# Subsonic time-periodic solution to compressible Euler equations with damping
in a bounded domain

Peng Qu e-mail<EMAIL_ADDRESS>School of Mathematical Sciences, Shanghai Key
Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai
200433 China  Huimin Yu e-mail<EMAIL_ADDRESS>Department of mathematics,
Shandong Normal University, Jinan 250014 China  Xiaomin Zhang e-mail:
<EMAIL_ADDRESS>Department of mathematics, Shandong Normal University,
Jinan 250014 China

Abstract: In this paper, we consider the one-dimensional isentropic
compressible Euler equations with linear damping $\beta(t,x)\rho u$ in a
bounded domain, which can be used to describe the process of compressible
flows through a porous medium. And the model is imposed a dissipative subsonic
time-periodic boundary condition. Our main results reveal that the time-
periodic boundary can trigger a unique subsonic time-periodic smooth solution
which is stable under small perturbations on initial data. Moreover, the time-
periodic solution possesses higher regularity and stability provided a higher
regular boundary condition.
Keywords: Isentropic compressible Euler equations, time-periodic boundary,
source term, global existence, stability, subsonic flow, time-periodic
solutions
Mathematics Subject Classification 2010: 35B10, 35A01, 35Q31.

##  1 Introduction

In this paper, we consider the one-dimensional compressible Euler equations
for isentropic flow with linear damping in Eulerian coordinates

$\left\\{\begin{aligned} &\partial_{t}\rho+\partial_{x}(\rho u)=0,\\\
&\partial_{t}(\rho u)+\partial_{x}(\rho u^{2}+p)=\beta(t,x)\rho
u,\end{aligned}\right.$ (1.1)

where temporal and spatial variables $(t,x)\in
D=\\{t,x)|t\in\mathbb{R}_{+},x\in[0,L]\\}$. $\rho,u$ and
$p(\rho)=A\rho^{\gamma}$ are the density, velocity and pressure respectively.
Here the adiabatic gas exponent $\gamma>1$ and pressure coefficient $A$ is
normalized to $1$. Damping coefficient $\beta=\beta(t,x)$ is a non-positive
$C^{2}$ smooth function.
In recent years, much effort has been made on the time-periodic solutions to
the viscous fluids equations and the hyperbolic conservation laws, see for
example [1, 5, 6, 7, 2, 9, 12, 13, 8]. However, the time-periodic solutions
mentioned above are usually caused by the time-periodic external forces. As
far as we know, there is little work to consider the problem with time-
periodical boundary. In 2019, Yuan [16] studied the existence and high-
frequency limiting behavior of supersonic time-periodic solutions to the 1-D
isentropic compressible Euler equations (i.e. $\beta(t,x)\equiv 0$) with time-
periodic inflow boundary conditions. Supersonic is an essential assumption in
[16], which implies that all characteristic propagate forward in both space
and time. Therefore, due to the finite propagation speed, Yuan obtained the
existence of time-periodic solution after the start-up time $T_{0}>0$, which
are needed for the boundary condition to effect the whole domain. From a
physical point of view, the force produced by the wall of duct [11] or porous
medium can be regarded as some source term added to the Euler equations. A
natural question is: what kind of effects can the friction term do on the
time-periodic solution$?$ [14, 15] considered the existence and stability of
supersonic time periodic flows for the compressible Euler equations with
friction term. Due to wave interactions between families of different
directions, it is well known that the case with subsonic boundary conditions
is more complicated for compressible Euler equation. Motivated by [10], in
which Qu considered the time-periodic solutions triggered by a kind of
dissipative time-periodic boundary condition for the general quasilinear
hyperbolic systems, we consider the subsonic time-periodical solutions of
isentropic Euler equation with linear damping and the damping coefficient
$\beta(t,x)$ satisfy the following hypothesis:
_(H): there exist two constants $\beta_{*}<0$ and $T_{*}>0$ such that_

$\displaystyle\beta_{*}\leq\beta(t,x)\leq 0,$ (1.2)
$\displaystyle|\partial_{t}\beta(t,x)|,|\partial_{x}\beta(t,x)|\leq\epsilon,$
(1.3) $\displaystyle\beta(t+T_{*},x)=\beta(t,x),$ (1.4)

_where the small constant $\epsilon$ will be determined later._

Obviously, (1.2)-(1.4) hold if $\beta(t,x)$ is a negative constant.
It should be noted that: the third author Zhang [17] studied the same problem
with $\beta(t,x)=\beta(t)$. However, [17] needs the smallness of the friction
coefficient $\beta(t)$ and its integral needs to be zero in one cycle, i.e.

$\displaystyle\int_{t}^{t+T_{*}}\beta(s)ds=0,$

which can not be satisfied for general constants. We remove these two
restrictions and include the constant friction coefficient case in this paper.
The main idea used in this paper is that we select a special linearized
iteration scheme (3.3) and (3.5). The advantage of this modified iteration
scheme is to make better use of the nonstrictly diagonally dominant structure
brought by the linear damping. While the iteration scheme used in [17] can
only utilize the smallness of the source term. [18] also studied this problem
with nonlinear damping.
The rest of this paper is organized as follows. In Section 2, we first
introduce the Riemann invariants of the homogeneous compressible Euler
equations and give the main results: Theorem 2.1–Theorem 2.4. In Section 3, we
use the linearized iteration method to prove Theorem 2.1. In Section 4, we use
the inductive method to prove Theorem 2.2. In Section 5 and Section 6, we
consider the higher regularity and stability of the time-periodic solution.

##  2 Prelimilaries and Main Results

In this section, we make some transformations on equations (1.1) and give the
main results of this paper.
Firstly, it is easy to check that the two eigenvalues of system (1.1) are

$\lambda_{1}=u-c,\quad\lambda_{2}=u+c.$

Then there holds

$\displaystyle\lambda_{1}(\underline{\rho},0)<0<\lambda_{2}(\underline{\rho},0)$

and

$\displaystyle\lambda_{1}(\rho,u)<0<\lambda_{2}(\rho,u),\quad(\rho,u)\in\Im$
(2.1)

for any positive constant $\underline{\rho}>0$ and a small neighborhood $\Im$
of $(\underline{\rho},0)$.
Using the Riemann invariants $m$ and $n$ defined by

$m=\frac{1}{2}(u-\frac{2}{\gamma-1}c),\quad
n=\frac{1}{2}(u+\frac{2}{\gamma-1}c),$ (2.2)

the system (1.1) becomes the following diagonal form

$\displaystyle\left\\{\begin{aligned}
m_{t}+\lambda_{1}(m,n)m_{x}=\frac{\beta}{2}(m+n),\\\
n_{t}+\lambda_{2}(m,n)n_{x}=\frac{\beta}{2}(m+n),\end{aligned}\right.$ (2.3)

where

$c=\sqrt{\frac{\partial
p}{\partial\rho}}=\sqrt{\gamma}\rho^{\frac{\gamma-1}{2}},$
$\lambda_{1}(m,n)=\frac{\gamma+1}{2}m+\frac{3-\gamma}{2}n,\quad\lambda_{2}(m,n)=\frac{3-\gamma}{2}m+\frac{\gamma+1}{2}n.$

We impose the system (1.1) or (2.3) with the following initial data and
boundary conditions

$\displaystyle t=0:\quad m(0,x)$
$\displaystyle=m_{0}(x),~{}~{}n(0,x)=n_{0}(x),$ (2.4) $\displaystyle x=0:\quad
n(t,0)$ $\displaystyle=n_{b}(t)+\kappa_{2}(m(t,0)-\underline{m}),$ (2.5)
$\displaystyle x=L:\quad m(t,L)$
$\displaystyle=m_{b}(t)+\kappa_{1}(n(t,L)-\underline{n}),$ (2.6)

where the constants $|\kappa_{1}|<1$, $|\kappa_{2}|<1$ and
$-\underline{m}=\underline{n}=\frac{\sqrt{\gamma}}{\gamma-1}\underline{\rho}^{\frac{\gamma-1}{2}}$.
Moreover, in (2.5)-(2.6), we assume $m_{b}(t),n_{b}(t)$ are two periodic
functions with the period $T_{*}>0$, namely,

$m_{b}(t+T_{*})=m_{b}(t),\quad n_{b}(t+T_{*})=n_{b}(t).$

(2.5)-(2.6) is a kind of dissipative boundary condition in the sense of [3].
Denote the perturbation variable

$\phi(t,x)=(\phi_{1}(t,x),\phi_{2}(t,x))^{\top}\mathop{=}\limits^{def.}(m(t,x)-\underline{m},n(t,x)-\underline{n})^{\top}$

and let $\Phi=(\underline{m},\underline{n})^{\top}$, then equations (2.3) can
be rewritten as

$\displaystyle\left\\{\begin{aligned}
\partial_{t}\phi_{1}+\lambda_{1}(\phi+\Phi)\partial_{x}\phi_{1}=\frac{\beta}{2}(\phi_{1}+\phi_{2}),\\\
\partial_{t}\phi_{2}+\lambda_{2}(\phi+\Phi)\partial_{x}\phi_{2}=\frac{\beta}{2}(\phi_{1}+\phi_{2}),\end{aligned}\right.$
(2.7)

where
$\lambda_{1}(\Phi)=-\underline{c}=-\sqrt{\gamma}\underline{\rho}^{\frac{\gamma-1}{2}},\lambda_{2}(\Phi)=\underline{c}=\sqrt{\gamma}\underline{\rho}^{\frac{\gamma-1}{2}}.$
Correspondingly, the initial data and boundary conditions (2.4)-(2.6) become

$\displaystyle t=0:~{}~{}\phi(0,x)$
$\displaystyle=\phi_{0}(x)=(\phi_{1_{0}}(x),\phi_{2_{0}}(x))^{\top}$
$\displaystyle\hskip
36.98866pt=(m_{0}(x)-\underline{m},n_{0}(x)-\underline{n})^{\top},$ (2.8)
$\displaystyle x=0:~{}~{}\phi_{2}(t,0)$
$\displaystyle=\phi_{2_{b}}(t)+\kappa_{2}\phi_{1}(t,0),\quad t\geq 0,$ (2.9)
$\displaystyle x=L:~{}~{}\phi_{1}(t,L)$
$\displaystyle=\phi_{1_{b}}(t)+\kappa_{1}\phi_{2}(t,L),\quad t\geq 0,$ (2.10)

where
$\phi_{1_{b}}(t)=m_{b}(t)-\underline{m},\phi_{2_{b}}(t)=n_{b}(t)-\underline{n}$
satisfying

$\phi_{i_{b}}(t+T_{*})=\phi_{i_{b}}(t),\quad t>0,i=1,2.$

Obviously, we have the following two facts:
$\mathbf{1}:~{}$By (2.1), we get

$\displaystyle\lambda_{1}(\phi+\Phi)<0<\lambda_{2}(\phi+\Phi),\quad\forall\phi\in\Psi,$

where $\Psi$ is a small neighborhood of $O=(0,0)^{\top}$ corresponding to
$\Im$.
$\mathbf{2}:~{}$Define
$\nu_{i}(\phi+\Phi)=\lambda_{i}^{-1}(\phi+\Phi)(i=1,2)$, then there exists a
positive constant $A_{0}$, such that

$\displaystyle\mathop{\max}\limits_{i=1,2}\mathop{\sup}\limits_{\phi\in\Psi}|\nu_{i}(\phi+\Phi)|\leq
A_{0}.$ (2.11)

Next, we give the main results in the following theorems.

######  Theorem 2.1.

(Existence of time-periodic solutions) There exists a small enough constant
$\epsilon_{1}>0$ and a constant $C_{E}>0$ such that for any given
$0<\epsilon<\epsilon_{1}$, and any given $C^{1}$ smooth functions
$\phi_{i_{b}}(t)(i=1,2)$ satisfying

$\displaystyle\phi_{i_{b}}(t+T_{*})=\phi_{i_{b}}(t),\quad t>0,i=1,2,$ (2.12)
$\displaystyle\|\phi_{i_{b}}(t)\|_{C^{1}(\mathbb{R}_{+})}\leq\epsilon,$ (2.13)

there exists a $C^{1}$ smooth function $\phi_{0}(x)$ satisfying

$\displaystyle\|\phi_{0}\|_{C^{1}([0,L])}\leq C_{E}\epsilon,$ (2.14)

such that the initial-boundary value problem (2.7)-(2.10) admits a $C^{1}$
time-periodic solution $\phi=\phi^{(T_{*})}(t,x)$ on
$D=\\{(t,x)|t\in\mathbb{R}_{+},x\in[0,L]\\}$ which satisfies

$\displaystyle\phi^{(T_{*})}(t+T_{*},x)$
$\displaystyle=\phi^{(T_{*})}(t,x),\quad\forall(t,x)\in D$ (2.15)

and

$\displaystyle\|\phi^{(T_{*})}\|_{C^{1}(D)}$ $\displaystyle\leq
C_{E}\epsilon.$ (2.16)

######  Theorem 2.2.

(Stability of time-periodic solutions) There exists a small constant
$\epsilon_{2}\in(0,\epsilon_{1})$ and a constant $C_{S}>0$, such that for any
given $\epsilon\in(0,\epsilon_{2})$ and any given $C^{1}$ smooth functions
$\phi_{0}=\phi_{0}(x)$ and $\phi_{i_{b}}(t)(i=1,2)$ satisfying (2.14) and
(2.12)-(2.13) with certain compatibilities, the initial-boundary value problem
(2.7)-(2.10) has a unique global $C^{1}$ classical solution $\phi=\phi(t,x)$
on $D=\\{(t,x)|t\in\mathbb{R}_{+},x\in[0,L]\\}$ satisfying

$\displaystyle\|\phi(t,\cdot)-\phi^{(T_{*})}(t,\cdot)\|_{C^{0}}\leq
C_{S}\epsilon\xi^{[t/T_{0}]},\quad\forall t\geq 0,$ (2.17)

where $\phi^{(T_{*})}$, depending on $\phi_{i_{b}}(t)(i=1,2)$, is the time-
periodic solution given through Theorem 2.1, $\xi\in(0,1)$ is a constant and
$T_{0}=\mathop{\max}\limits_{i=1,2}\mathop{\sup}\limits_{\phi\in\Phi}\frac{L}{|\lambda_{i}(\phi+\underline{\phi})|}$.

The uniqueness of the time-periodic solution is a direct consequence from
Theorem 2.2 by taking $t\rightarrow+\infty$.

######  Corollary 2.1.

(Uniqueness of the time-periodic solution) There exists a constant
$\epsilon_{3}\in(0,\epsilon_{2})$, such that for any given
$\epsilon\in(0,\epsilon_{3})$ and any given $C^{1}$ smooth functions
$\phi_{i_{b}}(t)(i=1,2)$ satisfying (2.12)-(2.13), the corresponding time-
periodic solution $\phi=\phi^{(T_{*})}(t,x)$ obtained in Theorem 2.1 is
unique.

######  Theorem 2.3.

(Regularity of the time-periodic solution) If $\phi_{i_{b}}(t)(i=1,2)$ satisfy
(2.12)-(2.13) and possess further $W^{2,\infty}$ regularity with

$\displaystyle\mathop{\max}\limits_{i=1,2}\|m_{i_{b}}^{\prime\prime}(t)\|_{L^{\infty}}\leq
M_{2}<+\infty,$ (2.18)

then there exist constants $C_{R}>0$ and $\epsilon_{4}\in(0,\epsilon_{1})$,
such that for any given $\epsilon\in(0,\epsilon_{4})$, the time-periodic
solution $\phi=\phi^{(T_{*})}(t,x)$ provided by Theorem 2.1 is also a
$W^{2,\infty}$ function with

$\displaystyle\max\\{\|\partial_{t}^{2}\phi^{(T_{*})}\|_{L^{\infty}},\|\partial_{t}\partial_{x}\phi^{(T_{*})}\|_{L^{\infty}},\|\partial_{x}^{2}\phi^{(T_{*})}\|_{L^{\infty}}\\}\leq(1+A_{0})^{2}C_{R}<+\infty,$
(2.19)

where $A_{0}$ is defined in (2.11).

######  Theorem 2.4.

(Stabilization around the time-periodic solution) Assume that (2.18) holds,
then there exist constants $C_{S}^{*}>0$ and
$\epsilon_{5}\in(0,\min\\{\epsilon_{2},\epsilon_{4}\\})$, such that for any
given $\epsilon\in(0,\epsilon_{5})$, we have not only the $C^{0}$ convergence
result (2.17) in Theorem 2.2, but also the $C^{1}$ exponential convergence as

$\displaystyle\max\\{\|\partial_{t}\phi(t,\cdot)-\partial_{t}\phi^{(T_{*})}(t,\cdot)\|_{C^{0}},\|\partial_{x}\phi(t,\cdot)-\partial_{x}\phi^{(T_{*})}(t,\cdot)\|_{C^{0}}\\}$
$\displaystyle\leq A_{0}C_{S}^{*}\epsilon\xi^{[t/T_{0}]},$
$\displaystyle\forall t\geq 0.$ (2.20)

##  3 Existence of Time-periodic Solutions

In this section, we give the proof of Theorem 2.1 by applying the linearized
iteration method.
Firstly, multiplying $\nu_{i}(\phi+\Phi)=\lambda_{i}^{-1}(\phi+\Phi)(i=1,2)$
on both sides of the $i$-th equation of (2.7) for $i=1,2$ and swapping the
positions of $t$ and $x$, we have

$\displaystyle\partial_{x}\phi_{1}+\nu_{1}(\phi+\Phi)\partial_{t}\phi_{1}=$
$\displaystyle\frac{\beta}{2}\nu_{1}(\Phi)\phi_{1}+\frac{\beta}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\Phi)\Big{)}\phi_{1}$
$\displaystyle+\frac{\beta}{2}\nu_{1}(\phi+\Phi)\phi_{2},$ (3.1)
$\displaystyle\partial_{x}\phi_{2}+\nu_{2}(\phi+\Phi)\partial_{t}\phi_{2}=$
$\displaystyle\frac{\beta}{2}\nu_{2}(\Phi)\phi_{2}+\frac{\beta}{2}\Big{(}\nu_{2}(\phi+\Phi)-\nu_{2}(\Phi)\Big{)}\phi_{2}$
$\displaystyle+\frac{\beta}{2}\nu_{2}(\phi+\Phi)\phi_{1}.$ (3.2)

Then using (3.1)-(3.2) and (2.9)-(2.10), we establish the following
"initial"-value problem of linearized system

$\displaystyle\partial_{x}\phi_{1}^{(l)}+\nu_{1}(\phi^{(l-1)}+\Phi)\partial_{t}\phi_{1}^{(l)}=$
$\displaystyle\frac{\beta}{2}\nu_{1}(\Phi)\phi_{1}^{(l)}+\frac{\beta}{2}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}\phi_{1}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}\nu_{1}(\phi^{(l-1)}+\Phi)\phi_{2}^{(l-1)},$
(3.3) $\displaystyle x=L:~{}~{}~{}~{}\phi_{1}^{(l)}(t,L)$
$\displaystyle=\phi_{1_{br}}(t)+\kappa_{1}\phi_{2_{r}}^{(l-1)}(t,L)$ (3.4)

and

$\displaystyle\partial_{x}\phi_{2}^{(l)}+\nu_{2}(\phi^{(l-1)}+\Phi)\partial_{t}\phi_{2}^{(l)}=$
$\displaystyle\frac{\beta}{2}\nu_{2}(\Phi)\phi_{2}^{(l)}+\frac{\beta}{2}\Big{(}\nu_{2}(\phi^{(l-1)}+\Phi)-\nu_{2}(\Phi)\Big{)}\phi_{2}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}\nu_{2}(\phi^{(l-1)}+\Phi)\phi_{1}^{(l-1)},$
(3.5) $\displaystyle x=0:~{}~{}~{}~{}\phi_{2}^{(l)}(t,0)$
$\displaystyle=\phi_{2_{bl}}(t)+\kappa_{2}\phi_{1_{l}}^{(l-1)}(t,0),$ (3.6)

where

$\displaystyle\phi_{1_{br}}(t)=\left\\{\begin{aligned} \phi_{1_{b}}(t),\quad
t\geq 0,\\\ \phi_{1_{b^{\prime}}}(t),\quad
t<0,\end{aligned}\right.\quad\phi_{2_{bl}}(t)=\left\\{\begin{aligned}
\phi_{2_{b}}(t),\quad t\geq 0,\\\ \phi_{2_{b^{\prime}}}(t),\quad
t<0\end{aligned}\right.$

are the time-periodic extensions of $\phi_{i_{b}}(t)(i=1,2)$.
Here we select a special linearized iteration scheme (3.3) and (3.5) to make
better use of the nonstrictly diagonally dominant structure brought by
damping.
For system (3.3)-(3.6), we start to iterate from

$\displaystyle\phi^{(0)}(t,x)=0$ (3.7)

and prove the following a priori estimates.

######  Proposition 3.1.

There exists a small enough constant $\epsilon_{1}>0$ and constants $M_{1}>0$,
$C_{1}>0$ and $\theta\in(0,1)$, such that for any given
$\epsilon\in(0,\epsilon_{1})$, the following estimates hold:

$\displaystyle\phi^{(l)}(t+T_{*},x)=\phi^{(l)}(t,x),\quad\forall(t,x)\in\mathbb{R}\times[0,L],l\in\mathbb{N}_{+},$
(3.8)
$\displaystyle\|\phi^{(l)}\|_{C^{1}}\leq(C_{1}+M_{1})\epsilon,\quad\forall
l\in\mathbb{N}_{+},$ (3.9)
$\displaystyle\|\phi^{(l)}-\phi^{(l-1)}\|_{C^{0}}\leq
M_{1}\epsilon\theta^{l},\quad\forall l\in\mathbb{N}_{+},$ (3.10)
$\displaystyle\mathop{\max}\limits_{i=1,2}\\{\varpi(\delta|\partial_{t}\phi_{i}^{(l)})+\varpi(\delta|\partial_{x}\phi_{i}^{(l)})\\}\leq(\frac{1}{3}+\frac{1}{2}A_{0})\Omega(\delta),\quad\forall
l\in\mathbb{N}_{+},$ (3.11)

where

$\displaystyle\|\phi^{(l)}\|_{C^{1}(D)}$
$\displaystyle\mathop{=}\limits^{def.}\mathop{\max}\limits_{i=1,2}\\{\|\phi_{i}^{(l)}\|_{C^{0}(D)},\|\partial_{t}\phi_{i}^{(l)}\|_{C^{0}(D)},\|\partial_{x}\phi_{i}^{(l)}\|_{C^{0}(D)}\\},$
$\displaystyle\varpi(\delta|h)$
$\displaystyle=\mathop{\sup}\limits_{\begin{subarray}{c}|t_{1}-t_{2}|\leq\delta\\\
|x_{1}-x_{2}|\leq\delta\end{subarray}}|h(t_{1},x_{1})-h(t_{2},x_{2})|,$

and $\Omega(\delta)$ is a continuous function of $\delta\in(0,1)$ which is
independent of $l$ and satisfies

$\mathop{\lim}\limits_{\delta\rightarrow 0^{+}}\Omega(\delta)=0.$

###### Proof.

We prove the a priori estimates (3.8)-(3.11) inductively, i.e., for each
$l\in\mathbb{N}_{+}$, we show

$\displaystyle\phi_{i}^{(l)}(t+T_{*},x)=\phi_{i}^{(l)}(t,x),\quad\forall(t,x)\in\mathbb{R}\times[0,L],\forall
i=1,2,$ (3.12)
$\displaystyle\mathop{\max}\limits_{i=1,2}\\{\|\phi_{i}^{(l)}\|_{C^{0}},\|\partial_{t}\phi_{i}^{(l)}\|_{C^{0}}\\}\leq
M_{1}\epsilon,$ (3.13)
$\displaystyle\mathop{\max}\limits_{i=1,2}\\{\|\partial_{x}\phi_{i}^{(l)}\|_{C^{0}}\\}\leq
C_{1}\epsilon,$ (3.14)
$\displaystyle\mathop{\max}\limits_{i=1,2}\|\phi_{i}^{(l)}-\phi_{i}^{(l-1)}\|_{C^{0}}\leq
M_{1}\epsilon\theta^{l},$ (3.15)
$\displaystyle\mathop{\max}\limits_{i=1,2}\varpi(\delta|\partial_{t}\phi_{i}^{(l)}(\cdot,x))\leq\frac{1}{8[A_{0}+1]}\Omega(\delta),\quad\forall
x\in[0,L]$ (3.16)

and

$\displaystyle\mathop{\max}\limits_{i=1,2}\\{\varpi(\delta|\partial_{t}\phi_{i}^{(l)})+\varpi(\delta|\partial_{x}\phi_{i}^{(l)})\\}\leq(\frac{1}{3}+\frac{1}{2}A_{0})\Omega(\delta)$
(3.17)

under the following hypothesis

$\displaystyle\phi_{i}^{(l-1)}(t+T_{*},x)=\phi_{i}^{(l-1)}(t,x),\quad\forall(t,x)\in\mathbb{R}\times[0,L],\forall
i=1,2,$ (3.18)
$\displaystyle\mathop{\max}\limits_{i=1,2}\\{\|\phi_{i}^{(l-1)}\|_{C^{0}},\|\partial_{t}\phi_{i}^{(l-1)}\|_{C^{0}}\\}\leq
M_{1}\epsilon,$ (3.19)
$\displaystyle\mathop{\max}\limits_{i=1,2}\\{\|\partial_{x}\phi_{i}^{(l-1)}\|_{C^{0}}\\}\leq
C_{1}\epsilon,$ (3.20)
$\displaystyle\mathop{\max}\limits_{i=1,2}\|\phi_{i}^{(l-1)}-\phi_{i}^{(l-2)}\|_{C^{0}}\leq
M_{1}\epsilon\theta^{l},\quad\forall l\geq 2,$ (3.21)
$\displaystyle\mathop{\max}\limits_{i=1,2}\varpi(\delta|\partial_{t}\phi_{i}^{(l-1)}(\cdot,x))\leq\frac{1}{8[A_{0}+1]}\Omega(\delta),\quad\forall
x\in[0,L]$ (3.22)

and

$\displaystyle\mathop{\max}\limits_{i=1,2}\\{\varpi(\delta|\partial_{t}\phi_{i}^{(l-1)})+\varpi(\delta|\partial_{x}\phi_{i}^{(l-1)})\\}\leq(\frac{1}{3}+\frac{1}{2}A_{0})\Omega(\delta),$
(3.23)

where $[A_{0}+1]$ represents the integer part of $A_{0}+1$ and

$\varpi(\delta|h(\cdot,x))=\mathop{\max}\limits_{|t_{1}-t_{2}|\leq\delta}|h(t_{1},x)-h(t_{2},x)|.$

In the remaining parts of this section, we will use several steps to check the
above estimates (3.12)-(3.17) one by one.
$\mathbf{Step~{}1}:~{}$ Transformations of the principal equations.
Define the characteristic curve $t=t_{i}^{(l)}(x;t_{0},x_{0})(i=1,2)$ as the
following form:

$\displaystyle\left\\{\begin{aligned}
&\frac{dt_{i}^{(l)}}{dx}(x;t_{0},x_{0})=\nu_{i}(\phi^{(l-1)}+\Phi)(t_{i}^{(l)}(x;t_{0},x_{0}),x),\\\
&t_{i}^{(l)}(x_{0};t_{0},x_{0})=t_{0}.\end{aligned}\right.$ (3.24)

Denote

$F_{1}(t,x)=e^{\int_{x}^{L}\frac{\beta(t,s)}{2}\nu_{1}(\Phi)ds},~{}~{}F_{2}(t,x)=e^{-\int_{0}^{x}\frac{\beta(t,s)}{2}\nu_{2}(\Phi)ds}.$

Noticing $\beta(t,x)\leq 0,\nu_{1}(\Phi)<0,\nu_{2}(\Phi)>0$ and $x\in[0,L]$,
we have

$\displaystyle F_{1}(t,x),F_{2}(t,x)\geq 1,$ (3.25)
$\displaystyle\frac{\partial F_{1}(t,x)}{\partial
x}=-\frac{\beta(t,x)}{2}\nu_{1}(\Phi)F_{1}(t,x)<0,$ (3.26)
$\displaystyle\frac{\partial F_{1}(t,x)}{\partial
t}=\int_{x}^{L}\frac{\partial_{t}\beta(t,s)}{2}\nu_{1}(\Phi)dsF_{1}(t,x),$
$\displaystyle\frac{\partial F_{2}(t,x)}{\partial
x}=-\frac{\beta(t,x)}{2}\nu_{2}(\Phi)F_{2}(t,x)>0,$ (3.27)
$\displaystyle\frac{\partial F_{2}(t,x)}{\partial
t}=-\int_{0}^{x}\frac{\partial_{t}\beta(t,s)}{2}\nu_{2}(\Phi)dsF_{2}(t,x).$

Furthermore, by (1.2) and (2.11), we get

$\displaystyle 1\leq F_{1}(t,x)\leq
e^{-\frac{\beta_{*}}{2}A_{0}L}\mathop{=}\limits^{def.}M_{0}>1,$ (3.28)
$\displaystyle 1\leq F_{2}(t,x)\leq M_{0}.$ (3.29)

Let $M_{1}=\frac{100}{1-\kappa}$ with
$\kappa=\max\\{|\kappa_{1}|,|\kappa_{2}|\\}<1$,then

$\displaystyle M_{1}\geq|\kappa_{1}|M_{1}+100,\quad
M_{1}\geq|\kappa_{2}|M_{1}+100.$ (3.30)

Now, we turn problem (3.3)-(3.6) into the system of $F_{i}(t,x)\phi_{i}^{(l)}$
as follows:

$\displaystyle\partial_{x}\Big{(}F_{1}\phi_{1}^{(l)}\Big{)}+\nu_{1}(\phi^{(l-1)}+\Phi)\partial_{t}\Big{(}F_{1}\phi_{1}^{(l)}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{\beta}{2}F_{1}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})$
$\displaystyle+\frac{\beta}{2}F_{1}\nu_{1}(\Phi)\phi_{2}^{(l-1)}+\nu_{1}(\phi^{(l-1)}+\Phi)F_{1}\phi_{1}^{(l)}\int_{x}^{L}\frac{\partial_{t}\beta(t,s)}{2}\nu_{1}(\Phi)ds,$
(3.31) $\displaystyle
x=L:~{}~{}~{}~{}F_{1}(t,L)\phi_{1}^{(l)}(t,L)=\phi_{1}^{(l)}(t,L)=\phi_{1_{br}}(t)+\kappa_{1}\phi_{2_{r}}^{(l-1)}(t,L),$
(3.32)
$\displaystyle\partial_{x}\Big{(}F_{2}\phi_{2}^{(l)}\Big{)}+\nu_{2}(\phi^{(l-1)}+\Phi)\partial_{t}\Big{(}F_{2}\phi_{2}^{(l)}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{\beta}{2}F_{2}\Big{(}\nu_{2}(\phi^{(l-1)}+\Phi)-\nu_{2}(\Phi)\Big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})$
$\displaystyle+\frac{\beta}{2}F_{2}\nu_{2}(\Phi)\phi_{1}^{(l-1)}-\nu_{2}(\phi^{(l-1)}+\Phi)F_{2}\phi_{2}^{(l)}\int_{0}^{x}\frac{\partial_{t}\beta(t,s)}{2}\nu_{2}(\Phi)ds,$
(3.33) $\displaystyle
x=0:~{}~{}~{}~{}F_{2}(t,0)\phi_{2}^{(l)}(t,0)=\phi_{2}^{(l)}(t,0)=\phi_{2_{bl}}(t)+\kappa_{2}\phi_{1_{l}}^{(l-1)}(t,0).$
(3.34)

By (1.4), (2.12) and (3.18), it is easy to know that if
$F_{i}(t,x)\phi_{i}^{(l)}(t,x)(i=1,2)$ solves problem (3.31)-(3.34), so does
$F_{i}(t+T_{*},x)\phi_{i}^{(l)}(t+T_{*},x)(i=1,2).$ Then we get
$F_{i}(t+T_{*},x)\phi_{i}^{(l)}(t+T_{*},x)=F_{i}(t,x)\phi_{i}^{(l)}(t,x)$ by
the uniqueness of this linear system. Thus (3.12) is proved.
$\mathbf{Step~{}2}:~{}$ The proof of (3.13) and (3.14).
Next, we prove the $C^{0}$ estimate for $\phi_{i}^{(l)}(i=1,2)$. By the aid of
conditions (2.13), (3.4), (3.6), (3.30), we get

$\displaystyle\|\phi_{1}^{(l)}(\cdot,L)\|_{C^{0}}\leq\epsilon+|\kappa_{1}|M_{1}\epsilon\leq
M_{1}\epsilon-99\epsilon,$ (3.35)
$\displaystyle\|\phi_{2}^{(l)}(\cdot,0)\|_{C^{0}}\leq
M_{1}\epsilon-99\epsilon.$ (3.36)

Then we integrate (3.31) along the $1$st characteristic curve
$t=t_{1}^{(l)}(y;t,x)$ from $L$ to $x$ to get

$\displaystyle
F_{1}(t,x)\phi_{1}^{(l)}(t,x)-F_{1}(t,L)\phi_{1}^{(l)}(t_{1}^{(l)}(L;t,x),L)$
$\displaystyle=$
$\displaystyle\int^{x}_{L}\Big{(}\frac{\beta(\tau,y)}{2}F_{1}(\tau,y)\big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})$
$\displaystyle+\nu_{1}(\phi^{(l-1)}+\Phi)F_{1}(\tau,y)\phi_{1}^{(l)}\int_{y}^{L}\frac{\partial_{t}\beta(\tau,s)}{2}\nu_{1}(\Phi)ds\Big{)}\big{|}_{\tau=t_{1}^{(l)}(y;t,x)}dy$
$\displaystyle+\int^{x}_{L}-\big{(}\frac{\partial}{\partial
y}F_{1}(t_{1}^{(l)}(y;t,x),y)\big{)}\phi_{2}^{(l-1)}(t_{1}^{(l)}(y;t,x),y)dy,$

where $F_{1}(t,L)=1$.
Using (1.2)-(1.3), (2.11), (3.19) and (3.35), we get

$\displaystyle\|\phi_{1}^{(l)}\|_{C^{0}}$
$\displaystyle\leq\frac{M_{1}\epsilon-99\epsilon}{F_{1}(t,x)}+C\epsilon^{2}+\frac{F_{1}(t,x)-1}{F_{1}(t,x)}M_{1}\epsilon$
$\displaystyle\leq\frac{M_{1}\epsilon-99\epsilon}{F_{1}(t,x)}+\frac{\epsilon}{F_{1}(t,x)}+\frac{F_{1}(t,x)-1}{F_{1}(t,x)}M_{1}\epsilon$
$\displaystyle=M_{1}\epsilon-\frac{98\epsilon}{F_{1}(t,x)}<M_{1}\epsilon,$
(3.37)

where in the above second inequality, we used
$C\epsilon^{2}<\frac{\epsilon}{F_{1}(t,x)}$ for a suitably small $\epsilon$.
Unless specified, in this paper $C$ denotes a generic constant.
Similarly, we can get

$\displaystyle\|\phi_{2}^{(l)}\|_{C^{0}}<M_{1}\epsilon.$ (3.38)

Now, we consider the temporal derivative estimates.
Let

$z_{i}^{(l)}=\partial_{t}\phi_{i}^{(l)},\quad i=1,2,\forall l\in\mathbb{N}.$

Differentiating equations (3.3), (3.5) with respect to $t$, we get

$\displaystyle\partial_{x}z_{1}^{(l)}+\nu_{1}(\phi^{(l-1)}+\Phi)\partial_{t}z_{1}^{(l)}$
$\displaystyle=$
$\displaystyle\frac{\beta}{2}\nu_{1}(\Phi)z_{1}^{(l)}+\frac{\partial_{t}\beta}{2}\nu_{1}(\Phi)\phi_{1}^{(l)}+\frac{\beta}{2}\nu_{1}(\Phi)z_{2}^{(l-1)}+\frac{\partial_{t}\beta}{2}\nu_{1}(\Phi)\phi_{2}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(z_{1}^{(l-1)}+z_{2}^{(l-1)})$
$\displaystyle+\frac{\partial_{t}\beta}{2}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})$
$\displaystyle+\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{1}(\phi^{(l-1)}+\Phi)}{\partial\phi_{j}}(z_{j}^{(l-1)}\phi_{1}^{(l-1)}+z_{j}^{(l-1)}\phi_{2}^{(l-1)})$
$\displaystyle-\sum_{j=1}^{2}\frac{\partial\nu_{1}(\phi^{(l-1)}+\Phi)}{\partial\phi_{j}}z_{j}^{(l-1)}z_{1}^{(l)},$
(3.39)

and

$\displaystyle\partial_{x}z_{2}^{(l)}+\nu_{2}(\phi^{(l-1)}+\Phi)\partial_{t}z_{2}^{(l)}$
$\displaystyle=$
$\displaystyle\frac{\beta}{2}\nu_{2}(\Phi)z_{2}^{(l)}+\frac{\partial_{t}\beta}{2}\nu_{2}(\Phi)\phi_{2}^{(l)}+\frac{\beta}{2}\nu_{2}(\Phi)z_{1}^{(l-1)}+\frac{\partial_{t}\beta}{2}\nu_{2}(\Phi)\phi_{1}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}\Big{(}\nu_{2}(\phi^{(l-1)}+\Phi)-\nu_{2}(\Phi)\Big{)}(z_{1}^{(l-1)}+z_{2}^{(l-1)})$
$\displaystyle+\frac{\partial_{t}\beta}{2}\Big{(}\nu_{2}(\phi^{(l-1)}+\Phi)-\nu_{2}(\Phi)\Big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})$
$\displaystyle+\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{2}(\phi^{(l-1)}+\Phi)}{\partial\phi_{j}}(z_{j}^{(l-1)}\phi_{1}^{(l-1)}+z_{j}^{(l-1)}\phi_{2}^{(l-1)})$
$\displaystyle-\sum_{j=1}^{2}\frac{\partial\nu_{2}(\phi^{(l-1)}+\Phi)}{\partial\phi_{j}}z_{j}^{(l-1)}z_{2}^{(l)}.$
(3.40)

Correspondingly, at the boundary $x=0$ and $x=L$, we have

$\displaystyle
x=L:~{}~{}~{}~{}z_{1}^{(l)}(t,L)=\phi^{\prime}_{1_{br}}(t)+\kappa_{1}z_{2_{r}}^{(l-1)}(t,L),\quad
t\in\mathbb{R},$ (3.41)

and

$\displaystyle
x=0:~{}~{}~{}~{}z_{2}^{(l)}(t,0)=\phi^{\prime}_{2_{bl}}(t)+\kappa_{2}z_{1_{l}}^{(l-1)}(t,0),\quad
t\in\mathbb{R}.$ (3.42)

By (2.13) and (3.19), we get

$\displaystyle\|z_{1}^{(l)}(\cdot,L)\|_{C^{0}}\leq\epsilon+|\kappa_{1}|M_{1}\epsilon\leq
M_{1}\epsilon-99\epsilon,$ (3.43)
$\displaystyle\|z_{2}^{(l)}(\cdot,0)\|_{C^{0}}\leq\epsilon+|\kappa_{2}|M_{1}\epsilon\leq
M_{1}\epsilon-99\epsilon.$ (3.44)

In order to overcome the complexity of linearized equations caused by the
quasilinear nature of the original system, we introduce the method of
"estimation by twice integration". We first get a rough estimate of
$z_{1}^{(l)}$ as follows.
Multiplying $\mathrm{sgn}(z_{1}^{(l)})$ on both sides of (3.39), we get

$\displaystyle\partial_{x}|z_{1}^{(l)}|+\nu_{1}(\phi^{(l-1)}+\Phi)\partial_{t}|z_{1}^{(l)}|$
$\displaystyle=$
$\displaystyle\Big{(}\frac{\beta}{2}\nu_{1}(\Phi)-\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}z_{j}^{(l-1)}\Big{)}|z_{1}^{(l)}|+\frac{\partial_{t}\beta}{2}\mathrm{sgn}(z_{1}^{(l)})\nu_{1}(\Phi)\phi_{1}^{(l)}$
$\displaystyle+\frac{\beta}{2}\mathrm{sgn}(z_{1}^{(l)})\nu_{1}(\Phi)z_{2}^{(l-1)}+\frac{\partial_{t}\beta}{2}\mathrm{sgn}(z_{1}^{(l)})\nu_{1}(\Phi)\phi_{2}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}\mathrm{sgn}(z_{1}^{(l)})\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(z_{1}^{(l-1)}+z_{2}^{(l-1)})$
$\displaystyle+\frac{\partial_{t}\beta}{2}\mathrm{sgn}(z_{1}^{(l)})\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})$
$\displaystyle+\frac{\beta}{2}\mathrm{sgn}(z_{1}^{(l)})\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(z_{j}^{(l-1)}\phi_{1}^{(l-1)}+z_{j}^{(l-1)}\phi_{2}^{(l-1)}).$

Using the sign of $\beta(t,x)$ in (1.2) and noticing (2.11), (3.19), it is
easy to see

$\frac{\beta}{2}\nu_{1}(\Phi)-\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}z_{j}^{(l-1)}>0.$

Then integrating the result along the $1$st characteristic curve
$t=t_{1}^{(l)}(y;t,x)$ from $L$ to $x$ and using (1.2)-(1.3), (2.11), (3.19)
and (3.43), one has

$\displaystyle|z_{1}^{(l)}(t,x)|\leq$
$\displaystyle|z_{1}^{(l)}(t_{1}^{(l)}(L;t,x),L)|+(\frac{|\beta|}{2}|\nu_{1}(\Phi)|+|\partial_{t}\beta||\nu_{1}(\Phi)|)|\int_{L}^{x}M_{1}\epsilon
dy|+C\epsilon^{2}$ $\displaystyle\leq$ $\displaystyle
C_{0}\epsilon,\quad\forall(t,x)\in\mathbb{R}\times[0,L],$ (3.45)

where $C_{0}>0$ is a constant independent of $l$.
Next we transform equation (3.39) into the following equation of
$F_{1}z_{1}^{(l)}$

$\displaystyle\partial_{x}(F_{1}z_{1}^{(l)})+\nu_{1}(\phi^{(l-1)}+\Phi)\partial_{t}(F_{1}z_{1}^{(l)})$
$\displaystyle=$
$\displaystyle\frac{\partial_{t}\beta}{2}F_{1}\nu_{1}(\Phi)\phi_{1}^{(l)}+\frac{\beta}{2}F_{1}\nu_{1}(\Phi)z_{2}^{(l-1)}+\frac{\partial_{t}\beta}{2}F_{1}\nu_{1}(\Phi)\phi_{2}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}F_{1}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(z_{1}^{(l-1)}+z_{2}^{(l-1)})$
$\displaystyle+\frac{\partial_{t}\beta}{2}F_{1}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})$
$\displaystyle+\frac{\beta}{2}F_{1}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(z_{j}^{(l-1)}\phi_{1}^{(l-1)}+z_{j}^{(l-1)}\phi_{2}^{(l-1)})-F_{1}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}z_{j}^{(l-1)}z_{1}^{(l)}$
$\displaystyle+\nu_{1}(\phi^{(l-1)}+\Phi)F_{1}z_{1}^{(l)}\int_{x}^{L}\frac{\partial_{t}\beta(t,s)}{2}\nu_{1}(\Phi)ds,$
(3.46)

then integrate it along the $1$st characteristic curve $t=t_{1}^{(l)}(y;t,x)$
from $L$ to $x$ to get

$\displaystyle
F_{1}(t,x)z_{1}^{(l)}(t,x)-F_{1}(t_{1}^{(l)}(L;t,x),L)z_{1}^{(l)}(t_{1}^{(l)}(L;t,x),L)$
$\displaystyle=$ $\displaystyle\int_{L}^{x}-(\frac{\partial}{\partial
y}F_{1}(t_{1}^{(l)}(y;t,x),y))z_{2}^{(l-1)}(t_{1}^{(l)}(y;t,x),y)dy$
$\displaystyle+\int_{L}^{x}\Big{(}\frac{\partial_{t}\beta}{2}F_{1}\nu_{1}(\Phi)\phi_{1}^{(l)}+\frac{\partial_{t}\beta}{2}F_{1}\nu_{1}(\Phi)\phi_{2}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}F_{1}\big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\big{)}(z_{1}^{(l-1)}+z_{2}^{(l-1)})$
$\displaystyle+\frac{\partial_{t}\beta}{2}F_{1}\big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})$
$\displaystyle+\frac{\beta}{2}F_{1}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(z_{j}^{(l-1)}\phi_{1}^{(l-1)}+z_{j}^{(l-1)}\phi_{2}^{(l-1)})-F_{1}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}z_{j}^{(l-1)}z_{1}^{(l)}$
$\displaystyle+\nu_{1}(\phi^{(l-1)}+\Phi)F_{1}z_{1}^{(l)}\int_{y}^{L}\frac{\partial_{t}\beta(\cdot,s)}{2}\nu_{1}(\Phi)ds\Big{)}(t_{1}^{(l)}(y;t,x),y)dy.$

By (1.2)-(1.3), (2.11), (3.19), (3.43) and the rough estimate (3.45), we get

$\displaystyle|z_{1}^{(l)}(t,x)|$
$\displaystyle\leq\frac{1}{F_{1}(t,x)}(M_{1}\epsilon-99\epsilon)+\frac{F_{1}(t,x)-1}{F_{1}(t,x)}M_{1}\epsilon+C\epsilon^{2}$
$\displaystyle\leq M_{1}\epsilon-\frac{99}{F_{1}(t,x)}\epsilon+C\epsilon^{2}$
$\displaystyle<M_{1}\epsilon.$ (3.47)

Similarly, we have

$\displaystyle|z_{2}^{(l)}(t,x)|<M_{1}\epsilon.$ (3.48)

By applying the equations (3.3), (3.5) and noting (1.2), (2.11), (3.19),
(3.37)-(3.38) and (3.47)-(3.48), we gain

$\displaystyle\|\partial_{x}\phi_{i}^{(l)}\|_{C^{0}}$ $\displaystyle\leq
A_{0}M_{1}\epsilon+\beta_{*}A_{0}M_{1}\epsilon+C\epsilon^{2}$
$\displaystyle\leq C_{1}\epsilon,$ (3.49)

where we choose the constant $C_{1}>A_{0}M_{1}+\beta_{*}A_{0}M_{1}$, which is
independent of $l$.
$\mathbf{Step~{}3}:~{}$ $\\{\phi_{i}^{(l)}\\}(i=1,2)$ is a Cauchy sequence in
$C^{0}$.
By (3.7), we get (3.15) for $l=1$ from (3.37)-(3.38) directly. Next we prove
(3.15) for $l\geq 2$. Select $\theta<1$ satisfying

$\theta>|\kappa_{1}|,~{}~{}\theta>|\kappa_{2}|.$

At the boundary $x=0$ and $x=L$, by (3.21), we have

$\displaystyle\|\phi_{1}^{(l)}(t,L)-\phi_{1}^{(l-1)}(t,L)\|_{C^{0}}$
$\displaystyle\leq|\kappa_{1}|\|\phi_{2}^{(l-1)}(t,L)-\phi_{2}^{(l-2)}(t,L)\|_{C^{0}}$
$\displaystyle\leq|\kappa_{1}|M_{1}\epsilon\theta^{l-1},$ (3.50)
$\displaystyle\|\phi_{2}^{(l)}(t,0)-\phi_{2}^{(l-1)}(t,0)\|_{C^{0}}$
$\displaystyle\leq|\kappa_{2}|\|\phi_{1}^{(l-1)}(t,0)-\phi_{1}^{(l-2)}(t,0)\|_{C^{0}}$
$\displaystyle\leq|\kappa_{2}|M_{1}\epsilon\theta^{l-1}.$ (3.51)

In the domain $D$, from (3.3), we get

$\displaystyle\partial_{x}(\phi_{1}^{(l)}-\phi_{1}^{(l-1)})+\nu_{1}(\phi^{(l-1)}+\Phi)\partial_{t}(\phi_{1}^{(l)}-\phi_{1}^{(l-1)})$
$\displaystyle=$
$\displaystyle\frac{\beta}{2}\nu_{1}(\Phi)(\phi_{1}^{(l)}-\phi_{1}^{(l-1)})-\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\phi^{(l-2)}+\Phi)\Big{)}\partial_{t}\phi_{1}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(\phi_{1}^{(l-1)}-\phi_{1}^{(l-2)})$
$\displaystyle+\frac{\beta}{2}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\phi^{(l-2)}+\Phi)\Big{)}\phi_{1}^{(l-2)}$
$\displaystyle+\frac{\beta}{2}\nu_{1}(\phi^{(l-1)}+\Phi)(\phi_{2}^{(l-1)}-\phi_{2}^{(l-2)})$
$\displaystyle+\frac{\beta}{2}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\phi^{(l-2)}+\Phi)\Big{)}\phi_{2}^{(l-2)},$

then we multiply $F_{1}(t,x)$ on both sides of the above equality to gain

$\displaystyle\partial_{x}\Big{(}F_{1}(\phi_{1}^{(l)}-\phi_{1}^{(l-1)})\Big{)}+\nu_{1}(\phi^{(l-1)}+\Phi)\partial_{t}\Big{(}F_{1}(\phi_{1}^{(l)}-\phi_{1}^{(l-1)})\Big{)}$
$\displaystyle=$ $\displaystyle-
F_{1}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\phi^{(l-2)}+\Phi)\Big{)}\partial_{t}\phi_{1}^{(l-1)}$
$\displaystyle+\nu_{1}(\phi^{(l-1)}+\Phi)F_{1}(\phi_{1}^{(l)}-\phi_{1}^{(l-1)})\int_{x}^{L}\frac{\partial_{t}\beta(t,s)}{2}\nu_{1}(\Phi)ds$
$\displaystyle+\frac{\beta}{2}F_{1}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(\phi_{1}^{(l-1)}-\phi_{1}^{(l-2)})$
$\displaystyle+\frac{\beta}{2}F_{1}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\phi^{(l-2)}+\Phi)\Big{)}\phi_{1}^{(l-2)}$
$\displaystyle+\frac{\beta}{2}F_{1}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}(\phi_{2}^{(l-1)}-\phi_{2}^{(l-2)})$
$\displaystyle+\frac{\beta}{2}F_{1}\nu_{1}(\Phi)(\phi_{2}^{(l-1)}-\phi_{2}^{(l-2)})$
$\displaystyle+\frac{\beta}{2}F_{1}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\phi^{(l-2)}+\Phi)\Big{)}\phi_{2}^{(l-2)},$

and integrate it along the $1$st characteristic curve $t=t_{1}^{(l)}(y;t,x)$
from $L$ to $x$ to get

$\displaystyle F_{1}(t,x)(\phi_{1}^{(l)}(t,x)-\phi_{1}^{(l-1)}(t,x))$
$\displaystyle=$ $\displaystyle
F_{1}(t,L)(\phi_{1}^{(l)}(t_{1}^{(l)}(L;t,x),L)-\phi_{1}^{(l-1)}(t_{1}^{(l)}(L;t,x),L))$
$\displaystyle+\int_{L}^{x}\Big{(}-F_{1}\big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\phi^{(l-2)}+\Phi)\big{)}\partial_{t}\phi_{1}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}F_{1}\big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\big{)}(\phi_{1}^{(l-1)}-\phi_{1}^{(l-2)})$
$\displaystyle+\frac{\beta}{2}F_{1}\big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\phi^{(l-2)}+\Phi)\big{)}\phi_{1}^{(l-2)}$
$\displaystyle+\frac{\beta}{2}F_{1}\big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\big{)}(\phi_{2}^{(l-1)}-\phi_{2}^{(l-2)})$
$\displaystyle+\frac{\beta}{2}F_{1}(\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\phi^{(l-2)}+\Phi))\phi_{2}^{(l-2)}$
$\displaystyle+\nu_{1}(\phi^{(l-1)}+\Phi)F_{1}(\phi_{1}^{(l)}-\phi_{1}^{(l-1)})\int_{y}^{L}\frac{\partial_{t}\beta(\cdot,s)}{2}\nu_{1}(\Phi)ds\Big{)}(t_{1}^{(l)}(y;t,x),y)dy$
$\displaystyle+\int_{L}^{x}-(\frac{\partial}{\partial
y}F_{1}(\cdot,y))(\phi_{2}^{(l-1)}-\phi_{2}^{(l-2)})(t_{1}^{(l)}(y;t,x),y)dy.$

By (3.19), (3.21) and (3.50), we have

$\displaystyle\|(\phi_{1}^{(l)}-\phi_{1}^{(l-1)})\|_{C^{0}}$
$\displaystyle\leq\frac{|\kappa_{1}|M_{1}\epsilon\theta^{l-1}}{F_{1}(t,x)}+\frac{F_{1}(t,x)-1}{F_{1}(t,x)}M_{1}\epsilon\theta^{l-1}+C\epsilon
M_{1}\epsilon\theta^{l-1}$ $\displaystyle\leq
M_{1}\epsilon\theta^{l-1}-\frac{1-|\kappa_{1}|}{F_{1}(t,x)}M_{1}\epsilon\theta^{l-1}+C\epsilon
M_{1}\epsilon\theta^{l-1}$ $\displaystyle\leq M_{1}\epsilon\theta^{l}.$ (3.52)

Similarly, we have

$\displaystyle\|(\phi_{2}^{(l)}-\phi_{2}^{(l-1)})\|_{C^{0}}\leq
M_{1}\epsilon\theta^{l}.$ (3.53)

$\mathbf{Step~{}4}:~{}$ The proof of (3.16) and (3.17).
Now, we show the modulus of continuity for $z_{i}^{(l)}(i=1,2)$ on the
temporal direction (3.16), which is very important to prove (3.17).
For $\delta\in(0,1)$, we choose

$\displaystyle\Omega(\delta)=\frac{24}{1-\kappa}[A_{0}+1](\sqrt{\epsilon}\delta+\varpi(\delta|\phi^{\prime}_{1_{b}})+\varpi(\delta|\phi^{\prime}_{2_{b}})+\varpi(\delta|\partial_{t}\beta)).$
(3.54)

Since $\varpi(\delta|\phi^{\prime}_{i_{b}})(i=1,2)$ and
$\varpi(\delta|\partial_{t}\beta)$ are monotonically increasing, bounded and
continuous concave functions of $\delta$ and
$\mathop{\lim}\limits_{\delta\rightarrow
0^{+}}\varpi(\delta|\phi^{\prime}_{i_{b}})=0$, then $\Omega(\delta)$ is also
such a function and

$\mathop{\lim}\limits_{\delta\rightarrow 0^{+}}\Omega(\delta)=0.$

At the boundary $x=L$, for any given $t_{1},t_{2}\in\mathbb{R}$ with
$|t_{1}-t_{2}|\leq\delta\ll 1$, one has

$|z_{1}^{(l)}(t_{1},L)-z_{1}^{(l)}(t_{2},L)|\leq|\phi^{\prime}_{1_{br}}(t_{1})-\phi^{\prime}_{1_{br}}(t_{2})|+|\kappa_{1}||z_{2_{r}}^{(l-1)}(t_{1},L)-z_{2_{r}}^{(l-1)}(t_{2},L)|.$

Thus, by (3.22) and (3.54), we get

$\displaystyle\varpi(\delta|z_{1}^{(l)}(\cdot,L))$
$\displaystyle\leq\varpi(\delta|\phi^{\prime}_{1_{b}})+\kappa\varpi(\delta|z_{2}^{(l-1)}(\cdot,L))$
$\displaystyle<\frac{1-\kappa}{24[A_{0}+1]}\Omega(\delta)+\frac{\kappa}{8[A_{0}+1]}\Omega(\delta)$
$\displaystyle=(\frac{1}{24[A_{0}+1]}+\frac{\kappa}{12[A_{0}+1]})\Omega(\delta)$
$\displaystyle<\frac{1}{8[A_{0}+1]}\Omega(\delta).$ (3.55)

In the domain $D$, for any $x\in[0,L]$ and $t_{1},t_{2}\in\mathbb{R}$ with
$|t_{1}-t_{2}|\leq\delta$, by the definition of the characteristic curve, one
has

$t_{1}^{(l)}(y;t_{*},x)=\int_{x}^{y}\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(\tilde{y};t_{*},x),\tilde{y})+\Phi)d\tilde{y}+t_{*}.$

Thus,

$\displaystyle|t_{1}^{(l)}(y;t_{1},x)-t_{1}^{(l)}(y;t_{2},x)|$
$\displaystyle\leq$
$\displaystyle|t_{1}-t_{2}|+\int_{x}^{y}|\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(\tilde{y};t_{1},x),\tilde{y})+\Phi)-\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(\tilde{y};t_{2},x),\tilde{y})+\Phi)|d\tilde{y}$
$\displaystyle\leq$
$\displaystyle|t_{1}-t_{2}|+\int_{x}^{y}\sum_{j=1}^{2}|\frac{\partial\nu_{1}}{\partial\phi_{j}}|\|z_{j}^{(l-1)}\|_{C^{0}}|t_{1}^{(l)}(\tilde{y};t_{1},x)-t_{1}^{(l)}(\tilde{y};t_{2},x)|d\tilde{y},$

then by the Gronwall’s inequality and (3.19), we have

$\displaystyle|t_{1}^{(l)}(y;t_{1},x)-t_{1}^{(l)}(y;t_{2},x)|\leq
e^{C\epsilon}|t_{1}-t_{2}|\leq(1+\sqrt{\epsilon})|t_{1}-t_{2}|.$ (3.56)

Noticing the concavity of $\Omega(\delta)$, we have

$\frac{1}{1+\sqrt{\epsilon}}\Omega((1+\sqrt{\epsilon})\delta)+\frac{\sqrt{\epsilon}}{1+\sqrt{\epsilon}}\Omega(0)\leq\Omega(\delta),$

then

$\Omega((1+\sqrt{\epsilon})\delta)\leq(1+\sqrt{\epsilon})\Omega(\delta).$

Thus, by (3.22), we get

$\displaystyle|z_{i}^{(l-1)}(t_{1}^{(l)}(y;t_{1},x),y)-z_{i}^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)|$
$\displaystyle\leq$
$\displaystyle\frac{1}{8[A_{0}+1]}\Omega((1+\sqrt{\epsilon})\delta)$
$\displaystyle\leq$
$\displaystyle\frac{1}{8[A_{0}+1]}(1+\sqrt{\epsilon})\Omega(\delta).$ (3.57)

Integrating (3.39) along $t=t_{1}^{(l)}(y;t_{1},x)$ and
$t=t_{1}^{(l)}(y;t_{2},x)$ respectively and then subtracting the two results,
we get

$\displaystyle z_{1}^{(l)}(t_{1},x)-z_{1}^{(l)}(t_{2},x)$ $\displaystyle=$
$\displaystyle
z_{1}^{(l)}(t_{1}^{(l)}(L;t_{1},x),L)-z_{1}^{(l)}(t_{1}^{(l)}(L;t_{2},x),L)$
$\displaystyle+\int_{L}^{x}\frac{\beta}{2}\nu_{1}(\Phi)z_{1}^{(l)}\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}dy+\int_{L}^{x}\frac{\beta}{2}\nu_{1}(\Phi)z_{2}^{(l-1)}\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\partial_{t}\beta(t_{1}^{(l)}(y;t_{1},x),y)-\partial_{t}\beta(t_{1}^{(l)}(y;t_{2},x),y)}{2}\nu_{1}(\Phi)\phi_{1}^{(l)}\Big{|}_{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\partial_{t}\beta(t_{1}^{(l)}(y;t_{2},x),y)}{2}\nu_{1}(\Phi)\phi_{1}^{(l)}\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\partial_{t}\beta(t_{1}^{(l)}(y;t_{1},x),y)-\partial_{t}\beta(t_{1}^{(l)}(y;t_{2},x),y)}{2}\Big{(}\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{1},x),y)+\Phi)$
$\displaystyle-\nu_{1}(\Phi)\Big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})\Big{|}_{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\partial_{t}\beta(t_{1}^{(l)}(y;t_{2},x),y)}{2}\Big{(}\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{1},x),y)+\Phi)$
$\displaystyle-\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)\Big{)}(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})\Big{|}_{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\partial_{t}\beta(t_{1}^{(l)}(y;t_{2},x),y)}{2}\Big{(}\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)-\nu_{1}(\Phi)\Big{)}$
$\displaystyle(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\partial_{t}\beta(t_{1}^{(l)}(y;t_{1},x),y)-\partial_{t}\beta(t_{1}^{(l)}(y;t_{2},x),y)}{2}\nu_{1}(\Phi)\phi_{2}^{(l-1)}\Big{|}_{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\partial_{t}\beta(t_{1}^{(l)}(y;t_{2},x),y)}{2}\nu_{1}(\Phi)\phi_{2}^{(l-1)}\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\beta}{2}\Big{(}\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{1},x),y)+\Phi)-\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)\Big{)}$
$\displaystyle(z_{1}^{(l-1)}+z_{2}^{(l-1)})\Big{|}_{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\beta}{2}\Big{(}\nu_{1}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)-\nu_{1}(\Phi)\Big{)}(z_{1}^{(l-1)}+z_{2}^{(l-1)})\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\beta}{2}\sum_{j=1}^{2}\Big{(}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{1},x),y)+\Phi)-\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)\Big{)}$
$\displaystyle(z_{j}^{(l-1)}\phi_{1}^{(l-1)}+z_{j}^{(l-1)}\phi_{2}^{(l-1)})\Big{|}_{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)z_{j}^{(l-1)}\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}$
$\displaystyle(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})\Big{|}_{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)z_{j}^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)$
$\displaystyle(\phi_{1}^{(l-1)}+\phi_{2}^{(l-1)})\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\sum_{j=1}^{2}\Big{(}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{1},x),y)+\Phi)-\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)\Big{)}$
$\displaystyle(z_{j}^{(l-1)}z_{1}^{(l)})\Big{|}_{(t_{1}^{(l)}(y;t_{1},x),y)}dy$
$\displaystyle+\int_{L}^{x}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)z_{j}^{(l-1)}\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}z_{1}^{(l)}(t_{1}^{(l)}(y;t_{1},x),y)dy$
$\displaystyle+\int_{L}^{x}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)+\Phi)z_{j}^{(l-1)}(t_{1}^{(l)}(y;t_{2},x),y)z_{1}^{(l)}\Big{|}_{(t_{1}^{(l)}(y;t_{2},x),y)}^{(t_{1}^{(l)}(y;t_{1},x),y)}dy.$

By (1.2)-(1.3), (2.11), (3.19), (3.55)-(3.57) and Gronwall’s inequality, we
have

$\displaystyle|z_{1}^{(l)}(t_{1},x)-z_{1}^{(l)}(t_{2},x)|\leq\frac{1}{8[A_{0}+1]}C_{2}\Omega(\delta),$
(3.58)

where $C_{2}>0$ is a constant independent of $l$.
Using the integral expression (3.46) of $F_{1}(t,x)z_{1}^{(l)}(t,x)$ and then
by (1.2)-(1.3), (2.11), (3.19), and (3.55)-(3.58),we have

$\displaystyle|z_{1}^{(l)}(t_{1},x)-z_{1}^{(l)}(t_{2},x)|$ $\displaystyle\leq$
$\displaystyle\frac{1}{F_{1}(t,x)}(\frac{1}{24[A_{0}+1]}+\frac{1}{12[A_{0}+1]}\kappa)(1+\sqrt{\epsilon})\Omega(\delta)$
$\displaystyle+\frac{F_{1}(t,x)-1}{F_{1}(t,x)}\frac{1}{8[A_{0}+1]}(1+\sqrt{\epsilon})\Omega(\delta)+C\epsilon^{2}(1+\sqrt{\epsilon})|t_{1}-t_{2}|$
$\displaystyle+C\epsilon\frac{1+\sqrt{\epsilon}}{8[A_{0}+1]}\Omega(\delta)+\frac{1}{8[A_{0}+1]}C\epsilon\Omega(\delta)$
$\displaystyle\leq$
$\displaystyle\frac{1}{8[A_{0}+1]}\Omega(\delta)-(\frac{1-\kappa}{12[A_{0}+1]}-\sqrt{\epsilon})\Omega(\delta)$
$\displaystyle<$ $\displaystyle\frac{1}{8[A_{0}+1]}\Omega(\delta).$ (3.59)

Finally, we prove (3.17). We first consider the special case that two given
points $(t_{1},x_{1})$ and $(t_{2},x_{2})$ with
$|t_{1}-t_{2}|\leq\delta,|x_{1}-x_{2}|\leq\delta$ locate on the same
characteristic curve $t=t_{1}^{(l)}(x;t_{0},x_{0})$, namely,
$t_{2}=t_{1}^{(l)}(x_{2};t_{1},x_{1})$. Using the similar method of (3.47), we
can get

$\displaystyle|z_{1}^{(l)}(t_{1},x_{1})-z_{1}^{(l)}(t_{2},x_{2})|\leq
C\epsilon\delta\leq\frac{1}{12}\Omega(\delta).$ (3.60)

Then, for general two points $(t_{1},x_{1})$ and $(t_{2},x_{2})$ with
$|t_{1}-t_{2}|\leq\delta,|x_{1}-x_{2}|\leq\delta$, we can choose a point
$(t_{3},x_{1})$ locating on the $1$st characteristic curve passing through
$(t_{2},x_{2})$, namely, $t_{3}=t_{1}^{(l)}(x_{1};t_{2},x_{2})$.
By definition (3.24) and (2.11), we have

$|t_{3}-t_{2}|\leq|\nu_{1}||x_{1}-x_{2}|\leq A_{0}\delta,$

and thus

$|t_{3}-t_{1}|\leq|t_{3}-t_{2}|+|t_{2}-t_{1}|\leq(A_{0}+1)\delta.$

Now we combine estimates (3.59)-(3.60) to get

$\displaystyle|z_{1}^{(l)}(t_{1},x_{1})-z_{1}^{(l)}(t_{2},x_{2})|$
$\displaystyle\leq$
$\displaystyle|z_{1}^{(l)}(t_{1},x_{1})-z_{1}^{(l)}(\frac{[A_{0}+1]t_{1}+t_{3}}{A_{0}+1},x_{1})|$
$\displaystyle+|z_{1}^{(l)}(\frac{[A_{0}+1]t_{1}+t_{3}}{A_{0}+1},x_{1})-z_{1}^{(l)}(\frac{([A_{0}+1]-1)t_{1}+2t_{3}}{A_{0}+1},x_{1})|$
$\displaystyle+\ldots+|z_{1}^{(l)}(\frac{t_{1}+[A_{0}+1]t_{3}}{A_{0}+1},x_{1})-z_{1}^{(l)}(t_{3},x_{1})|+|z_{1}^{(l)}(t_{3},x_{1})-z_{1}^{(l)}(t_{2},x_{2})|$
$\displaystyle\leq$
$\displaystyle\frac{[A_{0}+1]+1}{8[A_{0}+1]}\Omega(\delta)+\frac{1}{12}\Omega(\delta)$
$\displaystyle\leq$ $\displaystyle\frac{1}{3}\Omega(\delta).$ (3.61)

The combination of (3.60) and (3.61) leads to

$\displaystyle\varpi(\delta|z_{1}^{(l)})\leq\frac{1}{3}\Omega(\delta).$ (3.62)

In a similar way, we obtain

$\displaystyle\varpi(\delta|z_{2}^{(l)})\leq\frac{1}{3}\Omega(\delta).$ (3.63)

By the aid of equations (3.3), (3.5) and by (1.2), (2.11), (3.19), (3.23),
(3.54) and (3.62)-(3.63), we have

$\displaystyle\varpi(\delta|\partial_{x}\phi_{i}^{(l)})$ $\displaystyle\leq
A_{0}\frac{1}{3}\Omega(\delta)+C\epsilon^{2}\delta+C\epsilon\delta$
$\displaystyle\leq A_{0}\frac{1}{2}\Omega(\delta),\quad i=1,2.$ (3.64)

Thus, (3.62)-(3.64) indicate (3.17). ∎

With the help of Proposition 3.1 and the similar arguments as in [10], the
proof of Theorem 2.1 could be presented, here we omit the details.

##  4 Stability of the Time-periodic Solution

In this section, we give the proof of Theorem 2.2 to consider the stability of
the time-periodic solution obtained in Theorem 2.1. For the sake of proving
the existence of the classical solutions $\phi=\phi(t,x)$, we only need to
prove the following Lemma 4.1 on the basis of the existence and uniqueness of
local $C^{1}$ solution for the mixed initial-boundary value problem for
quasilinear hyperbolic system(cf. Chapter $4$ in [4]). Using the method in
[3], we can give the proof of Lemma 4.1. Here we omit the details.

######  Lemma 4.1.

There exists a small constant $\varepsilon_{6}>0$, for any given
$\varepsilon\in(0,\varepsilon_{6})$, there exists
$\sigma=\sigma(\varepsilon)>0$ such that if

$\displaystyle\|\phi_{i_{b}}\|_{C^{1}(\mathbb{R}_{+})}\leq\sigma,\quad i=1,2,$
$\displaystyle\|\phi_{0}\|_{C^{1}[0,L]}\leq\sigma,$

then the $C^{1}$ solution $\phi=\phi(t,x)$ to the initial-boundary value
problem (2.7)-(2.10) satisfies

$\displaystyle\|\phi\|_{C^{1}(D)}\leq\varepsilon.$ (4.1)

Now, we prove (2.17) inductively. For any $t_{*}>0$ and $N\in\mathbb{N}$, we
prove

$\displaystyle\mathop{\max}\limits_{i=1,2}\|\phi_{i}(t,\cdot)-\phi_{i}^{(T_{*})}(t,\cdot)\|_{C^{0}}\leq
C_{S}\epsilon\xi^{N+1},\quad\forall t\in[t_{*}+T_{0},t_{*}+2T_{0}],$ (4.2)

under the hypothesis

$\displaystyle\mathop{\max}\limits_{i=1,2}\|\phi_{i}(t,\cdot)-\phi_{i}^{(T_{*})}(t,\cdot)\|_{C^{0}}\leq
C_{S}\epsilon\xi^{N},\quad\forall t\in[t_{*},t_{*}+T_{0}],$ (4.3)

where $\xi\in(0,1)$ is a constant to be determined later and
$\phi_{i}^{(T_{*})}(t,x),i=1,2$ is the time-periodic solution obtained in
Theorem 2.1. Let

$\theta(t)=\mathop{\max}\limits_{1\leq i\leq
2}\mathop{\sup}\limits_{x\in[0,L]}|\phi_{i}(t,x)-\phi_{i}^{(T_{*})}(t,x)|.$

Obviously, $\theta(t)$ is continuous and

$\theta(t_{*}+T_{0})\leq C_{S}\varepsilon\xi^{N}$

follows (4.3). Then it’s just necessary to prove

$\displaystyle\theta(t)\leq C_{S}\epsilon\xi^{N+1},\quad\forall
t\in[t_{*}+T_{0},\tau]$ (4.4)

under the assumption

$\displaystyle\theta(t)\leq C_{S}\epsilon\xi^{N},\quad\forall
t\in[t_{*},\tau]$ (4.5)

for any $\tau\in[t_{*}+T_{0},t_{*}+2T_{0}]$.
At the boundary $x=L$, one has

$\phi_{1}(t,L)-\phi_{1}^{(T_{*})}(t,L)=\kappa_{1}(\phi_{2}(t,L)-\phi_{2}^{(T_{*})}(t,L)),$

then from (4.5), we have

$\displaystyle|\phi_{1}(t,L)-\phi_{1}^{(T_{*})}(t,L)|\leq|\kappa_{1}|C_{S}\epsilon\xi^{N}.$
(4.6)

Similarly, at the boundary $x=0$, we get

$\displaystyle|\phi_{2}(t,0)-\phi_{2}^{(T_{*})}(t,0)|\leq|\kappa_{2}|C_{S}\epsilon\xi^{N}.$
(4.7)

As for the interior estimates, we have

$\displaystyle\partial_{x}\phi_{1}^{(T_{*})}+\nu_{1}(\phi^{(T_{*})}+\Phi)\partial_{t}\phi_{1}^{(T_{*})}=$
$\displaystyle\frac{\beta(t,x)}{2}\nu_{1}(\Phi)\phi_{1}^{(T_{*})}+\frac{\beta(t,x)}{2}(\nu_{1}(\phi^{(T_{*})}+\Phi)$
$\displaystyle-\nu_{1}(\Phi))\phi_{1}^{(T_{*})}+\frac{\beta(t,x)}{2}\nu_{1}(\phi^{(T_{*})}+\Phi)\phi_{2}^{(T_{*})}.$
(4.8)

Then by (3.1) and (4.8), we get

$\displaystyle\partial_{x}(\phi_{1}-\phi_{1}^{(T_{*})})+\nu_{1}(\phi+\Phi)\partial_{t}(\phi_{1}-\phi_{1}^{(T_{*})})$
$\displaystyle=$
$\displaystyle-\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}\partial_{t}\phi_{1}^{(T_{*})}+\frac{\beta(t,x)}{2}\nu_{1}(\Phi)(\phi_{1}-\phi_{1}^{(T_{*})})$
$\displaystyle+\frac{\beta(t,x)}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\Phi)\Big{)}\Big{(}(\phi_{1}+\phi_{2})-(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})\Big{)}$
$\displaystyle+\frac{\beta(t,x)}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})$
$\displaystyle+\frac{\beta(t,x)}{2}\nu_{1}(\Phi)(\phi_{2}-\phi_{2}^{(T_{*})}).$
(4.9)

Multiplying $F_{1}(t,x)$ on both sides of (4.9), we gain

$\displaystyle\partial_{x}\Big{(}F_{1}(\phi_{1}-\phi_{1}^{(T_{*})})\Big{)}+\nu_{1}(\phi+\Phi)\partial_{t}\Big{(}F_{1}(\phi_{1}-\phi_{1}^{(T_{*})})\Big{)}$
$\displaystyle=$ $\displaystyle-
F_{1}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}\partial_{t}\phi_{1}^{(T_{*})}$
$\displaystyle+\nu_{1}(\phi+\Phi)F_{1}(\phi_{1}-\phi_{1}^{(T_{*})})\int_{x}^{L}\frac{\partial_{t}\beta(t,s)}{2}\nu_{1}(\Phi)ds$
$\displaystyle+\frac{\beta(t,x)}{2}F_{1}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\Phi)\Big{)}\Big{(}(\phi_{1}+\phi_{2})-(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})\Big{)}$
$\displaystyle+\frac{\beta(t,x)}{2}F_{1}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})$
$\displaystyle+\frac{\beta(t,x)}{2}F_{1}\nu_{1}(\Phi)(\phi_{2}-\phi_{2}^{(T_{*})}),$

then integrate the result along the $1$st characteristic curve
$t=t_{1}(x;\hat{t},\hat{x})$ defined by

$\displaystyle\left\\{\begin{aligned}
&\frac{dt_{1}}{dx}(x;\hat{t},\hat{x})=\nu_{1}(\phi(t_{1}(x;\hat{t},\hat{x}),x)+\Phi),\\\
&t_{1}(\hat{x};\hat{t},\hat{x})=\hat{t}.\end{aligned}\right.$ (4.10)

Noting
$T_{0}=L\mathop{\max}\limits_{i=1,2}\mathop{\sup}\limits_{\phi\in\Psi}|\nu_{i}(\phi+\Phi)|$,
for each points $(\hat{t},\hat{x})\in[t_{*}+T_{0},\tau]\times[0,L]$, the
backward curve $t=t_{1}(x;\hat{t},\hat{x})$ will intersect the boundary in a
time interval shorter than $T_{0}$, namely,

$t_{1}(L;\hat{t},\hat{x})\in[\hat{t}-T_{0},\hat{t}]\subseteq[t_{*},\tau],\quad\forall(\hat{t},\hat{x})\in[t_{*}+T_{0},\tau]\times[0,L],$

and thus we can use estimates (4.6)-(4.7) on the boundary.
Using (1.2), (2.16) and (4.5)-(4.6), we get

$\displaystyle|\phi_{1}(\hat{t},\hat{x})-\phi_{1}^{(T_{*})}(\hat{t},\hat{x})|$
$\displaystyle\leq$
$\displaystyle\frac{|\kappa_{1}|}{F_{1}(\hat{t},\hat{x})}C_{S}\epsilon\xi^{N}+C\epsilon
C_{S}\epsilon\xi^{N}+\frac{F_{1}(\hat{t},\hat{x})-1}{F_{1}(\hat{t},\hat{x})}C_{S}\epsilon\xi^{N}$
$\displaystyle\leq$ $\displaystyle C_{S}\epsilon\xi^{N+1}.$

Here we choose a constant $0<\xi<1$ satisfying

$\frac{|\kappa_{1}|}{F_{1}(\hat{t},\hat{x})}+C\epsilon+\frac{F_{1}(\hat{t},\hat{x})-1}{F_{1}(\hat{t},\hat{x})}\leq\xi.$

Similarly, we get

$\displaystyle|\phi_{2}(\hat{t},\hat{x})-\phi_{2}^{(T_{*})}(\hat{t},\hat{x})|\leq
C_{S}\epsilon\xi^{N+1}.$

Thus, we have

$\theta(\hat{t})\leq C_{S}\epsilon\xi^{N+1}.$

Since $\hat{t}\in[t_{*}+T_{0},\tau]$ is arbitrary, we get (4.4).

##  5 Regularity of the Time-periodic Solution

In this section, we will prove higher regularity of the time-periodic
solutions, provided that all boundary functions $m_{i_{b}}(t)(i=1,2)$ possess
higher regularity.
In order to get the regularity of $\phi^{(T_{*})}$, we use the iteration
scheme (3.3)-(3.6) introduced in Section 3 and prove the following
proposition.

######  Proposition 5.1.

For the iteration scheme (3.3)-(3.6), assuming that (2.18) holds, then exists
a large enough constant $C_{R}>0$, such that for any given
$l\in\mathbb{N}_{+}$, we have

$\displaystyle\|\partial_{t}^{2}\phi_{i}^{(l)}\|_{L^{\infty}}\leq C_{R},$
(5.1)
$\displaystyle\|\partial_{t}\partial_{x}\phi_{i}^{(l)}\|_{L^{\infty}}\leq
A_{0}C_{R},$ (5.2)
$\displaystyle\|\partial_{x}^{2}\phi_{i}^{(l)}\|_{L^{\infty}}\leq
A_{0}^{2}C_{R}$ (5.3)

under the hypothesis

$\displaystyle\|\partial_{t}^{2}\phi_{i}^{(l-1)}\|_{L^{\infty}}\leq
C_{R}<+\infty,$ (5.4)
$\displaystyle\|\partial_{t}\partial_{x}\phi_{i}^{(l-1)}\|_{L^{\infty}}\leq
A_{0}C_{R},$ (5.5)
$\displaystyle\|\partial_{x}^{2}\phi_{i}^{(l-1)}\|_{L^{\infty}}\leq
A_{0}^{2}C_{R}.$ (5.6)

###### Proof.

Since we use actually the same sequence constructed in Section 3. By
Proposition 3.1, we already have (3.13)-(3.17) for each $l$ and especially,

$\displaystyle\|\phi^{(l)}\|_{C^{1}}\leq(C_{1}+M_{1})\epsilon,\quad\|\phi^{(l-1)}\|_{C^{1}}\leq(C_{1}+M_{1})\epsilon.$
(5.7)

Let

$\varphi_{i}^{(l)}=\partial_{t}z_{i}^{(l)}=\partial_{t}^{2}\phi_{i}^{(l)},\quad
i=1,2;l\in\mathbb{N}_{+}.$

First, by the aid of boundary conditions (3.41) and (3.42) and using (2.18)
and (5.4), we have

$\displaystyle|\varphi_{1}^{(l)}(t,L)|\leq M_{2}+|\kappa_{1}|C_{R},$ (5.8)
$\displaystyle|\varphi_{2}^{(l)}(t,0)|\leq M_{2}+|\kappa_{2}|C_{R}.$ (5.9)

Then we use a method similar to (3.47) to get

$\displaystyle\|\varphi_{1}^{(l)}(t,x)\|_{L^{\infty}}\leq\hbar_{1}C_{R},$
(5.10)

where constant $0<\hbar_{1}<1$ is independent of $l$.
Similarly, we have

$\displaystyle\|\varphi_{2}^{(l)}(t,x)\|_{L^{\infty}}\leq\hbar_{1}C_{R}.$
(5.11)

Next, using (3.39) and (3.40) and by (1.2)-(1.3), (2.11), (3.13) and
(5.10)-(5.11), we get

$\displaystyle\|\partial_{x}\partial_{t}\phi_{i}^{(l)}|_{L^{\infty}}$
$\displaystyle\leq A_{0}\hbar_{1}C_{R}+C\epsilon+C\epsilon^{2}$
$\displaystyle\leq A_{0}\hbar_{2}C_{R},$ (5.12)

where $\hbar_{2}\in(\hbar_{1},1)$ is a constant independent of $l$.
Then taking the spatial derivative to (3.3), we have

$\displaystyle\partial_{x}^{2}\phi_{1}^{(l)}=$
$\displaystyle-\nu_{1}(\phi^{(l-1)}+\Phi)\partial_{x}\partial_{t}\phi_{1}^{(l)}-\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}+\Phi)\partial_{x}\phi_{j}^{(l-1)}z_{1}^{(l)}$
$\displaystyle+\frac{\beta}{2}\nu_{1}(\Phi)\partial_{x}\phi_{1}^{(l)}+\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}+\Phi)\partial_{x}\phi_{j}^{(l-1)}\phi_{1}^{(l-1)}$
$\displaystyle+\frac{\partial_{x}\beta}{2}\nu_{1}(\Phi)\phi_{1}^{(l)}+\frac{\beta}{2}(\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi))\partial_{x}\phi_{1}^{(l-1)}$
$\displaystyle+\frac{\partial_{x}\beta}{2}\Big{(}\nu_{1}(\phi^{(l-1)}+\Phi)-\nu_{1}(\Phi)\Big{)}\phi_{1}^{(l-1)}+\frac{\beta}{2}\nu_{1}(\phi^{(l-1)}+\Phi)\partial_{x}\phi_{2}^{(l-1)}$
$\displaystyle+\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(l-1)}+\Phi)\partial_{x}\phi_{j}^{(l-1)}\phi_{2}^{(l-1)}+\frac{\partial_{x}\beta}{2}\nu_{1}(\phi^{(l-1)}+\Phi)\phi_{2}^{(l-1)}.$

Thus, using (1.2)-(1.3), (2.11), (3.13)-(3.14) (5.7) and (5.12), we get

$\displaystyle\|\partial_{x}^{2}\phi_{1}^{(l)}\|_{L^{\infty}}\leq$
$\displaystyle A_{0}^{2}\hbar_{2}C_{R}+C\epsilon$ $\displaystyle\leq$
$\displaystyle A_{0}^{2}C_{R}.$ (5.13)

In a similar way, we also get

$\displaystyle\|\partial_{x}^{2}\phi_{2}^{(l)}\|_{L^{\infty}}\leq
A_{0}^{2}C_{R}.$ (5.14)

∎

$\mathbf{The~{}Proof~{}of~{}Theorem~{}2.3.}$ By (5.1)-(5.3), we know that
$\\{\phi^{(l)}\\}_{l=1}^{\infty}$ is uniformly $W^{2,\infty}$ bounded and then
$weak^{*}$ convergent. Moreover, noting that $\\{\phi^{(l)}\\}_{l=1}^{\infty}$
converges strongly to $\phi^{(T_{*})}$ in $C^{1}$, so we get the
$W^{2,\infty}$ regularity of $\phi^{(T_{*})}$.

##  6 Boundary Stabilization around the Time-periodic Solution

In this section, we will give the proof of Theorem 2.4.
Noting that we have got the $C^{0}$ exponential convergence in Theorem 2.2 as
follows:

$\displaystyle\|\phi_{i}(t,\cdot)-\phi_{i}^{(T_{*})}(t,\cdot)\|_{C^{0}}\leq
C_{S}\epsilon\xi^{N},\quad\forall t\in[NT_{0},(N+1)T_{0}),\forall
l\in\mathbb{N}_{+},$ (6.1)

which also shows that

$\displaystyle\|\phi_{i}(t,\cdot)-\phi_{i}^{(T_{*})}(t,\cdot)\|_{C^{0}}\leq
C_{S}\epsilon\xi^{N+1},\quad\forall t\in[(N+1)T_{0},(N+2)T_{0}),\forall
l\in\mathbb{N}_{+}.$ (6.2)

Moreover, by Theorem 2.1, Theorem 2.3 and Lemma 4.1, we have

$\displaystyle\|\phi\|_{C^{1}}\leq
C_{3}\epsilon,~{}~{}\|\phi^{(T_{*})}\|_{C^{1}}\leq
C_{E}\epsilon,~{}~{}\|\phi^{(T_{*})}\|_{W^{2,\infty}}\leq(1+A_{0})^{2}C_{R}.$
(6.3)

By the continuity, we will inductively get the estimates for the convergence
of the first derivatives, namely, for each $N\in\mathbb{N}_{+}$ and
$\tau\in[(N+1)T_{0},(N+2)T_{0}]$, we will prove

$\displaystyle\|\partial_{t}\phi_{i}(t,\cdot)-\partial_{t}\phi_{i}^{(T_{*})}(t,\cdot)\|_{C^{0}}\leq
C_{S}^{*}\epsilon\xi^{N+1},\quad\forall t\in[(N+1)T_{0},\tau],\forall
l\in\mathbb{N}_{+},$ (6.4)
$\displaystyle\|\partial_{x}\phi_{i}(t,\cdot)-\partial_{x}\phi_{i}^{(T_{*})}(t,\cdot)\|_{C^{0}}\leq
A_{0}C_{S}^{*}\epsilon\xi^{N+1},\quad\forall t\in[(N+1)T_{0},\tau],\forall
l\in\mathbb{N}_{+}$ (6.5)

under the assumption

$\displaystyle\|\partial_{t}\phi_{i}(t,\cdot)-\partial_{t}\phi_{i}^{(T_{*})}(t,\cdot)\|_{C^{0}}\leq
C_{S}^{*}\epsilon\xi^{N},\quad\forall t\in[NT_{0},\tau],\forall
l\in\mathbb{N}_{+},$ (6.6)
$\displaystyle\|\partial_{x}\phi_{i}(t,\cdot)-\partial_{x}\phi_{i}^{(T_{*})}(t,\cdot)\|_{C^{0}}\leq
A_{0}C_{S}^{*}\epsilon\xi^{N},\quad\forall t\in[NT_{0},\tau],\forall
l\in\mathbb{N}_{+}.$ (6.7)

Let

$z_{i}=\partial_{t}\phi_{i},\quad w_{i}=\partial_{x}\phi_{i}$

and

$z_{i}^{(T_{*})}=\partial_{t}\phi_{i}^{(T_{*})},\quad
w_{i}^{(T_{*})}=\partial_{x}\phi_{i}^{(T_{*})}.$

Taking the temporal derivative on boundary conditions (2.9)-(2.10), we get

$\displaystyle z_{2}(t,0)=\phi^{\prime}_{2_{b}}(t)+\kappa_{2}z_{1}(t,0),\quad
t>0,$ $\displaystyle
z_{1}(t,L)=\phi^{\prime}_{1_{b}}(t)+\kappa_{1}z_{2}(t,L),\quad t>0$

and

$\displaystyle
z_{2}^{(T_{*})}(t,0)=\phi^{\prime}_{2_{b}}(t)+\kappa_{2}z_{1}^{(T_{*})}(t,0),\quad
t>0,$ $\displaystyle
z_{1}^{(T_{*})}(t,L)=\phi^{\prime}_{1_{b}}(t)+\kappa_{1}z_{2}^{(T_{*})}(t,L),\quad
t>0.$

Thus, on the boundary $x=0$, we have

$\displaystyle\mathop{\sup}\limits_{t\in[NT_{0},\tau]}|z_{2}(t,0)-z_{2}^{(T_{*})}(t,0)|$
$\displaystyle\leq|\kappa_{2}||z_{1}(t,0)-z_{1}^{(T_{*})}(t,0)|$
$\displaystyle\leq|\kappa_{2}|C_{S}^{*}\epsilon\xi^{N}.$ (6.8)

Similarly, on $x=L$, we have

$\displaystyle\mathop{\sup}\limits_{t\in[NT_{0},\tau]}|z_{1}(t,L)-z_{1}^{(T_{*})}(t,L)|\leq|\kappa_{1}|C_{S}^{*}\epsilon\xi^{N}.$
(6.9)

In the domain, we take the temporal derivative of (3.1) and (4.8) to get

$\displaystyle\partial_{x}z_{1}+\nu_{1}(\phi+\Phi)\partial_{t}z_{1}$
$\displaystyle=$
$\displaystyle\frac{\beta}{2}\nu_{1}(\Phi)z_{1}+\frac{\partial_{t}\beta}{2}\nu_{1}(\Phi)\phi_{1}+\frac{\beta}{2}\nu_{1}(\Phi)z_{2}+\frac{\partial_{t}\beta}{2}\nu_{1}(\Phi)\phi_{2}$
$\displaystyle+\frac{\beta}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\Phi)\Big{)}(z_{1}+z_{2})$
$\displaystyle+\frac{\partial_{t}\beta}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\Phi)\Big{)}(\phi_{1}+\phi_{2})$
$\displaystyle+\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(z_{j}\phi_{1}+z_{j}\phi_{2})-\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}z_{j}z_{1},$
(6.10)
$\displaystyle\partial_{x}z_{1}^{(T_{*})}+\nu_{1}(\phi^{(T_{*})}+\Phi)\partial_{t}z_{1}^{(T_{*})}$
$\displaystyle=$
$\displaystyle\frac{\beta}{2}\nu_{1}(\Phi)z_{1}^{(T_{*})}+\frac{\partial_{t}\beta}{2}\nu_{1}(\Phi)\phi_{1}^{(T_{*})}+\frac{\beta}{2}\nu_{1}(\Phi)z_{2}^{(T_{*})}+\frac{\partial_{t}\beta}{2}\nu_{1}(\Phi)\phi_{2}^{(T_{*})}$
$\displaystyle+\frac{\beta}{2}\Big{(}\nu_{1}(\phi^{(T_{*})}+\Phi)-\nu_{1}(\Phi)\Big{)}(z_{1}^{(T_{*})}+z_{2}^{(T_{*})})$
$\displaystyle+\frac{\partial_{t}\beta}{2}\Big{(}\nu_{1}(\phi^{(T_{*})}+\Phi)-\nu_{1}(\Phi)\Big{)}(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})$
$\displaystyle+\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(z_{j}^{(T_{*})}\phi_{1}^{(T_{*})}+z_{j}^{(T_{*})}\phi_{2}^{(T_{*})})-\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}z_{j}^{(T_{*})}z_{1}^{(T_{*})}.$
(6.11)

Furthermore, by (6.10)-(6.11), we get

$\displaystyle\partial_{x}(z_{1}-z_{1}^{(T_{*})})+\nu_{1}(\phi+\Phi)\partial_{t}(z_{1}-z_{1}^{(T_{*})})$
$\displaystyle=$
$\displaystyle-\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}\partial_{t}z_{1}^{(T_{*})}+\frac{\beta}{2}\nu_{1}(\Phi)(z_{1}-z_{1}^{(T_{*})})$
$\displaystyle+\frac{\partial_{t}\beta}{2}\nu_{1}(\Phi)(\phi_{1}-\phi_{1}^{(T_{*})})+\frac{\beta}{2}\nu_{1}(\Phi)(z_{2}-z_{2}^{(T_{*})})+\frac{\partial_{t}\beta}{2}\nu_{1}(\Phi)(\phi_{2}-\phi_{2}^{(T_{*})})$
$\displaystyle+\frac{\beta}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\Phi)\Big{)}\Big{(}(z_{1}+z_{2})-(z_{1}^{(T_{*})}+z_{2}^{(T_{*})})\Big{)}$
$\displaystyle+\frac{\beta}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}(z_{1}^{(T_{*})}+z_{2}^{(T_{*})})$
$\displaystyle+\frac{\partial_{t}\beta}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\Phi)\Big{)}\Big{(}(\phi_{1}+\phi_{2})-(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})\Big{)}$
$\displaystyle+\frac{\partial_{t}\beta}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})$
$\displaystyle+\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)z_{j}\Big{(}(\phi_{1}+\phi_{2})-(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})\Big{)}$
$\displaystyle+\frac{\beta}{2}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)(z_{j}-z_{j}^{(T_{*})})(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})$
$\displaystyle+\frac{\beta}{2}\sum_{j=1}^{2}\Big{(}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)-\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(T_{*})}+\Phi)\Big{)}(z_{j}^{(T_{*})})(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})$
$\displaystyle-\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)z_{j}(z_{1}-z_{1}^{(T_{*})})-\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)(z_{j}-z_{j}^{(T_{*})})z_{1}^{(T_{*})}$
$\displaystyle-\sum_{j=1}^{2}\Big{(}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)-\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(T_{*})}+\Phi)\Big{)}z_{j}^{(T_{*})}z_{1}^{(T_{*})}.$
(6.12)

And we multiply $F_{1}(t,x)$ on both sides of (6.12) to gain

$\displaystyle\partial_{x}\Big{(}F_{1}(z_{1}-z_{1}^{(T_{*})})\Big{)}+\nu_{1}(\phi+\Phi)\partial_{t}\Big{(}F_{1}(z_{1}-z_{1}^{(T_{*})})\Big{)}$
$\displaystyle=$ $\displaystyle-
F_{1}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}\partial_{t}z_{1}^{(T_{*})}$
$\displaystyle+\nu_{1}(\phi+\Phi)F_{1}(z_{1}-z_{1}^{(T_{*})})\int_{x}^{L}\frac{\partial_{t}\beta(t,s)}{2}\nu_{1}(\Phi)ds$
$\displaystyle+\frac{\partial_{t}\beta}{2}F_{1}\nu_{1}(\Phi)(\phi_{1}-\phi_{1}^{(T_{*})})+\frac{\partial
F_{1}}{\partial x}(z_{2}-z_{2}^{(T_{*})})$
$\displaystyle+\frac{\partial_{t}\beta}{2}F_{1}\nu_{1}(\Phi)(\phi_{2}-\phi_{2}^{(T_{*})})$
$\displaystyle+\frac{\beta}{2}F_{1}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\Phi)\Big{)}\Big{(}(z_{1}+z_{2})-(z_{1}^{(T_{*})}+z_{2}^{(T_{*})})\Big{)}$
$\displaystyle+\frac{\beta}{2}F_{1}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}(z_{1}^{(T_{*})}+z_{2}^{(T_{*})})$
$\displaystyle+\frac{\partial_{t}\beta}{2}F_{1}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\Phi)\Big{)}\Big{(}(\phi_{1}+\phi_{2})-(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})\Big{)}$
$\displaystyle+\frac{\partial_{t}\beta}{2}F_{1}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})$
$\displaystyle+\frac{\beta}{2}F_{1}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)z_{j}\Big{(}(\phi_{1}+\phi_{2})-(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})\Big{)}$
$\displaystyle+\frac{\beta}{2}F_{1}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)(z_{j}-z_{j}^{(T_{*})})(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})$
$\displaystyle+\frac{\beta}{2}F_{1}\sum_{j=1}^{2}\Big{(}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)-\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(T_{*})}+\Phi)\Big{)}(z_{j}^{(T_{*})})(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})$
$\displaystyle-
F_{1}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)z_{j}(z_{1}-z_{1}^{(T_{*})})$
$\displaystyle-
F_{1}\sum_{j=1}^{2}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)(z_{j}-z_{j}^{(T_{*})})z_{1}^{(T_{*})}$
$\displaystyle-
F_{1}\sum_{j=1}^{2}\Big{(}\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi+\Phi)-\frac{\partial\nu_{1}}{\partial\phi_{j}}(\phi^{(T_{*})}+\Phi)\Big{)}z_{j}^{(T_{*})}z_{1}^{(T_{*})}.$

Then we integrate it along the corresponding backward characteristic curve
$t=t_{1}(x;\hat{t},\hat{x})$ defined by (4.9) and by (1.2)-(1.3), (2.11),
(6.1)-(6.3), (6.6)-(6.7) and (6.9), we have

$\displaystyle|z_{1}(\hat{t},\hat{x})-z_{1}^{(T_{*})}(\hat{t},\hat{x})|$
$\displaystyle\leq$
$\displaystyle\frac{|\kappa_{1}|C_{S}^{*}\epsilon\xi^{N}}{F_{1}(\hat{t},\hat{x})}+C\epsilon
C_{S}\epsilon\xi^{N}+C\epsilon C_{S}^{*}\epsilon\xi^{N}$
$\displaystyle+\frac{F_{1}(\hat{t},\hat{x})-1}{F_{1}(\hat{t},\hat{x})}C_{S}^{*}\epsilon\xi^{N}+F_{1}(\hat{t},\hat{x})\mathop{\sup}\limits_{\phi\in\Psi}|\nabla\nu_{1}|C_{R}C_{S}\epsilon\xi^{N}.$

Here we choose constant

$C_{S}^{*}>30\mathop{\max}\limits_{1\leq i\leq
2}\mathop{\sup}\limits_{\phi\in\Psi}|\nabla\nu_{i}|C_{R}C_{S}+30\beta_{*}A_{0}C_{S}+C_{S},$

then we can get

$\displaystyle|z_{1}(\hat{t},\hat{x})-z_{1}^{(T_{*})}(\hat{t},\hat{x})|\leq\hbar_{3}C_{S}^{*}\epsilon\xi^{N+1},\quad\forall(\hat{t},\hat{x})\in[(N+1)T_{0},\tau]\times[0,L],$

where constant $0<\hbar_{3}<1$.
Similarly, we get

$\displaystyle|z_{2}(\hat{t},\hat{x})-z_{2}^{(T_{*})}(\hat{t},\hat{x})|\leq\hbar_{3}C_{S}^{*}\epsilon\xi^{N+1},\quad\forall(\hat{t},\hat{x})\in[(N+1)T_{0},\tau]\times[0,L].$

Thus, by the arbitrariness of $\hat{t}\in[(N+1)T_{0},\tau]$, we have

$\displaystyle\|z_{i}(t,\cdot)-z_{i}^{(T_{*})}\|_{C^{0}}\leq\hbar_{3}C_{S}^{*}\epsilon\xi^{N+1},\quad\forall
t\in[(N+1)T_{0},\tau].$ (6.13)

Using (3.1) and (4.8), we get

$\displaystyle w_{1}-w_{1}^{(T_{*})}$ $\displaystyle=$
$\displaystyle\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}z_{1}-\nu_{1}(\phi^{(T_{*})}+\Phi)(z_{1}-z_{1}^{(T_{*})})$
$\displaystyle+\frac{\beta}{2}\nu_{1}(\Phi)(\phi_{1}-\phi_{1}^{(T_{*})})+\frac{\beta}{2}\Big{(}\nu_{1}(\phi+\Phi)-\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{)}(\phi_{1}+\phi_{2})$
$\displaystyle+\frac{\beta}{2}\nu_{1}(\phi^{(T_{*})}+\Phi)\Big{(}(\phi_{1}+\phi_{2})-(\phi_{1}^{(T_{*})}+\phi_{2}^{(T_{*})})\Big{)}$
$\displaystyle+\frac{\beta}{2}\nu_{1}(\Phi)(\phi_{2}-\phi_{2}^{(T_{*})}),$

then by (1.2), (2.11), (6.1)-(6.3) and (6.13), we have

$\displaystyle\|w_{1}(t,\cdot)-w_{1}^{(T_{*})}(t,\cdot)\|$ $\displaystyle\leq$
$\displaystyle A_{0}\hbar_{3}C_{S}^{*}\epsilon\xi^{N+1}+C\epsilon
C_{S}\epsilon\xi^{N}+2\beta_{*}A_{0}C_{S}\epsilon\xi^{N}$ $\displaystyle\leq$
$\displaystyle A_{0}C_{S}^{*}\epsilon\xi^{N+1}.$ (6.14)

This indicates that we complete the proof of Theorem 2.4.

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##  Appendix A Gas-phase metallicities

We derive the gas phase metallicity of hCOSMOS galaxies following the method
described in Zahid et al. (2013) and Sohn et al. (2019), which is based on
measuring the emission line strengths. We first derive the model continuum of
each galaxy based on the stellar population synthesis model by Bruzual &
Charlot (2003). Then, we fit each emission line with a Gaussian from the
continuum subtracted spectrum. The uncertainties in the line flux measurements
were also computed using standard error propagation from the uncertainties in
the spectrum. We then apply dust extinction correction based on Cardelli et
al. (1989) extinction curve.

We apply the Kobulnicky & Kewley (2004) technique that determines the gas
metallicities (i.e., 12 + log (O/H)) of galaxies. This technique uses $R_{23}$
and $O_{32}$ indices that are defined as:

$R_{23}=\frac{[\rm OII]\lambda 3727+[\rm OIII]\lambda 4959+[\rm OIII]\lambda
5007}{H\beta},$ (7)

and

$O_{32}=\frac{[\rm OIII]\lambda 4959+[\rm OIII]\lambda 5007}{[\rm OII]\lambda
3727}.$ (8)

Here, we use the line fluxes of each emission line. We note that [OII]$\lambda
3727$ indicates the sum [OII]$\lambda 3726+\lambda 3729$ doublet, which is
hardly resolved in Hectospec spectra. We also use 1.33 times [OIII]$\lambda
5007$ as the sum of [OIII] line flux based on the assumption that the flux
ratio [OIII]$\lambda 4959$ / [OIII]$\lambda 5007$ is equal to 3 (Osterbrock &
Ferland 2006). We finally determine the metallicity based on the relative
positions of $R_{23}$ and $O_{23}$ indices with respect to model grids with an
intrinsic measurement uncertainty of $\sim 0.1$ dex (Kobulnicky & Kewley
2004). All of our sources lack $H\alpha$ while vast majority of sources
($>85\%$) have $H\beta$ emission strength of $\sim 0.2-0.3$ dex lower than the
median of SF hCOSMOS galaxies, confirming their quiescent nature.

##  Appendix B SED fitting systematics

In order to evaluate our SED modelling method and explore the eventual biases,
we produce the simulated data set and fit it using the exact same method that
we applied to our observed galaxies. The purpose of using simulations is to
analyse eventual observational effects on our SED fitting results. To achieve
this goal, we use the functionality available in CIGALE to create mock
catalogue of objects for each galaxy for which the physical parameters are
known. We build the simulated sample by adopting the best-fit SED model for
each fitted QG from our sample. As a result, the procedure gives one
artificial model per galaxy. The best SED model per object (with known
parameters) is then integrated through the same set of filters as in our
observational sample.We then perturb the input fluxes from the best SEDs by
adding a randomised noise, following a Gaussian distribution with $\sigma$
corresponding to the observed uncertainty per each photometric band. The SED
fitting of galaxies from the mock catalogue is further performed with the
exact same choice of physical models and their input parameters as for our
real data. This procedure allows to compare the results of the Bayesian
analysis provided by CIGALE on the mock catalogue to the input parameters used
to build it.

The difference (on a log-scale) between the input physical properties and the
best output parameters of the simulated catalogue is shown in Figure B.1. All
values presented on y-axes are plotted as a function of input $D_{\rm n}4000$,
as we want to ensure that our fitting methods does not produce significant
systematics as a function of stellar population age. We consider that a strong
constraint on a given parameter is obtained if difference between the input
and output approaches zero, followed by the small average dispersion. We find
that for all the physical quantities analysed in this work, the dispersion of
recovered values (input-output) follows the normal distribution, with more
than $80\%$ of sources lying within the mean offset of $\pm 0.25$. As
expected, slightly dispersed distribution is observed for $t_{\rm quench}$.
Nevertheless, despite the known difficulty in fully constraining this
parameter (see e.g., Ciesla et al. 2021), the overall trend with $D_{\rm
n}4000$ is flat, which assures that our choice of SFH is well-suited for most
of our QGs. Therefore, we conclude that our SED fitting procedure properly
recovers the true physical properties of our objects and does not introduce
any significant systematics.

Figure 10: For each panel: results of our mock analysis where we quantify the
difference between the input parameters used to build the simulated catalogue
in CIGALE, and results of the SED fitting of this mock catalogue. The offset
between the simulated and ”observed” physical parameters with CIGALE is
expressed on y-axis as a $\log$ scale. From top to bottom: stellar mass, dust
mass and time since quenching modelled with use of a flexible star formation
history. The offset between the ”true” and ”recovered” values is evaluated as
a function of input $D_{\rm n}4000$ from simulated catalogues. We consider
that a good constraint on a given parameter is obtained if difference between
the input and output approaches zero, which is indicated by the solid blue
lines. The trends following the binned means are displayed with red circles
with corresponding $1\sigma$ errors, while dotted red lines represent 0.25 dex
offset from the zero. Our analysis shows a good overall agreement between the
input and output values even in simulated galaxies with higher $D_{\rm
n}4000$.

##  Appendix C Testing the different selections of QGs

As demonstrated by many studies investigating QGs, all samples that meet the
certain quiescent galaxy criteria produce some level of contamination from SF
interlopers (see i.e., Moresco et al. 2013, Siudek et al. 2018, Leja et al.
2019, Bisigello et al. 2020, Zhang et al. 2022). Here we test up to which
level our results may depend on the method used for selecting the dusty QGs.
To accomplish this, we apply some additional, commonly used definitions of
quiescence to the main sample of 548 QGs analysed in this work. Namely, we
consider QG selection based on UVJ rest-frame colours (e.g., Muzzin et al.
2013 and Schreiber et al. 2015), and selection based on more conservative
spectroscopic criteria ($D_{\rm n}4000>1.6$, e.g., Brinchmann et al. 2004,
Gallazzi et al. 2005). In this way, we quantify the level of potential
contaminants, assuming that our alternative selection criteria is providing a
”clean” sample of QGs. Normally, this is a very conservative assumption, as it
is known that no selection can return a pure sample of QGs. For example, for
the full hCOSMOS catalogue, Damjanov et al. (2018) used the selection proposed
by Muzzin et al. (2013) to separate SF and QGs. They found $\sim 10\%$ of
objects with $D_{\rm n}4000<1.5$ that would be considered QGs based on UVJ
selection. In contrast, for the colour-selected SF galaxies, they discover
$13\%$ of objects that would be considered QG based on spectroscopic criteria
($D_{\rm n}4000>1.5$).

Here we perform the similar comparison with the goal to check if selection of
dusty QGs introduces significantly larger number of contaminants. For the main
UVJ colour selection, we consider Muzzin et al. 2013, who proposed the
following criteria to separate star-forming and passively evolving galaxies at
$0<z<0.6$:

$\displaystyle U-V>1.3$ (9) $\displaystyle V-J<1.5$ $\displaystyle
U-V>0.88\times(V-J)+0.69$

We also check another popular, redshift invariant UVJ selection proposed in
Schreiber et al. (2015). It is defined as:

$\displaystyle U-V>1.3$ (10) $\displaystyle V-J<1.6$ $\displaystyle
U-V>0.88\times(V-J)+0.49$

In this analysis, we stay conservative and consider a ”contaminant” any
outlier at the intersection of our main selection ($D_{\rm n}4000>1.5$) and
the alternative one. We then compare the resulting distributions of some key
physical parameters (specific dust masses, stellar ages, and the offset from
the MS) for such selected QG samples to the distributions from our parent
sample presented in the main text. Results of this analysis are shown in Table
C.1 and Fig.C.1.

Regardless of the criteria, it is evident that the vast majority of our 548
dusty QGs reside in the quiescent population region defined by galaxy rest-
frame UVJ colours. The fraction of our dusty QGs that satisfy these additional
criteria is $83\%$ for the UVJ selection of Muzzin et al. (2013), and $94\%$
in the case of the UVJ selection proposed by Schreiber et al. (2015).
Furthermore, if we impose more strict spectroscopic criterion ($D_{\rm
n}4000>1.6$), we would narrow down our parent sample to 420 objects, or $77\%$
of the initial sample. Therefore, the sample ”purity” in all three cases is
very high, ranging from $77\%$ to $94\%$ (as the fraction of contaminants
varies between $6-23\%$, see Table C.1). We thus ensure that the selection of
IR-detectable ”dusty QGs” does not produce a significant impact on the sample
purity compared to the samples that have no IR-detection. It is worth noting
that the selection by UVJ criteria proposed in Schreiber et al. (2015) has
recently been applied for sampling the ”dusty QGs” at $z\sim 1.5$ in the
GOODS-S field (Blánquez-Sesé et al. 2023). Therefore, our parent sample, which
almost entirely satisfies that selection, can be used as a benchmark for
probing the dust evolution against the stacking samples analysed at
higher-$z$’s.

Table 2: Cross-section between the parent sample of 548 QGs with other
selection criteria of QGs

Quiescent selection criteria | Number of cross-matches |  $\%$ of outliers
---|---|---
| with the main sample |
UVJ (Muzzin et al. 2013) | 453 | $17\%$
UVJ (Schreiber et al. 2015) | 518 | $6\%$
$D_{\rm n}4000>1.6$ | 420 | $23\%$

Figure 11: Upper panel: Distributions of $M_{\rm dust}/M_{\star}$,
$\Delta_{\rm MS}$ and stellar age for our parent sample (solid black line),
against the samples satisfying additional criteria of quiescence: UVJ
selection proposed in Muzzin et al. (2013) (dark red line) and $D_{\rm
n}4000>1.6$ (orange line); Lower panel: Positions of 548 dusty QGs from this
work in the UVJ rest-frame colour plane. Regions defined with solid and dashed
lines correspond to criteria indicated in the legend. Coloured circles
represent QGs with $1.5<D_{\rm n}4000<1.6$ and can be considered potential
contaminants; right: Evolution of specific dust mass as a function of
$M_{\star}$. The meaning of symbols is the same as in the left panel. Best
linear fits are shown for eQGs and sQGs for two different cases: the main
selection that includes 548 QGs (dark grey lines), and selection fulfilling
stricter spectroscopic criterion $D_{\rm n}4000>1.6$ (salmon coloured lines).

Upper panel of Fig.C.1 shows that, irrespective of applied definitions of
quiescence, specific dust masses, stellar ages and $\Delta_{\rm MS}$ follow
the same distribution as our main sample of QGs selected as $D_{\rm
n}4000>1.5$. Strong agreements in dust-related properties among samples
selected in different ways, ensure that the selection of IR-detectable ”dusty
QGs” does not produce significant impact on the sample purity. As expected,
removal of these outliers produces a relative decrease in the number of
recently quenched objects from the tail the specific dust mass distribution.
However, we check and find that ”outliers” have all characteristics of QGs, no
detection of $H\alpha$ emission lines, and very prominent negative offset from
the MS (median of $\log(\Delta_{\rm MS})=-0.82$). As illustrated in the lower
right panel of Fig.C.1, our conclusions regarding morphological impact on the
evolution of dust-related parameters are entirely preserved even if we change
the selection criteria. Therefore, we conclude that the findings and
conclusions in this paper are not affected by the definition of quiescence.


##

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