^{q_{m}(y)}|W_{\mu_{0}}^{m}|^{2}(y,s)dy+\int_{\mathbb{R}}(3\mu_{0}-\frac{3}{2}-e^{\frac{s}{2}}\beta_{\tau}\partial_{y}\mu_{m}(w)+Q_{m})e^{q_{m}(y)}|W_{\mu_{0}}^{m}|^{2}(y,s)dy$
(8.6) $\displaystyle=$ $\displaystyle
2\int_{\mathbb{R}}e^{q_{m}(y)}(\mathbb{F}_{\mu_{0}}^{m}\cdot
W_{\mu_{0}}^{m})(y,s)dy$ $\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}}e^{q_{m}(y)}|W_{\mu_{0}}^{m}|^{2}(y,s)dy+\int_{\mathbb{R}}e^{q_{m}(y)}|\mathbb{F}_{\mu_{0}}^{m}|^{2}(y,s)dy.$

Then (8.4) comes from (8.3) and (8.6). The estimate (8.5) can be obtained by a
standard energy estimate associated with the equation (3.22) for $m=n$ and
$\mu=\mu_{0}$. ∎

Next we analyze the structure of $\mathbb{F}_{\mu_{0}}^{m}$.

######  Lemma 8.3.

For $\mu_{0}$ satisfying (4.5) and $1\leq m\leq n-1$, then
$\mathbb{F}_{\mu_{0}}^{m}$ in (3.22) satisfies

$\displaystyle\int_{\mathbb{R}}|\mathbb{F}_{\mu_{0}}^{m}|^{2}(y,s)dy$
$\displaystyle\leq
4nM^{-\frac{1}{16}}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{j}|^{2}(y,s)dy+2nM^{\frac{1}{16}}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|\eta^{-\frac{1}{3}}(y)W_{\mu_{0}}^{j}|^{2}(y,s)dy$
(8.7)
$\displaystyle+2nM^{\frac{1}{4}}e^{-2s}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy+e^{-(3+\frac{1}{4})s}.$

###### Proof.

First, due to (3.3), (5.15), (5.54), (6.18), (6.22) and (7.4), the estimate in
(6.12) can be improved as

$|e^{\frac{s}{2}}\beta_{\tau}[A(w),\frac{\partial\gamma_{j}(w)}{\partial
w}]\partial_{y}W|\leq
M^{\frac{1}{64}}\eta^{-\frac{1}{3}}(y)(1-\delta_{j}^{n}).$ (8.8)

Thus, we can obtain from (3.22), (6.11) and (8.8) that

$\displaystyle\int_{\mathbb{R}}|\mathbb{F}_{\mu_{0}}^{m}|^{2}(y,s)dy$
$\displaystyle\leq
M^{\frac{1}{16}}e^{-s}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{j}|^{2}(y,s)dy+M^{\frac{1}{16}}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|\eta^{-\frac{1}{3}}(y)W_{\mu_{0}}^{j}|^{2}(y,s)dy$
(8.9) $\displaystyle+2\int_{\mathbb{R}}|\ell_{m}\cdot
F_{\mu_{0}}|^{2}(y,s)dy.$

Due to $\ell_{k}^{n}(w)=0$ and $a_{kn}(w)=0$ ($1\leq k\leq n-1$) by
(1.10)-(1.11), for $F_{\mu_{0}}$ in (3.16a), one then has

$\displaystyle|\ell_{m}\cdot F_{\mu_{0}}|\leq$
$\displaystyle\sum\limits_{1\leq\beta\leq\mu_{0}}2C_{\mu_{0}}^{\beta}e^{\frac{s}{2}}|\ell_{m}\cdot\partial_{y}^{\beta}A(w)\partial_{y}^{\mu_{0}+1-\beta}W|$
(8.10) $\displaystyle\leq$
$\displaystyle\sum\limits_{1\leq\beta\leq\mu_{0}}\sum\limits_{1\leq
q\leq\mu_{0}-\beta+1}2C_{\mu_{0}}^{\beta}\|\partial_{w}^{q}A(w)\|_{L^{\infty}}e^{\frac{s}{2}}\sum\limits_{k=1}^{n-1}I_{\beta
qk}$ $\displaystyle\leq$
$\displaystyle\left(M^{\frac{1}{16}}e^{s}\sum\limits_{1\leq\beta\leq\mu_{0}}\sum\limits_{1\leq
q\leq\mu_{0}-\beta+1}\sum\limits_{1\leq k\leq n-1}I_{\beta
qk}^{2}\right)^{1/2},$

where the last inequality comes from (3.3), (4.1)-(4.3), and $I_{\beta qk}$
satisfies

$I_{\beta qk}=\sum\limits_{\gamma_{1}+\cdots+\gamma_{q}=\mu_{0}-\beta+1,\
\gamma_{j}\geq 1\ (1\leq j\leq
q)}|\partial_{y}^{\gamma_{1}}W|\cdots|\partial_{y}^{\gamma_{q}}W|\cdot|\partial_{y}^{\beta}W_{k}|.$
(8.11)

Note that the estimates for ${I_{\beta qk}}^{\prime}s$ in (8.11) are taken in
Lemma A.2 for $1\leq k\leq n-1$. Substituting the three type estimates in
(A.3) into (8.10) shows

$\displaystyle\int_{\mathbb{R}}|\ell_{m}\cdot F_{\mu_{0}}|^{2}(y,s)dy$ (8.12)
$\displaystyle\leq$ $\displaystyle
2nM^{\frac{1}{16}}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|\eta^{-\frac{1}{3}}(y)W_{\mu_{0}}^{j}|^{2}(y,s)dy+2nM^{-\frac{1}{16}}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{j}|^{2}(y,s)dy$
$\displaystyle+4nM^{\frac{3}{8}}e^{-2s}\int_{\mathbb{R}^{n}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy+M^{3\mu_{0}+2}e^{-(3+\frac{1}{2})s}\
(1\leq m\leq n-1).$

Therefore, (8.7) follows from (8.9)-(8.10) and (8.12), and then the proof of
Lemma 8.3 is finished.∎

######  Lemma 8.4.

Let $\mu_{0}$ satisfy (4.5), then for $\mathbb{F}_{\mu_{0}}^{n}$ in (3.22),
one has

$\int_{\mathbb{R}}|\mathbb{F}_{\mu_{0}}^{n}|^{2}(y,s)dy\leq\frac{102}{100}(\mu_{0}+1)^{2}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy+M^{\frac{1}{2}}e^{-s}.$
(8.13)

###### Proof.

Note that under the assumptions (1.10)-(1.11), $\ell_{n}(w)$ has the
decomposition

$\ell_{n}(w)={\bf e}_{n}^{\top}+\sum\limits_{j=1}^{n-1}c_{j}(w)\ell_{j}(w),$
(8.14)

where $c_{j}(w)=-{\bf e}_{n}^{\top}\cdot\gamma_{j}(w)\ (1\leq j\leq n-1)$.

Combining (8.14) with $\gamma_{n}(w)={\bf e}_{n}$ in (1.11), (3.16), (3.22)
shows

$\displaystyle|\mathbb{F}_{\mu_{0}}^{n}|=$ $\displaystyle|\ell_{n}\cdot
F_{\mu_{0}}|\leq\sum\limits_{j=1}^{n-1}|c_{j}(w)\ell_{j}\cdot
F_{\mu_{0}}|+|{\bf e}_{n}^{\top}\cdot F_{\mu_{0}}|$ (8.15) $\displaystyle\leq$
$\displaystyle\sum\limits_{j=1}^{n-1}|c_{j}(w)\ell_{j}\cdot
F_{\mu_{0}}|+e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{1\leq\beta\leq\mu_{0}}C_{\mu_{0}}^{\beta}|\partial_{y}^{\beta}\mu_{n}(w)\partial_{y}^{\mu_{0}-\beta+1}W_{n}|$
$\displaystyle+e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{j=1}^{n-1}\sum\limits_{1\leq\beta\leq\mu_{0}}|\partial_{y}^{\beta}(a_{nj}(w))\partial_{y}^{\mu_{0}-\beta+1}W_{j}|:=\sum\limits_{i=1}^{5}J_{i},$

where

$\begin{cases}J_{1}=e^{\frac{s}{2}}\beta_{\tau}(\mu_{0}+1)|\partial_{y}W_{n}\partial_{y}^{\mu_{0}}W_{n}|,\\\
J_{2}=e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{2\leq\beta\leq\mu_{0}-1}C_{\mu_{0}}^{\beta}|\partial_{y}^{\beta}W_{n}\partial_{y}^{\mu_{0}-\beta}W_{n}|,\\\
J_{3}=\sum\limits_{j=1}^{n-1}|c_{j}(w)\ell_{j}\cdot F_{\mu_{0}}|,\\\
J_{4}=e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{1\leq\beta\leq\mu_{0}}C_{\mu_{0}}^{\beta}|\partial_{y}^{\beta}(\mu_{n}(w)-W_{n})\partial_{y}^{\mu_{0}-\beta+1}W_{n}|,\\\
J_{5}=e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{j=1}^{n-1}\sum\limits_{1\leq\beta\leq\mu_{0}}|\partial_{y}^{\beta}(a_{nj}(w))\partial_{y}^{\mu_{0}-\beta+1}W_{j}|.\end{cases}$

It is derived from (8.15) that

$\int_{\mathbb{R}}|\mathbb{F}_{\mu_{0}}^{n}|^{2}(y,s)dy\leq\frac{101}{100}\int_{\mathbb{R}}J_{1}^{2}(y,s)dy+M^{\frac{1}{32}}\sum\limits_{k=2}^{5}\int_{\mathbb{R}}J_{k}^{2}(y,s)dy.$
(8.16)

In addition, it follows from (3.3), (5.54) and (7.2) that

$\int_{\mathbb{R}}J_{1}^{2}(y,s)dy\leq(1+\varepsilon^{\frac{1}{40}})(\mu_{0}+1)^{2}\int_{\mathbb{R}}|\partial_{y}^{\mu_{0}}W_{n}|^{2}(y,s)dy.$
(8.17)

For $J_{2}$, by Hölder inequality, (3.3) and (5.54), we have

$\displaystyle\int_{\mathbb{R}}J_{2}^{2}(y,s)dy$ $\displaystyle\leq
M^{\frac{1}{8(\mu_{0}-1)}}\int_{\mathbb{R}}|\partial_{y}^{\mu_{0}-1}W_{n}|^{2}(y,s)ds$
(8.18)
$\displaystyle+M^{\frac{1}{16(\mu_{0}-1)}}e^{s}\sum\limits_{1<\beta<\mu_{0}-2}\|\partial_{y}^{\beta}W_{n}(\cdot,s)\|_{L^{\frac{2(\mu_{0}-2)}{\beta-1}}}^{2}\|\partial_{y}^{\mu_{0}-\beta}W_{n}(\cdot,s)\|_{L^{\frac{2(\mu_{0}-2)}{\mu_{0}-\beta-1}}}^{2}.$

Note that by Lemma A.1,

$\displaystyle\|\partial_{y}^{\beta}W_{n}(\cdot,s)\|_{L^{\frac{2(\mu_{0}-2)}{\beta-1}}}\leq
M^{\frac{1}{64(\mu_{0}-1)}}\|\partial_{y}W_{n}(\cdot,s)\|_{L^{\infty}}^{\frac{\mu_{0}-\beta-1}{\mu_{0}-2}}\|\partial_{y}^{\mu_{0}-1}W_{n}(\cdot,s)\|_{L^{2}}^{\frac{\beta-1}{\mu_{0}-2}},$
$\displaystyle\|\partial_{y}^{\mu_{0}-\beta}W_{n}(\cdot,s)\|_{L^{\frac{2(\mu_{0}-2)}{\mu_{0}-\beta-1}}}\leq
M^{\frac{1}{64(\mu_{0}-1)}}\|\partial_{y}W_{n}(\cdot,s)\|_{L^{\infty}}^{\frac{\beta-1}{\mu_{0}-2}}\|\partial_{y}^{\mu_{0}-1}W_{n}(\cdot,s)\|_{L^{2}}^{\frac{\mu_{0}-\beta-1}{\mu_{0}-2}}.$

Then combining these two estimates with (3.3), (5.54), (8.18) and Lemma A.1
yields

$\displaystyle\int_{\mathbb{R}}J_{2}^{2}(y,s)dy$ $\displaystyle\leq
2M^{\frac{1}{8(\mu_{0}-1)}}\int_{\mathbb{R}}|\partial_{y}^{\mu_{0}-1}W_{n}|^{2}(y,s)ds$
(8.19) $\displaystyle\leq
M^{\frac{1}{4}(\mu_{0}-1)}\|\partial_{y}W_{n}(\cdot,s)\|_{L^{2}}^{\frac{2}{\mu_{0}-1}}\|\partial_{y}^{\mu_{0}}W_{n}(\cdot,s)\|_{L^{2}}^{2\frac{\mu_{0}-2}{\mu_{0}-1}}$
$\displaystyle\leq
M^{-\frac{1}{8(\mu_{0}-2)}}\int_{\mathbb{R}}|\partial_{y}^{\mu_{0}}W_{n}|^{2}(y,s)dy+2M^{\frac{3}{8}}e^{-s}.$

For $J_{3}$, due to $\gamma_{j}(0)={\bf e}_{j}\ (1\leq j\leq n-1)$ in (1.11),
one then derives from (3.3) and (4.1)-(4.3) that

$|c_{j}(w)|\leq M\|W(\cdot,s)\|_{L^{\infty}}\leq
M^{4}\varepsilon^{\frac{1}{3}}\leq\varepsilon^{\frac{1}{4}}\ (1\leq j\leq
n-1).$ (8.20)

On the other hand, it follows from (8.2), (8.12) and (8.20) that

$\int_{\mathbb{R}}J_{3}^{2}(y,s)dy\leq\varepsilon^{\frac{1}{6}}e^{-3s}.$
(8.21)

With respect to $J_{4}$, one has from (8.15) that

$\displaystyle J_{4}=$ $\displaystyle
e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{1\leq\beta\leq\mu_{0}}C_{\mu_{0}}^{\beta}|\partial_{y}^{\beta}(\mu_{n}(w)-W_{n})\partial_{y}^{\mu_{0}-\beta+1}W_{n}|$
(8.22) $\displaystyle\leq$ $\displaystyle
e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{1\leq\beta\leq\mu_{0}}C_{\mu_{0}}^{\beta}|\partial_{w_{n}}(\mu_{n}(w)-W_{n})|I_{\beta
1n}$ $\displaystyle+$ $\displaystyle
e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{1\leq\beta\leq\mu_{0}}C_{\mu_{0}}^{\beta}\sum\limits_{1\leq
k\leq n-1}|\partial_{w_{k}}(\mu_{n}(w)-W_{n})|I_{\beta 1k}$ $\displaystyle+$
$\displaystyle
e^{\frac{s}{2}}\beta_{\tau}\sum\limits_{1\leq\beta\leq\mu_{0}}\sum\limits_{2\leq
q\leq\mu_{0}-\beta+1}c_{\beta qn}(w)I_{\beta qn}$ $\displaystyle:=$
$\displaystyle\sum\limits_{1\leq\beta\leq\mu_{0}}(J_{41\beta}+J_{42\beta}+J_{43\beta}),$

where $|c_{\beta qn}(w)|\leq M$ due to (3.3) and (4.1)-(4.3). Since
$\partial_{w_{n}}\mu_{n}(0)=1$, by (3.3) and (4.1)-(4.3), we have

$|J_{41\beta}|\leq M\varepsilon^{\frac{1}{6}}e^{\frac{s}{2}}I_{\beta 1n}.$
(8.23)

In the similar and easier way,

$|J_{42\beta}|+|J_{43\beta}|\leq M\sum\limits_{1\leq k\leq n-1}I_{\beta
1k}+M^{2}\sum\limits_{2\leq q\leq\mu_{0}-\beta+1}I_{\beta qn}.$ (8.24)

With the help of Lemma A.2, Lemma A.3 and (8.2), we obtain from (8.22)-(8.24)
that

$\int_{\mathbb{R}}J_{4}^{2}(y,s)dy\leq\varepsilon^{\frac{1}{10}}e^{-s}.$
(8.25)

Analogously to the treatment of $J_{4}$ in (8.22)-(8.25), one has

$\int_{\mathbb{R}}J_{5}^{2}(y,s)dy\leq\varepsilon^{\frac{1}{10}}e^{-s}.$
(8.26)

Therefore, (8.13) comes from (8.17), (8.19), (8.21), (8.25), (8.26) and the
largeness of $M$. ∎

###  8.2 Energy estimates of $W_{\mu_{0}}^{n}$

First, we close the estimate of $W_{\mu_{0}}^{n}$ in (8.2). Substituting
(8.13) into (8.5) yields

$\frac{d}{ds}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy+\int_{\mathbb{R}}(\frac{98\mu_{0}-102}{100}-\frac{5}{2}-\beta_{\tau}e^{\frac{s}{2}}\partial_{y}\mu_{n}(w))|W_{\mu_{0}}^{n}|^{2}(y,s)dy\leq
M^{\frac{1}{2}}e^{-s}.$ (8.27)

When $\mu_{0}\geq 6$, it is derived from (1.10), (3.3), (5.15), (5.54),
(6.18), (6.22), (7.2) and (7.4) that

$\frac{98\mu_{0}-102}{100}-\frac{5}{2}-\beta_{\tau}e^{\frac{s}{2}}\partial_{y}\mu_{n}(w)\geq\frac{486}{100}-\frac{5}{2}-Me^{-s}-(1+\varepsilon^{\frac{1}{20}})>\frac{6}{5}.$
(8.28)

In addition, one has from (8.27)-(8.28) that

$\frac{d}{ds}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy+\frac{6}{5}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy\leq
M^{\frac{1}{2}}e^{-s}.$

This yields

$\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy\leq
e^{\frac{6}{5}(-\log\varepsilon-s)}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,-\log\varepsilon)dy+5M^{\frac{1}{2}}e^{-s}.$
(8.29)

Then it follows from (8.29), (3.3), (3.28) and (6.9) that

$\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy\leq 6M^{\frac{1}{2}}e^{-s}.$
(8.30)

###  8.3 Energy estimates of $W_{\mu_{0}}^{j}\ (1\leq j\leq n-1)$

We now close the estimates of $W_{\mu_{0}}^{j}\ (1\leq j\leq n-1)$ in (8.2).
Similarly to (8.28), there exists a minimal positive integer $\mu_{0}\geq 6$
such that for all $1\leq m\leq n-1$,

$3\mu_{0}-\frac{5}{2}-\beta_{\tau}e^{\frac{s}{2}}\partial_{y}\mu_{m}(w)\geq
3\mu_{0}-\frac{5}{2}-Me^{-s}-(1+\varepsilon^{\frac{1}{4}})|\partial_{w_{n}}\mu_{m}(0)|\geq
4.$ (8.31)

This also ensures the assumption (4.5) in turn.

In addition, it follows from (8.7) and (8.30) that

$\int_{\mathbb{R}}|\mathbb{F}_{m}^{\mu_{0}}|^{2}(y,s)dy\leq\frac{\delta}{2n}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{j}|^{2}(y,s)dy+c^{*}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|\eta^{-\frac{1}{3}}(y)W_{\mu_{0}}^{j}|^{2}(y,s)dy+24nM^{\frac{3}{4}}e^{-3s},$
(8.32)

where $\delta=8n^{2}M^{-\frac{1}{16}}$ and $c^{*}=2nM^{\frac{1}{16}}$. On the
other hand, there exist two positive constants $K$ and $K^{*}$ such that

$\eta^{-\frac{2}{3}}(y)\leq\frac{\delta}{2nc^{*}}\ (|y|\geq K),\
\eta^{-\frac{2}{3}}(y)\leq\frac{K^{*}}{nc^{*}}\ (|y|\leq K).$ (8.33)

It is derived from (8.32) and (8.33) that

$\int_{\mathbb{R}}|\mathbb{F}_{m}^{\mu_{0}}|^{2}(y,s)dy\leq\frac{\delta}{n}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}j}|^{2}(y,s)dy+\frac{K^{*}}{n}\sum\limits_{j=1}^{n-1}\int_{-K}^{K}|W_{\mu_{0}j}|^{2}(y,s)dy+24nM^{\frac{3}{4}}e^{-3s}.$
(8.34)

Next we determine the Lipschitz continuous function $q_{m}(y)\ (1\leq m\leq
n-1)$ in Lemma 8.2. Due to (1.10a) and $\mu_{n}(0)=0$, when $\mu_{m}(0)<0\
(1\leq m\leq i_{0}-1)$, then $q_{m}(y)$ is defined as

$q_{m}(y):=q_{m}^{-}(y)=\begin{cases}0,\ y\leq-K,\\\ \frac{y}{K}+1,\ -K\leq
y\leq K,\\\ 2,\ y\geq K.\end{cases}$ (8.35)

When $\mu_{m}(0)>0\ (i_{0}\leq m\leq n-1)$, $q_{m}(y)$ is defined as

$q_{m}(y):=q_{m}^{+}(y)=q_{m}^{-}(-y).$ (8.36)

According to the definitions (8.35)-(8.36), we have

$0\leq q_{m}(y)\leq 2.$ (8.37)

On the other hand, $Q_{m}(y,s)$ in (8.3) satisfies

$\displaystyle Q_{m}(y,s)$
$\displaystyle\geq\left(\frac{1}{K}\beta_{\tau}e^{\frac{s}{2}}|\mu_{m}(w)-\dot{\xi}(t)|-\frac{3}{2}K\right)\chi_{\\{|y|\leq
K\\}}$ (8.38)
$\displaystyle\geq\left(\frac{|\mu_{m}(0)|}{2K}e^{\frac{s}{2}}-\frac{3}{2}K\right)\chi_{\\{|y|\leq
K\\}}\geq K^{*}\chi_{\\{|y|\leq K\\}},$

where the last inequality comes from (1.10), (3.3), (5.15), (6.18), (7.6) and
the fact of $s\geq-\log\varepsilon$.

With (8.30)-(8.31), (8.34), (8.37)-(8.38) and the largeness of $M$, summing up
$m$ on both sides of (8.4) from $1$ to $n-1$ yield

$\displaystyle\frac{d}{ds}\sum\limits_{m=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{m}|^{2}(y,s)dy+\frac{7}{2}\sum\limits_{m=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{m}|^{2}(y,s)dy\leq
14n^{2}M^{\frac{3}{4}}e^{4-3s}.$

This shows

$\sum\limits_{m=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{m}|^{2}(y,s)dy\leq
e^{\frac{7}{2}(-\log\varepsilon-s)}\sum\limits_{m=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{m}|^{2}(y,-\log\varepsilon)dy+28n^{2}M^{\frac{3}{4}}e^{4-3s}.$
(8.39)

Then it comes from (3.28), (6.9) and (8.39) that

$\sum\limits_{m=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy\leq
30n^{2}e^{4}M^{\frac{3}{4}}e^{-3s}.$ (8.40)

Proof of Theorem 8.1: It is derived from (8.30), (8.40) and (6.10) that

$\sum\limits_{j=1}^{n-1}\|\partial_{y}^{\mu_{0}}W_{j}|^{2}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq(n-1)\sum\limits_{k=1}^{n-1}\|W_{\mu_{0}}^{k}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq\sqrt{30}n^{2}e^{2}M^{\frac{3}{8}}e^{-\frac{3}{2}s}$
(8.41)

and

$\|\partial_{y}^{\mu_{0}}W_{n}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq\sum\limits_{k=1}^{n}\|W_{\mu_{0}}^{k}(\cdot,s)\|_{L^{2}(\mathbb{R})}\leq\sqrt{30}n^{2}e^{2}M^{\frac{3}{8}}e^{-\frac{3}{2}s}+\sqrt{6}M^{\frac{1}{4}}e^{-\frac{s}{2}}.$
(8.42)

Then the estimates in (8.1) come from (8.41), (8.42) and the largeness of $M$.
Therefore, the proof of Theorem 8.1 is completed.

##  9 Proof of main theorems

In the section, we complete the proofs of Theorem 3.1 and Theorem 1.1.

###  9.1 Proof of Theorem 3.1

Based on the local existence of (1.9) (see [19]), we utilize the continuous
induction to prove Theorem 3.1. To this end, under the induction assumptions
(4.1)-(4.3) and (4.6) for suitably large $M>16$, the proof of Theorem 3.1 is
mainly reduced to recover the estimates (4.1)-(4.3) and (4.6) with the smaller
coefficient bounds via the bootstrap arguments.

The induction assumptions of $\kappa(t),\tau(t)$ and $\xi(t)$ in (4.1a) and
their derivatives in (4.1b) are recovered in (7.4)-(7.5), (7.7) and
(7.1)-(7.2), (7.6) with $M$ replaced by the smaller ones $M^{\frac{1}{2}},2$
and $2M^{\frac{3}{4}}$ respectively.

In the similar way, the assumptions (4.2) for $W_{0}$ as well as $\mathcal{W}$
are also recovered by (5.41), (5.55), (5.54), (5.32) and (5.44) with the
coefficients replaced by the smaller ones accordingly. In addition, the
assumptions (4.3) for $W_{j}\ (j\neq n)$ are also obtained by (6.18), (6.23),
(6.35) and (6.36) with the coefficient $M$ replaced by the smaller ones. In
addition, the energy assumptions (4.6) are obviously derived by (8.1) in
Theorem 8.1 with $M$ replaced by $M^{\frac{1}{2}}$.

Therefore, Theorem 3.1 is proved via the method of continuous induction.

###  9.2 Proof of Theorem 1.1

Due to (3.2) and Theorem 3.1, in order to complete the proof of Theorem 1.1,
we only need to verify the $C^{\frac{1}{3}}$ optimal regularity of $W_{0}$ and
(1.17).

First, we show that the optimal regularity of $W_{0}(y,s)$ is
$C^{\frac{1}{3}}$ with respect to the spatial variable.

By (3.2), for any $t\geq-\varepsilon$ and $x_{1},x_{2}\in\mathbb{R}$, set

$s=s(t),y_{i}=(x_{i}-\xi(t))e^{\frac{3s}{2}}\ (i=1,2).$ (9.1)

Then for any $\alpha>0$, by (3.3), we arrive at

$\frac{|w_{n}(x_{1},t)-w_{n}(x_{2},t)|}{|x_{1}-x_{2}|^{\alpha}}=e^{\frac{3\alpha
s}{2}-\frac{s}{2}}\frac{|W_{0}(y_{1},s)-W_{0}(y_{2},s)|}{|y_{1}-y_{2}|^{\alpha}}.$
(9.2)

When $\alpha=\frac{1}{3}$, it is derived from (9.2), (3.14b) and (5.54) that

$\displaystyle\sup\limits_{x_{1},x_{2}\in\mathbb{R},x_{1}\neq
x_{2}}\frac{|w_{n}(x_{1},t)-w_{n}(x_{2},t)|}{|x_{1}-x_{2}|^{\alpha}}$ (9.3)
$\displaystyle\leq$
$\displaystyle\sup\limits_{y_{1},y_{2}\in\mathbb{R},y_{1}\neq
y_{2}}\frac{|W_{0}(y_{1},s)-W_{0}(y_{2},s)|}{|y_{1}-y_{2}|^{\frac{1}{3}}}$
$\displaystyle=$ $\displaystyle\sup\limits_{y_{1},y_{2}\in\mathbb{R},y_{1}\neq
y_{2}}\frac{\displaystyle|\int_{y_{2}}^{y_{1}}\partial_{y}W_{0}(z,s)dz|}{|y_{1}-y_{2}|^{\frac{1}{3}}}$
$\displaystyle\leq$ $\displaystyle
2\sup\limits_{y_{1},y_{2}\in\mathbb{R},y_{1}\neq
y_{2}}\frac{\displaystyle|\int_{y_{2}}^{y_{1}}\eta^{-\frac{1}{3}}(z)dz|}{|y_{1}-y_{2}|^{\frac{1}{3}}}\leq
12.$

When $\frac{1}{3}<\alpha<1$, with (3.13), (3.18), (5.32) and
$y_{1}^{*}=1,y_{2}^{*}=0$ (Denote ($x_{1}^{*},x_{2}^{*})$ by the corresponding
transformation (9.1) respectively), we have

$\displaystyle\frac{|W_{0}(y_{1}^{*},s)-W_{0}(y_{2}^{*},s)|}{|y_{1}^{*}-y_{2}^{*}|^{\alpha}}$
$\displaystyle\geq|\overline{W}(1)|-|\mathcal{W}(1,s)-\mathcal{W}(0,s)|$ (9.4)
$\displaystyle\geq|\overline{W}(1)|-2\varepsilon^{\frac{1}{11}}>\frac{1}{2}|\overline{W}(1)|>0.$

Thus, (9.4) shows that when $\frac{1}{3}<\alpha<1$, for any $\overline{M}>0$,
there exists a constant $s_{0}\geq-\log\varepsilon$ (and corresponding $t_{0}$
by (9.1)) such that when $s\geq s_{0}$,

$e^{\frac{3\alpha
s}{2}-\frac{s}{2}}\frac{|W_{0}(y_{1}^{*},s)-W_{0}(y_{2}^{*},s)|}{|y_{1}^{*}-y_{2}^{*}|^{\alpha}}\geq\frac{1}{2}e^{\frac{3\alpha
s_{0}}{2}-\frac{s_{0}}{2}}|\overline{W}(1)|>\overline{M}.$ (9.5)

Due to the arbitrariness of $\overline{M}$, it implies from (9.2) and (9.5)
that $w_{n}(x,t)\notin C^{\alpha}$ with $\alpha>\frac{1}{3}$. Combining this
with (9.3) shows that the optimal regularity of $w_{n}(x,t)$ is
$C^{\frac{1}{3}}$ with respect to the spatial variable.

Therefore, (1) and (2) in Theorem 1.1 are obtained from Theorem 3.1 and the
above verification of $C^{\frac{1}{3}}$ regularity for $W_{0}$.

Next we prove (1.17). It is derived from (3.2), (3.3) and (3.15) that

$\partial_{x}w_{n}(\xi(t),t)=e^{s}\partial_{y}W_{0}(0,s)=-e^{s}=\frac{1}{\tau(t)-t}.$
(9.6)

By the definition of $\tau(t)$ in (3.1), we have $\tau(T^{*})=T^{*}$ and then

$|T^{*}-\tau(t)|=|\tau(T^{*})-\tau(t)|\leq\int_{t}^{T^{*}}|\dot{\tau}(z)|dz\lesssim\varepsilon^{\frac{1}{3}}(T^{*}-t),$
(9.7)

where the last inequality comes from (2) in Theorem 3.1.

Following (9.7), one has

$\displaystyle|\tau(t)-t|\leq|T^{*}-t|+|\tau(t)-T^{*}|\leq(1+\varepsilon^{\frac{1}{4}})|T^{*}-t|,$
(9.8)
$\displaystyle|\tau(t)-t|\geq|T^{*}-t|-|\tau(t)-T^{*}|\geq(1-\varepsilon^{\frac{1}{4}})|T^{*}-t|.$

Then, we derive from (9.6)-(9.8) that

$-2<(T^{*}-t)\partial_{x}w_{n}(\xi(t),t)<-\frac{1}{2}.$ (9.9)

Collecting (9.7), (9.9) and Theorem 3.1 yields (1.17) in Theorem 1.1 and then
the proof of Theorem 1.1 is completed.

##  A Appendix

In the Appendix, we introduce a useful interpolation inequality (see [1]) and
give its applications.

######  Lemma A.1.

(Gagliardo-Nirenberg-Sobolev inequality). Let $u:\mathbb{R}^{d}\to\mathbb{R}$.
Fix $1\leq q,r\leq\infty$ and $j,m\in\mathbb{N}$, and
$\frac{j}{m}\leq\alpha\leq 1$. If

$\frac{1}{p}=\frac{j}{d}+\alpha(\frac{1}{r}-\frac{m}{d})+\frac{1-\alpha}{q},$

then one has

$\|D^{j}u\|_{L^{p}}\leq C\|D^{m}u\|_{L^{r}}^{\alpha}\|u\|_{L^{q}}^{1-\alpha},$
(A.1)

where the positive constant $C$ depends on $d,m,r,q$ and $m$.

Next we estimate the $L^{2}-$norm of the terms ${I_{\beta qk}}^{\prime}s\
(1\leq k\leq n)$ with the expression as

$I_{\beta qk}=\sum\limits_{\gamma_{1}+\cdots+\gamma_{q}=\mu_{0}-\beta+1,\
\gamma_{j}\geq 1(1\leq j\leq
q)}|\partial_{y}^{\gamma_{1}}W|\cdots|\partial_{y}^{\gamma_{q}}W|\cdot|\partial_{y}^{\beta}W_{k}|.$
(A.2)

The estimates of ${I_{\beta qk}}^{\prime}s$ are considered in two cases:
$1\leq k\leq n-1$ and $k=n$.

######  Lemma A.2.

For $1\leq k\leq n-1$, we have

$\displaystyle\sum\limits_{\beta=1,\mu_{0}}\int_{\mathbb{R}}I_{\beta
1k}^{2}(y,s)dy$ $\displaystyle\qquad\qquad\leq
2e^{-s}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|\eta^{-\frac{1}{3}}(y)W_{\mu_{0}}^{j}|^{2}(y,s)dy+M^{\frac{1}{16}}e^{-3s}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy+M^{3}e^{-6s},$
(A.3a) $\displaystyle\sum\limits_{1<\beta<\mu_{0}}\int_{\mathbb{R}}I_{\beta
1k}^{2}(y,s)dy$ $\displaystyle\qquad\qquad\leq
2M^{-\frac{1}{8}}e^{-s}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{j}|^{2}(y,s)dy+M^{\frac{1}{4}}e^{-3s}\int_{\mathbb{R}}|W_{\mu_{0}}^{n}|^{2}(y,s)dy+M^{3}e^{-6s},$
(A.3b) $\displaystyle\int_{\mathbb{R}}I_{\beta qk}^{2}(y,s)\leq
M^{3\mu_{0}+1}e^{-(4+\frac{1}{4})s}\ (q\geq 2).$ (A.3c)

###### Proof.

For the proof of (A.3a), by (3.3), (5.54), (6.22), the expansion in (3.21),
(6.9)-(6.10) and (8.2), we have

$\displaystyle\int_{\mathbb{R}}(I^{2}_{\mu_{0}1k}+I^{2}_{11k})(y,s)dy$ (A.4)
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{R}}|\partial_{y}W|^{2}\cdot|\partial_{y}^{\mu_{0}}W_{k}|^{2}(y,s)dy+\int_{\mathbb{R}}|\partial_{y}W_{k}|^{2}|\partial_{y}^{\mu_{0}}W|^{2}(y,s)dy$
$\displaystyle\leq$ $\displaystyle
2e^{-s}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|\eta^{-\frac{1}{3}}(y)W_{\mu_{0}}^{j}|^{2}(y,s)dy+M^{\frac{1}{16}}e^{-3s}\sum\limits_{j=1}^{n}\int_{\mathbb{R}}|W_{\mu_{0}}^{j}|^{2}(y,s)dy.$

Combining (A.4) with (8.2) yields (A.3a).

With respect to the case of $1<\beta<\mu_{0}$ and $q=1$ in (A.2), it is
derived from (3.3), (5.54), (6.10), (6.22), (8.2), Hölder inequality and Lemma
A.1 that

$\displaystyle\sum\limits_{1<\beta<\mu_{0}}\int_{\mathbb{R}}I_{\beta
1k}^{2}(y,s)dy$ (A.5) $\displaystyle\leq$
$\displaystyle\sum\limits_{1<\beta<\mu_{0}}\int_{\mathbb{R}}|\partial_{y}^{\mu_{0}+1-\beta}W|^{2}|\partial_{y}^{\beta}W_{k}|^{2}(y,s)dy$
$\displaystyle\leq$
$\displaystyle\sum\limits_{1<\beta<\mu_{0}}\|\partial_{y}^{\beta}W_{k}(\cdot,s)\|_{L^{\frac{2(\mu_{0}-1)}{\beta-1}}}^{2}\|\partial_{y}^{\mu_{0}-\beta+1}W(\cdot,s)\|_{L^{\frac{2(\mu_{0}-1)}{\mu_{0}-\beta}}}^{2}$
$\displaystyle\leq$ $\displaystyle
M^{\frac{1}{16}}\sum\limits_{1<\beta<\mu_{0}}\|\partial_{y}^{\mu_{0}}W_{k}(\cdot,s)\|_{L^{2}}^{2\frac{\beta-1}{\mu_{0}-1}}\|\partial_{y}W_{k}(\cdot,s)\|_{L^{\infty}}^{2\frac{\mu_{0}-\beta}{\mu_{0}-1}}\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{2\frac{\mu_{0}-\beta}{\mu_{0}-1}}\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{2\frac{\beta-1}{\mu_{0}-1}}$
$\displaystyle\leq$ $\displaystyle
M^{\frac{1}{8}+\frac{\beta-1}{8(\mu_{0}-1)}}\|\partial_{y}W_{k}(\cdot,s)\|_{L^{\infty}}^{2}\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{2}+M^{-\frac{1}{8}}\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{2}\|\partial_{y}^{\mu_{0}}W_{k}(\cdot,s)\|_{L^{2}}^{2}$
$\displaystyle\leq$ $\displaystyle
2M^{-\frac{1}{8}}e^{-s}\sum\limits_{j=1}^{n-1}\int_{\mathbb{R}}|W_{\mu_{0}}^{j}|^{2}(y,s)dy+M^{\frac{1}{4}}e^{-3s}\sum\limits_{j=1}^{n}\int_{\mathbb{R}}|W_{\mu_{0}}^{j}|^{2}(y,s)dy.$

Then (A.3b) comes from (A.5) and (8.2).

For the easier cases of $I_{\beta qk}$ ($q\geq 2$) in (A.2), we will apply the
following three type estimates with the help of Lemma A.1:

$\displaystyle\|\partial_{y}^{\gamma_{j}}W(\cdot,s)\|_{L^{\infty}}\leq
M^{\frac{1}{16}}\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{\frac{\gamma_{j}-1}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{\frac{\mu_{0}-\frac{1}{2}-\gamma_{j}}{\mu_{0}-1-\frac{1}{2}}},$
(A.6a) $\displaystyle\|\partial_{y}^{\gamma_{j}}W(\cdot,s)\|_{L^{2}}\leq
M^{\frac{1}{16}}\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{\frac{\gamma_{j}-1-\frac{1}{2}}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{\frac{\mu_{0}-\gamma_{j}}{\mu_{0}-1-\frac{1}{2}}}\
(\gamma_{j}\geq 2),$ (A.6b)
$\displaystyle\|\partial_{y}^{\beta}W_{k}(\cdot,s)\|_{L^{2}}\leq
M^{\frac{1}{16}}\|\partial_{y}^{\mu_{0}}W_{k}(\cdot,s)\|_{L^{2}}^{\frac{\beta-1-\frac{1}{2}}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}W_{k}(\cdot,s)\|_{L^{\infty}}^{\frac{\mu_{0}-\beta}{\mu_{0}-1-\frac{1}{2}}}\
(\beta\geq 2).$ (A.6c)

When $\beta\geq 2$ and $q\geq 2$, substituting (A.6a) and (A.6c) into (A.2)
yields

$\displaystyle\int_{\mathbb{R}}I_{\beta qk}^{2}(y,s)dy$ (A.7) $\displaystyle=$
$\displaystyle\sum\limits_{\gamma_{1}+\cdots+\gamma_{q}=\mu_{0}-\beta+1,\
\gamma_{j}\geq 1(1\leq j\leq
q)}\|\partial_{y}^{\gamma_{1}}W(\cdot,s)\|_{L^{\infty}}^{2}\cdots\|\partial_{y}^{\gamma_{q}}W(\cdot,s)\|_{L^{\infty}}^{2}\|\partial_{y}^{\beta}W_{k}(\cdot,s)\|_{L^{2}}^{2}$
$\displaystyle\leq$ $\displaystyle
M^{\frac{q+1}{4}}\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{2\frac{\mu_{0}-\beta+1-q}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{2\frac{q(\mu_{0}-\frac{1}{2})-(\mu_{0}-\beta+1)}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}^{\mu_{0}}W_{k}(\cdot,s)\|_{L^{2}}^{2\frac{\beta-1-\frac{1}{2}}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}W_{k}(\cdot,s)\|_{L^{\infty}}^{2\frac{\mu_{0}-\beta}{\mu_{0}-1-\frac{1}{2}}}$
$\displaystyle\leq$ $\displaystyle
M^{\frac{q+1}{2}}\left(\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{2q}+\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{2q}\right)\left(\|\partial_{y}^{\mu_{0}}W_{k}(\cdot,s)\|_{L^{2}}^{2}+\|\partial_{y}W_{k}(\cdot,s)\|_{L^{\infty}}^{2}\right).$

Combining this with (4.6), (5.54) and (6.22) derives

$\int_{\mathbb{R}}I_{\beta qk}^{2}(y,s)dy\leq M^{3q+1}e^{-(q+3)s}\leq
M^{3\mu_{0}+1}e^{-(q+3)s}\ (q\geq 2,\beta\geq 2).$ (A.8)

When $\beta=1$ and $q\geq 3$, similarly to (A.8), we have

$\displaystyle\int_{\mathbb{R}}I_{1qk}^{2}(y,s)dy$ (A.9) $\displaystyle=$
$\displaystyle\sum\limits_{\gamma_{1}+\cdots+\gamma_{q}=\mu_{0}-\beta+1,\
\gamma_{j}\geq 1(1\leq j\leq
q)}\|\partial_{y}^{\gamma_{1}}W(\cdot,s)\|_{L^{\infty}}^{2}\cdots\|\partial_{y}^{\gamma_{q}}W(\cdot,s)\|_{L^{\infty}}^{2}\|\partial_{y}W_{k}(\cdot,s)\|_{L^{2}}^{2}$
$\displaystyle\leq$ $\displaystyle
M^{\frac{q+1}{4}}\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{2\frac{\mu_{0}-\beta+1-q}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{2\frac{q(\mu_{0}-\frac{1}{2})-(\mu_{0}-\beta+1)}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}W_{k}(\cdot,s)\|_{L^{2}}^{2}$
$\displaystyle\leq$ $\displaystyle
M^{\frac{q+1}{2}}\left(\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{2q}+\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{2q}\right)\|\partial_{y}W_{k}(\cdot,s)\|_{L^{2}}^{2}$
$\displaystyle\leq$ $\displaystyle M^{3q+1}e^{-(q+\frac{5}{4})s}\ (q\geq 3),$

where the last estimate comes from (4.6) and (6.35) with $\nu=\frac{7}{24}$.

When $\beta=1$ and $q=2$, without loss of generality, we assume
$\gamma_{2}\geq 2$ due to $\gamma_{1}+\gamma_{2}=\mu_{0}\geq 6$. Then we apply
(A.6a) and (A.6b) to control $\partial_{y}^{\gamma_{1}}W$ and
$\partial_{y}^{\gamma_{2}}W$ respectively and subsequently obtain the
following estimate with (4.6) and (6.22)

$\displaystyle\int_{\mathbb{R}}I_{1qk}^{2}(y,s)dy$ (A.10) $\displaystyle=$
$\displaystyle\|\partial_{y}^{\gamma_{1}}W(\cdot,s)\|_{L^{\infty}}^{2}\|\partial_{y}^{\gamma_{2}}W(\cdot,s)\|_{L^{2}}^{2}\|\partial_{y}W_{k}(\cdot,s)\|_{L^{\infty}}^{2}$
$\displaystyle\leq$ $\displaystyle
M^{\frac{1}{4}}\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{2\frac{\mu_{0}-\frac{1}{2}}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{2\frac{\mu_{0}-2-\frac{1}{2}}{\mu_{0}-1-\frac{1}{2}}}\|\partial_{y}W_{k}(\cdot,s)\|_{L^{\infty}}^{2}$
$\displaystyle\leq$ $\displaystyle M^{5}e^{-5s}.$

Thus, we get (A.3c) from (A.8)-(A.10) and the proof of Lemma A.2 is finished.∎

######  Lemma A.3.

For $I_{\beta qn}$, one has

$\int_{\mathbb{R}}I_{\beta qn}^{2}(y,s)dy\leq M^{3q+1}e^{-(q+1)s}.$ (A.11)

###### Proof.

As in Lemma A.2, the estimates on the two terms $I_{\mu_{0}1n}$ and $I_{11n}$
are crucial in the proof of (A.11). For these two terms, similarly to (A.4),
one has

$\int_{\mathbb{R}}(I_{\mu_{0}1n}^{2}+_{11n}^{2})(y,s)dy\leq
2\int_{\mathbb{R}}|\partial_{y}W|^{2}\cdot|\partial_{y}^{\mu_{0}}W|^{2}(y,s)dy\leq
2\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{2}\cdot\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{2}.$
(A.12)

When $1<\beta<\mu_{0}$, similarly to (A.7), we arrive at

$\int_{\mathbb{R}}I_{\beta qn}^{2}(y,s)dy\leq
M^{\frac{q+1}{2}}\left(\|\partial_{y}W(\cdot,s)\|_{L^{\infty}}^{2q}+\|\partial_{y}^{\mu_{0}}W(\cdot,s)\|_{L^{2}}^{2q}\right)\left(\|\partial_{y}W_{n}(\cdot,s)\|_{L^{\infty}}^{2}+\|\partial_{y}^{\mu_{0}}W_{n}(\cdot,s)\|_{L^{2}}^{2}\right).$
(A.13)

Combining (A.12)-(A.13) with (3.3), (4.6), (5.54) and (6.22) yields (A.11) and
the proof of Lemma A.3 is completed. ∎

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