# Exponential stabilization and continuous dependence of solutions on initial
data in different norms for space-time-varying linear parabolic PDEs

Qiaoling Chen1<EMAIL_ADDRESS>Jun Zheng1,2
<EMAIL_ADDRESS>Guchuan Zhu2<EMAIL_ADDRESS>1School of
Mathematics, Southwest Jiaotong University, Chengdu 611756, Sichuan, China
2Department of Electrical Engineering, Polytechnique Montréal, P.O. Box 6079,
Station Centre-Ville, Montreal, QC, Canada H3T 1J4

###### Abstract

For an arbitrary parameter $p\in[1,+\infty]$, we consider the problem of
exponential stabilization in the spatial $L^{p}$-norm, and $W^{1,p}$-norm,
respectively, for a class of anti-stable linear parabolic PDEs with space-
time-varying coefficients in the absence of a Gevrey-like condition, which is
often imposed on time-varying coefficients of PDEs and used to guarantee the
existence of smooth (w.r.t. the time variable) kernel functions in the
literature. Then, based on the obtained exponential stabilities, we show that
the solution of the considered system depends continuously on the
$L^{p}$-norm, and $W^{1,p}$-norm, respectively, of the initial data. In order
to obtain time-independent (and thus sufficiently smooth) kernel functions
without a Gevrey-like condition and deal with singularities arising in the
case of $p\in[1,2)$, we apply a combinatorial method, i.e., the combination of
backstepping and approximation of Lyapunov functionals (ALFs), to stabilize
the considered system and establish the continuous dependence of solutions on
initial data in different norms.

###### keywords:

Exponential stability; continuous dependence; backstepping; approximation of
Lyapunov functional; parabolic PDEs; time varying system.

, ,

##  1 Introduction

In the last decades, the backstepping method has proved to be a powerful tool
for the boundary control of distributed parameter systems governed by $1$-D
partial differential equations (PDEs); see, e.g., [1]. The main idea of the
method is to transform the anti-stable PDEs into stable target systems by
using a boundary feedback control and a change of variable of the PDEs, i.e.,
two integral transformations, so as to eliminate the anti-stable term of the
considered PDEs. The process of conducting stability analysis via this method
mainly consists of two steps: (i) proving the existence of the needed
transformation and its inverse in a certain function space, which are often
accordant with the existence and regularity of two kernel functions, and can
be proceeded in the standard way; and (ii) establishing the stability
estimates for the target systems, which may be based on different methods of
PDEs or functional analysis.

In the context of parabolic systems, significant progress has been obtained in
exponential stabilization in $L^{2}$-norm or $H^{1}$-norm for linear parabolic
PDEs by using the method of backstepping; see, e.g., [1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14]. In general, when transformations are applied, the
target systems are expected to be in a simple form, whose stability can be
established easily. For example, the target systems usually have the following
form:

$\displaystyle u_{t}(x,t)=u_{xx}(x,t)-\lambda u(x,t),$ (1)

where $\lambda$ is an arbitrary constant; see, e.g., [1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11]. In particular, for $\lambda>0$, the stability in different norms
of (1) has been well studied, and is used together with the inverse
transformations to obtain the stability in $L^{2}$-norm or $H^{1}$-norm for
the original parabolic PDEs. Moreover, since the systems are linear, it is
clear that the stability in $L^{2}$-norm or $H^{1}$-norm implies the
continuous dependence of solutions on the initial data in $L^{2}$-norm or
$H^{1}$-norm.

It should be mentioned that, for parabolic PDEs with time-varying
coefficients, certain Gevrey-like condition, i.e., the time-varying
coefficients of the reaction terms are supposed to be in Gevrey class, is
often imposed to guarantee the existence of sufficiently smooth (w.r.t. the
time variable) kernel functions when the method of backstepping is applied;
see, e.g., [6, 9, 10, 11]. It is reasonable to believe that the method of
backstepping along with the Gevrey-like condition can be adopted to the
exponential stabilization in $L^{p}$-norm or $W^{1,p}$-norm, and to the
establishment of continuous dependence of solutions on the initial data in
$L^{p}$-norm or $W^{1,p}$-norm, for linear parabolic PDEs with space-time-
varying coefficients whenever $p\in[1,+\infty]$. Nevertheless, it is
challenging to obtain sufficiently smooth (w.r.t. the time variable) kernel
functions via backstepping if no Gevrey-like condition is imposed on time-
varying coefficients. Indeed, for a target system chosen as (1), the equations
of kernel functions are relevant to the reaction coefficients of the original
systems, and the proof of existence and higher regularity of kernel functions
depends strongly on the application of the property of the reaction
coefficients, namely, a Gevrey-like condition seems to be needed; see, e.g.,
[6, 9, 10, 11].

An alternative is to expect that the equations of kernel functions are
independent of the time variable, which may extensively reduce the
complications of deriving kernel functions. In this case, a target system
should have the following form:

$\displaystyle u_{t}(x,t)=u_{xx}(x,t)-\lambda(x,t)u(x,t),$ (2)

where $\lambda(x,t)>0$ depends on the reaction coefficients of the original
system, and hence is space-time varying. However, for space-time varying
$\lambda(x,t)$, it is difficult to establish the stability and the continuous
dependence of solutions on the initial data involved with the spatial
$L^{p}$-norm or $W^{1,p}$-norm for system (2) whenever $p\in[1,+\infty]$.

Let us take an example by considering system (2) that is defined over
$(0,1)\times(0,+\infty)$, and with homogeneous Dirichlet boundary conditions
and $L^{p}$-initial data $u_{0}$. It is well-known that the Lyapunov method is
a well-suited tool for stability analysis of parabolic PDEs. In particular,
for the linear system (2), the functional

$V(u):=\int_{0}^{1}|u(x,t)|^{p}\text{d}x$

is an suitable Lyapunov candidate that can be used to establish the
exponential stability in $L^{p}$-norm having the form

$\displaystyle\|u[t]\|_{L^{p}(0,1)}\leq C_{1}e^{\sigma
t}\|u_{0}\|_{L^{p}(0,1)},\forall t\geq 0,$ (3)

when $p\geq 2$, where $C_{1}>0$ and $\sigma>0$ are some constants; see, e.g.,
[1, 15].

In the case of $p\in[1,2)$, it may be expected that the functional $V(u)$
could be used to establish the stability in $L^{p}$-norm of system (2).
However, as indicated in [16, 17], the choice of $V(u)$ may lead to
singularities when $p\in[1,2)$. Indeed,

  1. $\bullet$

for $p=1$, due to the fact that the function $g(s):=|s|$ is not differentiable
at $s=0$, the functional
${\frac{\text{d}}{\text{d}t}\int_{0}^{1}|u|\text{d}x}$ is singular;

  2. $\bullet$

for $1<p<2$, by integrating by parts, it holds (formally) that

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{0}^{1}|u|^{p}\text{d}x=$
$\displaystyle p\int_{0}^{1}|u|^{p-1}\text{sgn}(u)u_{t}\text{d}x$
$\displaystyle=$ $\displaystyle
p\int_{0}^{1}|u|^{p-1}\text{sgn}(u)(u_{xx}-\lambda u)\text{d}x$
$\displaystyle=$
$\displaystyle-p\int_{0}^{1}\frac{\text{d}}{\text{d}u}(|u|^{p-1}\text{sgn}(u))u_{x}^{2}\text{d}x-p\int_{0}^{1}\lambda|u|^{p}\text{d}x.$

But $\frac{\text{d}}{\text{d}u}(|u|^{p-1}\text{sgn}(u))$ is singular at $u=0$,
where $\text{sgn}(\cdot)$ denotes the standard sign function.

It may be also expected that the stability in $L^{p}$-norm could be
established by using the Sobolev embedding $L^{2}(0,1)\hookrightarrow
L^{p}(0,1)$. For instance, for $p\in[1,2)$, it follows from
$L^{2}(0,1)\hookrightarrow L^{p}(0,1)$ and (3) that

$\displaystyle\|u[t]\|_{L^{p}(0,1)}\leq$ $\displaystyle
C_{2}\|u[t]\|_{L^{2}(0,1)}\leq C_{1}C_{2}e^{\sigma
t}\|u_{0}\|_{L^{2}(0,1)},\forall t\geq 0,$

where $C_{2}>0$ is the Sobolev embedding constant111It can be also determined
by using the Hölder’s inequality.. However, the stability estimate obtained in
this way involves the $L^{2}$-norm of the initial data, which does not imply
the continuous dependence of solutions on the $L^{p}$-norm of the initial
data.

It is worth mentioning that, compared with the upper case of
$p\in[2,+\infty]$, the intermediate case of $p\in(1,2)$ and the lower case of
$p=1$ make the problem more meaningful. For example, many physical models lead
to PDEs with $L^{1}$-data and should be estimated in $L^{1}$-norm, or need to
be shown that the solutions are continuously dependent on the $L^{1}$-data;
see, e.g., [18, 19, 20, 21]. It is also worth noting that the study on the
continuous dependence of solutions on initial data for PDEs is important not
only for the theory of regularity of evolution equations, but also for the
analysis of stability and robustness of PDE systems, e.g., the input-to-state
stability (ISS) and relative stability; see [22] for instance, where by using
the De Giorgi iteration the continuous dependence of solutions on the
$L^{\infty}$-norm of initial data was established for a class of nonlinear
parabolic PDEs with Robin boundary conditions, and used to prove the ISS in
$L^{\infty}$-norm for the considered systems and the stability in
$L^{\infty}$-norm for a class of cascaded nonlinear parabolic systems,
respectively.

Nevertheless, due to the aforementioned difficulties, few literature concerns
the stabilization and continuous dependence of solutions on initial data in
the spatial $L^{p}$-norm (or $W^{1,p}$-norm) for anti-stable parabolic PDEs
with time-varying coefficients when $p$ is restricted to be in $[1,2)$ and
smooth kernel functions are adopted without a Gevrey-like condition in
backstepping. This is the main motivation of the work.

In this paper, without any Gevrey-like condition and for arbitrary
$p\in[1,+\infty]$, we consider the problem of exponential stabilization and
continuous dependence of solutions on initial data for a class of space-time-
varying linear parabolic PDEs, and establish for solutions the estimates
containing the spatial $L^{p}$-norm, and $W^{1,p}$-norm, of the initial data,
respectively. The method adopted in this paper is based on the combination of
backstepping and approximations of Lyapunov functionals (ALFs), among which
the latter were used in, e.g., [16, 17], to analyze stability of nonlinear
PDEs with integrable inputs. More specifically,

  1. $\bullet$

in order to eliminate the anti-stable term of the considered parabolic PDEs,
we design a boundary feedback control by using the method of backstepping;

  2. $\bullet$

in the absence of a Gevrey-like condition, we consider time independent kernel
functions, and transform the original system into certain target system that
has space-time-varying coefficients;

  3. $\bullet$

in order to deal with singularities that may arise in the case of $p\in[1,2)$,
we apply the technique of ALFs to analyze the stability and continuous
dependence of solutions on initial data in different norms for the target
system whenever $p\in[1,+\infty]$.

The main contribution of this paper is to apply such a combinatorial method to
establish various estimates of solutions for a class of anti-stable parabolic
PDEs, and is two-fold:

  1. $\bullet$

exponential stability and continuous dependence of solutions on initial data
are established in different norms;

  2. $\bullet$

kernel functions are allowed to be independent of the time variable, and
therefore the Gevrey-like condition can be avoided.

In the rest of the paper, we introduce first some basic notations. In Section
2, we present the problem formulation, main result and technical line of the
proof. In Section 3, by using the technique of ALFs, we analyze the
exponential stability and continuous dependence of solutions on initial data
in different norms for the target systems. In Section 4, we present a result
of well-posedness and regularity of kernel functions. In Section 5, we prove
the main result stated in Section 2. Some conclusions are given in Section 6.

#### Notation.

Let $\mathbb{N}_{0}$, $\mathbb{N}$, $\mathbb{R}$, $\mathbb{R}_{\geq 0}$ and
${\mathbb{R}_{>0}}$ be the set of nonnegative integers, positive integers,
real numbers, nonnegative real numbers, and positive real numbers,
respectively.

For $q\in[1,+\infty]$, the notations of $L^{q}(0,1)$ and $W^{1,q}(0,1)$ denote
the standard Lebesgue spaces and Sobolev spaces with elements defined over
$(0,1)$ and equipped with the norm ${\|v\|_{q}:=}\|v\|_{L^{q}(0,1)}$ and
${\|v\|_{1,q}}:=\|v\|_{W^{1,q}(0,1)}$, respectively; see, e.g., [23].

Let
$L^{\infty}((0,1)\times{\mathbb{R}_{>0}}):=\\{v:(0,1)\times{\mathbb{R}_{>0}}\rightarrow\mathbb{R}|~{}v$
is measurable and satisfies
$\operatorname{ess\,sup}_{(y,t)\in(0,1)\times{\mathbb{R}_{>0}}}|v{(y,t)}|<+\infty\\}$.

For $v:(0,1)\times{\mathbb{R}_{>0}}\rightarrow\mathbb{R}$, the notation $v[t]$
(or $v[y]$) denotes the profile at certain $t\in{\mathbb{R}_{>0}}$ (or
$y\in(0,1)$), i.e., $v[t](y)=v(y,t)$ (or $v[y](t)=v(y,t)$) for all $y\in(0,1)$
(or $t\in\mathbb{R}_{>0}$). For $T\in{\mathbb{R}_{>0}}$ and a Banach space $X$
equipped with the norm $\|\cdot\|_{X}$, let
$C\left([0,T];X\right):=\\{v:(0,1)\times[0,T]\rightarrow\mathbb{R}|~{}v[t]\in
X$ and satisfies $\max_{t\in[0,T]}\|v[t]\|_{X}<+\infty\\}$, which is a Banach
space equipped with the norm
$\|v\|_{C\left([0,T];X\right)}:=\max_{t\in[0,T]}\|v[t]\|_{X}$.

For $Q={\mathbb{R}_{\geq 0}}$ (or $[0,1]$), let $C^{1}\left([0,1]\times
Q\right):=\\{v:[0,1]\times Q\rightarrow\mathbb{R}|~{}v$ has continuous
derivatives up to order $1$$\\}$.

For $l\in{\mathbb{R}_{>0}}$ and $T\in{\mathbb{R}_{>0}}$, the notions of
$\mathcal{H}^{2+l}\left([0,1]\right)$ and
$\mathcal{H}^{2+l,1+\frac{l}{2}}\left([0,1]\times[0,T]\right)$ denotes Hölder
spaces that are defined by [24, Chapter I].

##  2 Problem Setting, Main Result and Technical Line

###  2.1 Problem Setting and Main Result

Given certain initial data $w_{0}$ , we study the problem of stabilization and
continuous dependence of solutions on initial data in different norms for the
linear parabolic PDEs with space and time dependent coefficients:

$\displaystyle w_{t}(x,t)=$ $\displaystyle
w_{xx}(x,t)+c(x,t)w(x,t)+\int_{0}^{x}w(y,t)f(x,y)\text{d}y,(x,t)\in(0,1)\times{\mathbb{R}_{>0}},$
(4a) $\displaystyle w_{x}(0,t)=$ $\displaystyle 0,t\in{\mathbb{R}_{>0}},$ (4b)
$\displaystyle w_{x}(1,t)=$ $\displaystyle U(t),t\in{\mathbb{R}_{>0}},$ (4c)
$\displaystyle w(x,0)=$ $\displaystyle w_{0}(x),x\in(0,1),$ (4d)

where $c:[0,1]\times{\mathbb{R}_{\geq 0}}\rightarrow\mathbb{R}$ and
$f:[0,1]\times[0,1]\rightarrow\mathbb{R}$ are given functions, and $U$ is the
control input to be determined to stabilize the system.

For the system (4), the reactive coefficient $c$ often represents
characteristic quantities for heat exchange and reaction, e.g., [25, 26].
Moreover, if $c$ is positive and sufficiently large, the open-loop system
(i.e., $U(t)=0$) is unstable; see, e.g., [1].

Throughout this paper, we assume that

$\displaystyle{p\in[1,+\infty)}\ \text{is\ an\ arbitrary\ constant,\ or}\
p=+\infty,$ $\displaystyle f\in C^{1}\left([0,1]\times[0,1]\right),c\in
C^{1}\left([0,1]\times{\mathbb{R}_{\geq 0}}\right)\cap
L^{\infty}((0,1)\times{\mathbb{R}_{>0}}).$

Moreover, assume that $c$ has a form

$\displaystyle c(x,t):=$ $\displaystyle c_{1}(x)+c_{2}(t)$ (5)

with some functions
$c_{1}:[0,1]\rightarrow\mathbb{R}~{}\text{and}~{}c_{2}:{\mathbb{R}_{\geq
0}}\rightarrow\mathbb{R}$.

It is worth noting that stabilization of anti-stable parabolic PDEs under such
kind of structural condition (5) was firstly considered in [11], where
boundary feedback controls were designed to exponentially stabilize equation
(4a) (with $f(x,t)\equiv 0$) in $L^{2}$ and $W^{1,2}$ norms for $c(x,t)\equiv
c_{1}(x)$ (see also [12] with $f(x,t)\not\equiv 0$), and for $c(x,t)\equiv
c_{2}(t)$, respectively.

Let $\lambda_{0}$ be an arbitrary constant satisfying

$\displaystyle\lambda_{0}>\sup_{(x,t)\in(0,1)\times{\mathbb{R}_{>0}}}c(x,t).$
(6)

Let $D:=\\{(x,y)\in\mathbb{R}^{2}|~{}0\leq y\leq x\leq 1\\}$ and

$\displaystyle{\mu(x,y):=\lambda_{0}-c_{1}(x)+c_{1}(y)},\forall(x,y)\in D.$
(7)

In order to design a boundary feedback law to stabilize system (4) in the
sense of different norms, we consider the kernel function
$k:D\rightarrow\mathbb{R}$ that is independent of the time variable $t$ and
satisfies

$\displaystyle k_{xx}(x,y)-k_{yy}(x,y)=$
$\displaystyle{\mu(x,y)}k(x,y)+f(x,y)+\int_{y}^{x}k(x,z)f(z,y)\text{d}z,$ (8a)
$\displaystyle 2\frac{\text{d}}{\text{d}x}\left(k(x,x)\right)=$
$\displaystyle{\lambda_{0}},$ (8b) $\displaystyle k_{y}(x,0)=$ $\displaystyle
0,$ (8c) $\displaystyle k(0,0)=$ $\displaystyle 0,$ (8d)

where $\frac{\text{d}}{\text{d}x}\left(k(x,x)\right):=k_{x}(x,x)+k_{y}(x,x)$.
The existence and regularity of the kernel function $k$ that satisfies (8)
will be specified in Section 4.

Define the boundary feedback control law as

$\displaystyle U(t):=-k(1,1)w(1,t)-\int_{0}^{1}k_{x}(1,y)w(y,t)\text{d}y.$ (9)

Let $\beta\in(0,1)$ be a fixed constant. Define for the initial data the set

$\displaystyle W_{0}:=\bigg{\\{}$ $\displaystyle
w|~{}w\in\mathcal{H}^{2+\beta}([0,1]),w_{x}(0)=0,w_{x}(1)=-k(1,1)w(1)-\int_{0}^{1}k_{x}(1,y)w(y)\text{d}y\bigg{\\}}.$

Concerning with the well-posedness and stability in different norms of system
(4), we have the following theorem, which is the main result obtained in this
paper.

######  Theorem 2.1.

Given initial data in $W_{0}$ and considering system (4) under the feedback
control law (9), the following statements hold true:

  1. (i)

system (4) admits a unique classical solution $w$, which belongs to
$\mathcal{H}^{2+\beta,1+\frac{\beta}{2}}([0,1]\times[0,T])$ and has the
derivative $w_{xt}\in L^{2}((0,1)\times(0,T))$ for any $T\in\mathbb{R}_{>0}$;

  2. (ii)

system (4) is exponentially stabilized in $L^{p}$-norm, and $W^{1,p}$-norm,
having the estimates

$\displaystyle\|w[t]\|_{p}\leq
C_{1}e^{-\underline{\lambda}t}\|w_{0}\|_{p},~{}\forall t\in{\mathbb{R}_{>0}},$
(10)

and

$\displaystyle\|w[t]\|_{{1,p}}\leq
C_{2}e^{-\underline{\lambda}t}\|w_{0}\|_{{1,p}},~{}\forall
t\in{\mathbb{R}_{>0}},$ (11)

respectively, where $w_{0}\in W_{0}$ denotes the initial data,
$\underline{\lambda}$ is a positive constant given by

$\displaystyle\underline{\lambda}:=\inf_{(x,t)\in(0,1)\times{\mathbb{R}_{>0}}}(\lambda_{0}-c(x,t)),$
(12)

and $C_{1},C_{2}$ are positive constants depending only on $p,k$ and the
solution $l$ of (17) for $p\in[1,+\infty)$, and on $k$ and $l$ for
$p=+\infty$, respectively;

  3. (iii)

the solution of system (4) is continuously dependent on the $L^{p}$-norm, and
$W^{1,p}$-norm, of the initial data, having the estimates

$\displaystyle\|w_{1}-w_{2}\|_{C([0,T];L^{p}(0,1))}\leq
C_{1}\|w_{01}-w_{02}\|_{p},\forall T\in\mathbb{R}_{>0},$

and

$\displaystyle\|w_{1}-w_{2}\|_{C([0,T];W^{1,p}(0,1))}\leq
C_{2}\|w_{01}-w_{02}\|_{1,p},\forall T\in\mathbb{R}_{>0},$

respectively, where $w_{i}$ denotes the solution of system (4) with the
initial data $w_{0i}\in W_{0}(i=1,2)$, and $C_{1},C_{2}$ are the same as in
(ii).

######  Remark 2.1.

It is worth mentioning that under certain Gevrey-like conditions of the time-
varying coefficient $c$, e.g., with the assumption that $c[x]\in
C^{\infty}({\mathbb{R}_{>0}})$ and satisfies the following inequality for some
positive constant $R$ and $i\in\mathbb{N}_{0}$:

$\displaystyle\sup_{t\in{\mathbb{R}_{>0}}}\left|\partial_{t}^{i}c(x,t)\right|\leq
R^{i+1}(i!)^{\alpha},\forall x\in[0,1],$

the exponential stabilization in $L^{2}$-norm was studied in, e.g., [6, 9, 10,
11], for a class of time-varying systems with different boundary feedback
controls. The kernel functions obtained in the existing literature depend on
the time variable and are differentiable (w.r.t. $t$) up to the order of $i$.
Since $i$ can be given arbitrarily, the kernel functions can be sufficiently
smooth.

In this paper, by using a combinatorial method, no Gevrey-like condition is
imposed on the time-varying coefficient $c$, and the stability is established
in $L^{p}$-norm or $W^{1,p}$-norm for any $p\in[1,+\infty]$. In addition, the
kernel functions considered in this paper are independent of the time variable
(and therefore naturally infinitely differentiable w.r.t. $t$), which reduce
the associated computations extensively.

###  2.2 Technical Line

The proof of Theorem 2.1 mainly consists of 3 steps that are included in
Section 4, Section 3 and Section 5, respectively.

Step 1: take into account an equivalent system by using transformations.

Indeed, in order to avoid using a Gevrey-like condition, we transform system
(4) into a target system that has also a space-time-varying coefficient. More
precisely, for $\lambda_{0}$ satisfying (6), let first

$\displaystyle\lambda(x,t):=\lambda_{0}-c\left(x,t\right).$ (13)

Then using the integral transformation

$\displaystyle u(x,t):=w(x,t)+\int_{0}^{x}k(x,y)w(y,t)\text{d}y,$ (14)

we transform system (4) into the following target system:

$\displaystyle
u_{t}(x,t)=u_{xx}(x,t)-\lambda(x,t)u(x,t),(x,t)\in(0,1)\times{\mathbb{R}_{>0}},$
(15a) $\displaystyle u_{x}(0,t)=0,t\in{\mathbb{R}_{>0}},$ (15b) $\displaystyle
u_{x}(1,t)=0,t\in{\mathbb{R}_{>0}},$ (15c) $\displaystyle
u(x,0)=u_{0}(x),x\in(0,1),$ (15d)

where $u_{0}(x):=w_{0}(x)+\int_{0}^{x}k(x,y)w_{0}(y)\text{d}y$.

Conversely, using the integral transformation

$\displaystyle w(x,t):=u(x,t)-\int_{0}^{x}l(x,y)u(y,t)\text{d}y,$ (16)

we transform system (15) into the original system (4), where
$l:D\rightarrow\mathbb{R}$ satisfies

$\displaystyle l_{xx}(x,y)-l_{yy}(x,y)=$
$\displaystyle{\phi(x,y)}l(x,y)+f(x,y)-\int_{y}^{x}l(x,z)f(z,y)\text{d}z,$
(17a) $\displaystyle 2\frac{\text{d}}{\text{d}x}\left(l(x,x)\right)=$
$\displaystyle\lambda_{0},$ (17b) $\displaystyle l_{y}(x,0)=$ $\displaystyle
0,$ (17c) $\displaystyle l(0,0)=$ $\displaystyle 0,$ (17d)

with $\phi(x,y):=-\lambda_{0}-c_{1}(x)+c_{1}(y)$.

The equivalence of the original system (4) and system (15) can be proved in a
direct way. For instance, we only show that if $w$ is the solution of system
(4), then $u$ is the solution of system (15). Indeed, the partial derivatives
of (14) w.r.t. $x$ and $t$ give

$\displaystyle u_{x}(0,t)=$ $\displaystyle w_{x}(0,t)+k(0,0)w(0,t),$
$\displaystyle u_{x}(1,t)=$
$\displaystyle{w_{x}(1,t)+k(1,1)w(1,t)+\int_{0}^{1}k_{x}(1,y)w(y,t)\text{d}y}$

and

$\displaystyle u_{t}(x,t)-u_{xx}(x,t)+\lambda(x,t)u(x,t)$ $\displaystyle=$
$\displaystyle
w(x,t)\bigg{(}-k_{y}(x,x)-\frac{\text{d}}{\text{d}x}\left(k(x,x)\right)-k_{x}(x,x){+\lambda(x,t)+c(x,t)}\bigg{)}+\int_{0}^{x}w(y,t)\bigg{(}k_{yy}(x,y)-k_{xx}(x,y)$
$\displaystyle{+\lambda(x,t)k(x,y)+c(y,t)k(x,y)}+f(x,y)+\int_{y}^{x}k(x,z)f(z,y)\text{d}z\bigg{)}\text{d}y$
$\displaystyle+k_{y}(x,0)w(0,t)+\int_{y}^{x}k(x,z)f(z,y)\text{d}z\bigg{)}\text{d}y+k_{y}(x,0)w(0,t)$
$\displaystyle=$ $\displaystyle
w(x,t)\bigg{(}-k_{y}(x,x)-\frac{\text{d}}{\text{d}x}\left(k(x,x)\right)-k_{x}(x,x){+\lambda_{0}}\bigg{)}+\int_{0}^{x}w(y,t)\bigg{(}k_{yy}(x,y)-k_{xx}(x,y)$
$\displaystyle+\left(\lambda_{0}-c_{1}(x)-c_{2}(t)\right)k(x,y)+\left(c_{1}(y)+c_{2}(t)\right)k(x,y)+f(x,y)+\int_{y}^{x}k(x,z)f(z,y)\text{d}z\bigg{)}\text{d}y+k_{y}(x,0)w(0,t)$
$\displaystyle=$ $\displaystyle
w(x,t)\bigg{(}-2\frac{\text{d}}{\text{d}x}\left(k(x,x)\right){+\lambda_{0}}\bigg{)}+\int_{0}^{x}w(y,t)\bigg{(}k_{yy}(x,y)-k_{xx}(x,y)+\mu(x,y)k(x,y)$
$\displaystyle+f(x,y)+\int_{y}^{x}k(x,z)f(z,y)\text{d}z\bigg{)}\text{d}y+k_{y}(x,0)w(0,t)$
$\displaystyle=$ $\displaystyle 0.$

Since $k$ satisfies (8) and the boundary control law $U(t)$ is defined by (9),
we deduce that $u$ satisfies (15).

Step 2: establish stability in $L^{p}$-norm, and $W^{1,p}$-norm, for the
target system (15) with initial data belonging to $W_{0}$ by using the
technique of ALFs.

As mentioned in Introduction, singularities may appear in the Lyapunov
arguments when $p\in[1,2)$. To overcome this obstacle, we apply the technique
of ALFs that was introduced in [16, 17] for the analysis of integral input-to-
state stability of nonlinear parabolic PDEs with external disturbances.
Specifically, for any $\tau\in{\mathbb{R}_{>0}}$, let

$\displaystyle\rho_{\tau}(s):=\left\\{\begin{aligned} &|s|,\ |s|\geq\tau,\\\
&-\frac{s^{4}}{8\tau^{3}}+\frac{3s^{2}}{4\tau}+\frac{3\tau}{8},\
|s|<\tau.\end{aligned}\right.$ (18)

that is $C^{2}$-continuous in $s$ and satisfies for any $s\in\mathbb{R}$:

$\displaystyle\rho_{\tau}^{\prime}(0)=0,~{}0\leq|s|\leq\rho_{\tau}(s),~{}|\rho_{\tau}^{\prime}(s)|\leq
1,$ (19a) $\displaystyle
0\leq\rho_{\tau}^{\prime\prime}(s)=\left\\{\begin{aligned} &0,\
|s|\geq\tau,\\\ &\frac{3}{2\tau}\left(1-\frac{s^{2}}{\tau^{2}}\right),\
|s|<\tau,\end{aligned}\right.$ (19b) $\displaystyle
0\leq\rho_{\tau}(s)-\frac{3\tau}{8}\leq\rho_{\tau}^{\prime}(s)s\leq\rho_{\tau}(s)\leq|s|+\frac{3\tau}{8}.$
(19c)

The second inequality of (19a) and the fourth inequality of (19c) guarantee
that

$\displaystyle\int_{0}^{1}\rho_{\tau}^{p}(v)\text{d}x\rightarrow\int_{0}^{1}|v|^{p}\text{d}x~{}~{}\text{as}~{}~{}\tau\rightarrow
0^{+},$
$\displaystyle\int_{0}^{1}\rho_{\tau}^{p}(v_{x})\text{d}x\rightarrow\int_{0}^{1}|v_{x}|^{p}\text{d}x~{}~{}\text{as}~{}~{}\tau\rightarrow
0^{+},$

for any $v\in W^{1,p}(0,1)$. Therefore, instead of choosing
$\int_{0}^{1}|v|^{p}\text{d}x$ or
$\int_{0}^{1}|v|^{p}\text{d}x+\int_{0}^{1}|v_{x}|^{p}\text{d}x$ as a Lyapunov
functional, we take into account the functional
$\int_{0}^{1}\rho_{\tau}^{p}(v)\text{d}x$ or
$\int_{0}^{1}\rho_{\tau}^{p}(v)\text{d}x+\int_{0}^{1}\rho_{\tau}^{p}(v_{x})\text{d}x$
to establish the stability in $L^{p}$-norm or $W^{1,p}$-norm for the target
system (15) in the Lyapunov arguments.

Step 3: establish the exponentially stability in $L^{p}$-norm, and
$W^{1,p}$-norm, for the original system (4) with initial data in $W_{0}$ by
using the inverse transformation (16), the estimates of $u$, and the
properties of the kernel functions $k$ and $l$.

##  3 Well-posedness and Stability of the Target System

In accordance with $W_{0}$, we define

$\displaystyle
U_{0}:=\\{u|~{}u\in\mathcal{H}^{2+\beta}([0,1]),u_{x}(0)=u_{x}(1)=0\\}$

for the initial data of the target system (15).

######  Proposition 3.1.

Consider system (15) with initial data in $U_{0}$. The following statements
hold true:

  1. (i)

system (15) admits a unique classical solution $u$, which belongs to
$\mathcal{H}^{2+\beta,1+\frac{\beta}{2}}([0,1]\times[0,T])$ and has the
derivative $u_{xt}\in L^{2}((0,1)\times(0,T))$ for any $T>0$;

  2. (ii)

system (15) is exponentially stable in $L^{p}$-norm, and $W^{1,p}$-norm,
having the estimates

$\displaystyle\|u[t]\|_{p}\leq$ $\displaystyle
e^{-\underline{\lambda}t}\|u_{0}\|_{p},~{}\forall t\in{\mathbb{R}_{>0}},$ (20)

and

$\displaystyle\|u[t]\|_{{1,p}}\leq$ $\displaystyle
e^{-\underline{\lambda}t}\|u_{0}\|_{{1,p}},~{}\forall t\in{\mathbb{R}_{>0}},$
(21)

respectively, where where $u$ denotes the solution of system (15)
corresponding to the initial data $u_{0}\in U_{0}$, $\underline{\lambda}$ is a
positive constant defined by (12);

  3. (iii)

the solution of system (15) depends continuously on the $L^{p}$-norm, and
$W^{1,p}$-norm, of the initial data, having the esimates

$\displaystyle\|u_{1}-u_{2}\|_{C([0,T];L^{p}(0,1))}\leq\|u_{01}-u_{02}\|_{L^{p}(0,1)},$
(22)

and

$\displaystyle\|u_{1}-u_{2}\|_{C([0,T];W^{1,p}(0,1))}\leq\|u_{01}-u_{02}\|_{W^{1,p}(0,1)},$
(23)

respectively, where $u_{i}$ denotes the solution of system (15) corresponding
to the initial data $u_{0i}\in U_{0}$, $i=1,2$.

Proof. The assertion (i) follows from [24, Chap. V, Theorem 7.4] and the proof
of [24, Chap. V, Lemma 7.2].

We adopt the technique of ALFs to prove the assertions (ii) and (iii).

Indeed, let $u$ be the solution of system (15) with the initial data $u_{0}\in
U_{0}$. For any $\tau\in{\mathbb{R}_{>0}}$, let
$\rho_{\tau}:\mathbb{R}\rightarrow\mathbb{R}_{>0}$ be defined by (18).

For $p\in[1,+\infty)$, by integrating by parts, we get

$\displaystyle\frac{1}{p}\frac{\text{d}}{\text{d}t}\int_{0}^{1}\rho^{p}_{\tau}(u)\text{d}x=$
$\displaystyle\int_{0}^{1}\rho^{p-1}_{\tau}(u)\rho_{\tau}^{\prime}(u)u_{t}\text{d}x$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\rho^{p-1}_{\tau}(u)\rho_{\tau}^{\prime}(u)\left(u_{xx}-\lambda
u\right)\text{d}x$ $\displaystyle=$
$\displaystyle(\rho^{p-1}_{\tau}(u)\rho_{\tau}^{\prime}(u)u_{x})|_{x=0}^{x=1}-\int_{0}^{1}\left((p-1)\rho_{\tau}^{p-2}(u)(\rho_{\tau}^{\prime}(u))^{2}+\rho_{\tau}^{p-1}(u)\rho_{\tau}^{\prime\prime}(u)\right)u_{x}^{2}\text{d}x$
$\displaystyle-\int_{0}^{1}\rho_{\tau}^{p-1}(u)\rho_{\tau}^{\prime}(u)\lambda
u\text{d}x.$ (24)

Note that

$\displaystyle\phi(s):=$
$\displaystyle(p-1)\rho_{\tau}^{p-2}(s)(\rho_{\tau}^{\prime}(s))^{2}+\rho^{p-1}_{\tau}(s)\rho_{\tau}^{\prime\prime}(s)\geq
0,{\forall s\in\mathbb{R}.}$ (25)

By (19c), (24) and (25), we deduce that

$\displaystyle\frac{1}{p}\frac{\text{d}}{\text{d}t}\int_{0}^{1}\rho_{\tau}^{p}(u)\text{d}x\leq$
$\displaystyle-\int_{0}^{1}\rho_{\tau}^{p-1}(u)\rho_{\tau}^{\prime}(u)\lambda
u\text{d}x$ $\displaystyle\leq$
$\displaystyle-\int_{0}^{1}{\lambda}\rho_{\tau}^{p-1}(u)\left(\rho_{\tau}(u)-\frac{3\tau}{8}\right)\text{d}x$
$\displaystyle=$
$\displaystyle-\int_{0}^{1}{\lambda}\rho_{\tau}^{p}(u)\text{d}x+\frac{3\tau}{8}\int_{0}^{1}{\lambda}\rho_{\tau}^{p-1}(u)\text{d}x,$

which along with (6), (12) and (13) implies

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{0}^{1}\rho_{\tau}^{p}(u)\text{d}x\leq$
$\displaystyle-\underline{\lambda}p\int_{0}^{1}\rho_{\tau}^{p}(u)\text{d}x+\frac{3}{8}\tau
p\int_{0}^{1}{\lambda}\rho_{\tau}^{p-1}(u)\text{d}x.$ (26)

By Lemma A.1 in Appendix, for any $t\in[0,T]$, it holds that

$\displaystyle\int_{0}^{1}\rho_{\tau}^{p}(v(x,t))\text{d}x\leq$ $\displaystyle
e^{-\underline{\lambda}pt}\int_{0}^{1}\rho_{\tau}^{p}(u_{0}(x))\text{d}x+\tau\int_{0}^{t}e^{-\underline{\lambda}p(t-s)}\psi_{1}(s)\text{d}s,$
(27)

where
$\psi_{1}(s):=\frac{3}{8}p\int_{0}^{1}{\lambda}(x,s)\rho_{\tau}^{p-1}(u(x,s))\text{d}x$.

Letting $\tau\rightarrow 0^{+}$, we infer from (27) that

$\displaystyle\|u[t]\|_{p}\leq$ $\displaystyle
e^{-\underline{\lambda}t}\|u_{0}\|_{p},~{}\forall t\in[0,T],$ (28)

which is (20) in the assertion (ii) for $p\in[1,+\infty)$.

Analogously, for $p\in[1,+\infty)$, we calculate

$\displaystyle\frac{1}{p}\frac{\text{d}}{\text{d}t}\int_{0}^{1}\rho_{\tau}^{p}(u_{x})\text{d}x=$
$\displaystyle\int_{0}^{1}\rho_{\tau}^{p-1}(u_{x})\rho_{\tau}^{\prime}(u_{x}){u_{xt}}\text{d}x$
$\displaystyle=$
$\displaystyle\left(\rho_{\tau}^{p-1}(u_{x})\rho_{\tau}^{\prime}(u_{x})u_{t}\right)|_{x=0}^{x=1}\
-\int_{0}^{1}\left(\rho_{\tau}^{p-1}(u_{x})\rho_{\tau}^{\prime}(u_{x})\right)_{x}u_{t}\text{d}x$
$\displaystyle=$
$\displaystyle-\int_{0}^{1}\left(\rho_{\tau}^{p-1}(u_{x})\rho_{\tau}^{\prime}(u_{x})\right)_{x}\left(u_{xx}-\lambda
u\right)\text{d}x$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}\phi(u_{x})u_{xx}^{2}\text{d}x+\int_{0}^{1}\left(\rho_{\tau}^{p-1}(u_{x})\rho_{\tau}^{\prime}(u_{x})\right)_{x}\lambda
u\text{d}x$ $\displaystyle=$
$\displaystyle-\int_{0}^{1}\phi(u_{x})u_{xx}^{2}\text{d}x-\int_{0}^{1}\rho_{\tau}^{p-1}(u_{x})\rho_{\tau}^{\prime}(u_{x})\lambda
u_{x}\text{d}x.$ (29)

By (19c), (25) and (29), we get

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{0}^{1}\rho_{\tau}^{p}(u_{x})\text{d}x\leq-\underline{\lambda}p\int_{0}^{1}\rho_{\tau}^{p}(u_{x})\text{d}x+\frac{3}{8}\tau
p\int_{0}^{1}{\lambda}\rho_{\tau}^{p-1}(u_{x})\text{d}x.$ (30)

By (26) and (30), we have

$\displaystyle\frac{\text{d}}{\text{d}t}\left(\int_{0}^{1}\rho_{\tau}^{p}(u)\text{d}x+\int_{0}^{1}\rho_{\tau}^{p}(u_{x})\text{d}x\right)\leq$
$\displaystyle-\underline{\lambda}p\int_{0}^{1}\left(\rho_{\tau}^{p}(u)+\rho_{\tau}^{p}(u_{x})\right)\text{d}x+\frac{3}{8}\tau
p\int_{0}^{1}{\lambda}\left(\rho_{\tau}^{p-1}(u)+\rho_{\tau}^{p-1}(u_{x})\right)\text{d}x.$
(31)

By Lemma A.1 in Appendix, for any $t\in[0,T]$, it holds that

$\displaystyle\int_{0}^{1}\left(\rho_{\tau}^{p}(u(x,t))+\rho_{\tau}^{p}(u_{x}(x,t))\right)\text{d}x\leq
e^{-\underline{\lambda}pt}\int_{0}^{1}\left(\rho_{\tau}^{p}(u_{0}(x))+\rho_{\tau}^{p}(u_{x}(x,0))\right)\text{d}x+\tau\int_{0}^{t}e^{-\underline{\lambda}p(t-s)}\psi_{2}(s)\text{d}s,$
(32)

where
$\psi_{2}(s):=\frac{3}{8}p\int_{0}^{1}{\lambda}\left(\rho_{\tau}^{p-1}(u(x,s))+\rho_{\tau}^{p-1}(u_{x}(x,s)\right)\text{d}x$.

Letting $\tau\rightarrow 0^{+}$, we infer from (32) that

$\displaystyle\|u[t]\|_{1,p}\leq$ $\displaystyle
e^{-\underline{\lambda}t}\|u_{0}\|_{1,p},~{}\forall t\in[0,T],$ (33)

which is (21) in the assertion (ii) for $p\in[1,+\infty)$.

Now letting $p\rightarrow+\infty$, by the classical result of functional
analysis, e.g., [27, pp.74], we deduce that (20) and (21) also hold true for
$p=+\infty$.

In order to prove the assertion (iii), letting $v:=u_{1}-u_{2}$ and
$v_{0}:=u_{01}-u_{02}$, we infer from the linearity of system (15) and the
assertion (ii) that

$\displaystyle\|v[t]\|_{p}\leq$ $\displaystyle
e^{-\underline{\lambda}t}\|v_{0}\|_{p},~{}\forall t\in{\mathbb{R}_{>0}},$

and

$\displaystyle\|v[t]\|_{{1,p}}\leq$ $\displaystyle
e^{-\underline{\lambda}t}\|v_{0}\|_{{1,p}},~{}\forall t\in{\mathbb{R}_{>0}},$

hold true for all $p\in[1,+\infty]$. It follows that

$\displaystyle\max_{t\in[0,T]}\|v[t]\|_{p}\leq$
$\displaystyle\|v_{0}\|_{p},\forall T\in\mathbb{R}_{>0},$

and

$\displaystyle\max_{t\in[0,T]}\|v[t]\|_{1,p}\leq$
$\displaystyle\|v_{0}\|_{1,p},\forall T\in\mathbb{R}_{>0}.$

Therefore, the assertion (iii) is true. The proof of Proposition 3.1 is
complete. $\hfill\blacksquare$

##  4 Well-posedness and Regularity of Kernel Functions

In this section, we prove the existence, uniqueness and regularity of kernel
functions $k$ and $l$ by using the method of successive approximation as in,
e.g., [1, 11, 10]. Although the proof is standard, we would like to provide
details for the completeness. In particular, as indicated later in Remark 4.1,
we fix a gap that exists in the proof of uniqueness of kernel functions in the
existing literature.

######  Proposition 4.1.

The problem (8), and (17), has a unique solution $k$ and $l$, respectively,
both of which belong to $\mathcal{H}^{2+\beta}\left(D\right)$.

Proof. Let $\xi:=x+y$ and $\eta:=x-y.$ It follows that $\eta\in[0,1]$ and
$\xi\in[\eta,2-\eta]$. Let

$\displaystyle
G(\xi,\eta):=k(x,y)=k\left(\frac{\xi+\eta}{2},\frac{\xi-\eta}{2}\right).$

Then (8) is changed into

$\displaystyle 4G_{\xi\eta}(\xi,\eta)=$
$\displaystyle{\widetilde{\mu}(\xi,\eta)}G(\xi,\eta)+f\left(\frac{\xi+\eta}{2},\frac{\xi-\eta}{2}\right)+\int_{\frac{\xi-\eta}{2}}^{\frac{\xi+\eta}{2}}\widetilde{\mathcal{G}}(z,\xi,\eta)\text{d}z,$
(34a) $\displaystyle G(\xi,0)=$ $\displaystyle{\frac{\lambda_{0}}{4}}\xi,$
(34b) $\displaystyle G_{\xi}(\xi,\xi)=$ $\displaystyle G_{\eta}(\xi,\xi),$
(34c) $\displaystyle G(0,0)=$ $\displaystyle 0,$ (34d)

where

$\displaystyle{\widetilde{\mu}(\xi,\eta):=\mu(x,y)=\lambda_{0}-c_{1}\left(\frac{\xi+\eta}{2}\right)+c_{1}\left(\frac{\xi-\eta}{2}\right)},$
$\displaystyle\widetilde{\mathcal{G}}(z,\xi,\eta):=f\left(z,\frac{\xi-\eta}{2}\right)G\left(\frac{\xi+\eta}{2}+z,\frac{\xi+\eta}{2}-z\right).$

Integrating (34a) w.r.t. $\eta$ from $0$ to $\eta$ and differentiating (34b)
w.r.t. $\xi$ give

$\displaystyle G_{\xi}(\xi,\eta)=$
$\displaystyle{\frac{\lambda_{0}}{4}}+{{\frac{{1}}{4}}\int_{0}^{\eta}\widetilde{\mu}(\xi,s)G(\xi,s)\text{d}s}+\frac{1}{4}\int_{0}^{\eta}f\left(\frac{\xi+s}{2},\frac{\xi-s}{2}\right)\text{d}s+\frac{1}{4}\int_{0}^{\eta}\int_{\xi}^{\xi+\eta-s}\widehat{G}(\tau,s,\xi)\text{d}\tau\text{d}s,$
(35)

where
$\widehat{G}(\tau,s,\xi):=f\left(\frac{\tau-s}{2},\xi-\frac{\tau+s}{2}\right)G(\tau,s)$.

Integrating (35) w.r.t. $\xi$ from $\eta$ to $\xi$ gives

$\displaystyle G(\xi,\eta)=$ $\displaystyle
G(\eta,\eta)+{\frac{\lambda_{0}}{4}}(\xi-\eta)+{{\frac{{1}}{4}}\int_{\eta}^{\xi}\int_{0}^{\eta}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau$
$\displaystyle+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z.$
(36)

By (34c), we have

$\displaystyle\frac{\text{d}}{\text{d}\xi}\left(G(\xi,\xi)\right)=2G_{\xi}(\xi,\xi).$
(37)

Substituting $\eta=\xi$ into (35), then integrating it over $[0,\xi]$ and
applying (34d) and (37), we obtain

$\displaystyle G(\xi,\xi)=$
$\displaystyle{\frac{\lambda_{0}}{2}}\xi+{{\frac{{1}}{2}}\int_{0}^{\xi}\int_{0}^{\tau}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{2}\int_{0}^{\xi}\int_{0}^{\tau}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau$
$\displaystyle+\frac{1}{2}\int_{0}^{\xi}\int_{0}^{z}\int_{z}^{2z-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z,$

which gives

$\displaystyle G(\eta,\eta)=$
$\displaystyle{\frac{\lambda_{0}}{2}}\eta+{{\frac{{1}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{\tau}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau$
$\displaystyle+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z.$
(38)

We deduce by (4) and (4) that

$\displaystyle G(\xi,\eta)=$
$\displaystyle{\frac{\lambda_{0}}{4}}\left(\xi+\eta\right)+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{\tau}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau$
$\displaystyle+{{\frac{{1}}{4}}\int_{\eta}^{\xi}\int_{0}^{\eta}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+{{\frac{{1}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z$
$\displaystyle+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}\widehat{G}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z.$
(39)

We rewrite the integral equation (4) as

$\displaystyle G(\xi,\eta)=G_{0}(\xi,\eta)+\Phi_{G}(\xi,\eta),$ (40)

where we defined

$\displaystyle G_{0}(\xi,\eta):=$
$\displaystyle{\frac{\lambda_{0}}{4}}\left(\xi+\eta\right)+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{\tau}f\left(\frac{\tau+s}{2},\frac{\tau-s}{2}\right)\text{d}s\text{d}\tau,$

and $\Phi_{G}(\xi,\eta)$ via

$\displaystyle{\Phi_{H}(\xi,\eta)}:=$
$\displaystyle{{\frac{{1}}{4}}\int_{\eta}^{\xi}\int_{0}^{\eta}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+{{\frac{{1}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}\widetilde{\mu}(\tau,s)G(\tau,s)\text{d}s\text{d}\tau}+\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}\widehat{H}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z$
$\displaystyle+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}\widehat{H}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z,$
$\displaystyle\widehat{H}(\xi,\eta,z):=$ $\displaystyle
f\left(\frac{\xi-\eta}{2},z-\frac{\xi+\eta}{2}\right)H(\xi,\eta),$ (41)

for any $\eta\in[0,1]$, $\xi\in[\eta,2-\eta]$, $z\in[\eta,2-\eta]$, and any
function $H$.

Define the sequence $\\{G_{n}(\xi,\eta)\\}$ with $n\in\mathbb{N}_{0}$ via

$\displaystyle G_{n+1}(\xi,\eta):=G_{0}(\xi,\eta)+{\Phi_{G_{n}}}(\xi,\eta),$
(42)

and let $\Delta G_{n}(\xi,\eta):=G_{n+1}(\xi,\eta)-G_{n}(\xi,\eta)$. It holds
that

$\displaystyle\Delta G_{n+1}(\xi,\eta)={\Phi_{\Delta G_{n}}}(\xi,\eta),$

and

$\displaystyle G_{n+1}{(\xi,\eta)}=G_{0}{(\xi,\eta)}+\sum_{j=0}^{n}\Delta
G_{j}(\xi,\eta).$ (43)

From (42), we see that if $\\{G_{n}(\xi,\eta)\\}$ converges uniformly w.r.t.
$(\xi,\eta)$ when $n\rightarrow\infty$, then
$G(\xi,\eta):=\lim_{n\to\infty}G_{n}(\xi,\eta)$ is the solution of the
integral equation (40), and therefore the existence of a kernel function $k$
of (8) is guaranteed. In addition, in view of (43), the convergence of
$\\{G_{n}(\xi,\eta)\\}$ is equivalent to the convergence of the series
$\sum_{n=0}^{\infty}\Delta G_{n}(\xi,\eta)$. So it suffices to show that the
series $\sum_{n=0}^{\infty}\Delta G_{n}(\xi,\eta)$ is uniformly convergent
w.r.t. $(\xi,\eta)$. Furthermore, recalling the Weierstrass M-test, it
suffices to show that the following inequality

$\displaystyle\left|\Delta
G_{n}(\xi,\eta)\right|\leq\frac{M^{n+2}}{(n+1)!}\left(\xi+\eta\right)^{n+1},$
(44)

holds true for all $n\in\mathbb{N}_{0}$, where

$\displaystyle M:=$
$\displaystyle{\frac{\lambda_{1}+\overline{f}}{2}},\lambda_{1}:={\max\left\\{|\lambda_{0}|,\max_{(x,y)\in[0,1]\times[0,1]}|\lambda_{0}-c_{1}(x)+c_{1}(y)|\right\\}},{\overline{f}}:=\max_{(x,y)\in[0,1]\times[0,1]}|f(x,y)|.$

Now we prove (44) by induction. First of all, since

$\displaystyle\left|G_{0}(\xi,\eta)\right|$
$\displaystyle\leq{\frac{|\lambda_{0}|}{4}}\left(\xi+\eta\right)+\frac{{\overline{f}}}{4}(\xi\eta-\eta^{2})+\frac{{\overline{f}}}{4}\eta^{2}$
$\displaystyle\leq{\frac{|\lambda_{0}|}{4}}\left(\xi+\eta\right)+\frac{{\overline{f}}}{4}\xi\eta$
$\displaystyle\leq{\frac{|\lambda_{0}|}{4}}\times
2+\frac{{\overline{f}}}{4}\times 2\times 1$ $\displaystyle\leq M,$

it follows that

$\displaystyle\left|\Delta G_{0}(\xi,\eta)\right|=$
$\displaystyle\left|\Phi_{G_{0}}(\xi,\eta)\right|$ $\displaystyle\leq$
$\displaystyle\frac{{\lambda_{1}}}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}M\text{d}s\text{d}\tau+{\frac{{\lambda_{1}}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}M\text{d}s\text{d}\tau+\frac{\overline{f}}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}M\text{d}\tau\text{d}s\text{d}z+\frac{\overline{f}}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}M\text{d}\tau\text{d}s\text{d}z$
$\displaystyle=$ $\displaystyle
M\frac{{\lambda_{1}}}{4}(\xi\eta-\eta^{2})+M\frac{{\lambda_{1}}}{4}\eta^{2}+M\frac{\overline{f}}{8}(\xi\eta^{2}-\eta^{3})+M\frac{\overline{f}}{12}\eta^{3}$
$\displaystyle=$ $\displaystyle
M\left(\frac{{\lambda_{1}}}{4}\xi\eta+\frac{{\overline{f}}}{8}\xi\eta^{2}-\frac{{\overline{f}}}{24}\eta^{3}\right)$
$\displaystyle\leq$ $\displaystyle M\left(\frac{{\lambda_{1}}}{4}\xi\times
1+\frac{{\overline{f}}}{8}{\times 2\times 1\times\eta-0}\right)$
$\displaystyle\leq$
$\displaystyle{M\left(\frac{{\lambda_{1}}}{4}+\frac{\overline{f}}{4}\right)(\xi+\eta)}$
$\displaystyle\leq$ $\displaystyle M^{2}\left(\xi+\eta\right),$ (45)

which shows that (44) holds for $n=0$.

Supposing that the inequality (44) holds true for a general $n\in\mathbb{N}$,
we need to show that (44) also holds true for $n+1$. Indeed, we get

$\displaystyle\left|\Delta G_{n+1}(\xi,\eta)\right|=$
$\displaystyle\left|{\Phi_{\Delta
G_{n}}}(\xi,\eta)\right|=|I_{1}+I_{2}+I_{3}+I_{4}|,$ (46)

where

$\displaystyle I_{1}:=$
$\displaystyle{\frac{{1}}{4}}\int_{\eta}^{\xi}\int_{0}^{\eta}{\widetilde{\mu}(\tau,s)}\Delta
G_{n}(\tau,s)\text{d}s\text{d}\tau,$ $\displaystyle I_{2}:=$
$\displaystyle{\frac{{1}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}{\widetilde{\mu}(\tau,s)}\Delta
G_{n}(\tau,s)\text{d}s\text{d}\tau,$ $\displaystyle I_{3}:=$
$\displaystyle\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}\widehat{\Delta
G_{n}}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z,$ $\displaystyle I_{4}:=$
$\displaystyle\frac{1}{2}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}\widehat{\Delta
G_{n}}(\tau,s,z)\text{d}\tau\text{d}s\text{d}z,$

with $\widehat{\Delta G_{n}}$ defined via (41).

Since (44) holds true for $n$, it holds that

$\displaystyle|I_{1}|\leq$
$\displaystyle\frac{{\lambda_{1}}M^{n+2}}{4(n+1)!}\int_{\eta}^{\xi}\int_{0}^{\eta}(\tau+s)^{n+1}\text{d}s\text{d}\tau$
$\displaystyle=$
$\displaystyle\frac{{\lambda_{1}}M^{n+2}}{4(n+1)!}\frac{1}{n+2}\int_{\eta}^{\xi}{\left((\tau+\eta)^{n+2}-\tau^{n+2}\right)}\text{d}\tau$
$\displaystyle\leq$
$\displaystyle\frac{{\lambda_{1}}M^{n+2}}{4(n+1)!}\frac{1}{n+2}\int_{\eta}^{\xi}(\tau+\eta)^{n+2}\text{d}\tau$
$\displaystyle=$
$\displaystyle\frac{{\lambda_{1}}M^{n+2}}{4(n+1)!}\frac{1}{(n+2)(n+3)}\left((\xi+\eta)^{n+3}-(2\eta)^{n+3}\right),$
$\displaystyle|I_{2}|\leq$
$\displaystyle\frac{{\lambda_{1}}M^{n+2}}{2(n+1)!}\int_{0}^{\eta}\int_{0}^{\tau}(\tau+s)^{n+1}\text{d}s\text{d}\tau$
$\displaystyle\leq$
$\displaystyle\frac{{\lambda_{1}}M^{n+2}}{2(n+1)!}\frac{1}{n+2}\int_{0}^{\eta}(2\tau)^{n+2}\text{d}\tau$
$\displaystyle=$
$\displaystyle\frac{{\lambda_{1}}M^{n+2}}{4(n+1)!}\frac{1}{(n+2)(n+3)}(2\eta)^{n+3},$
$\displaystyle|I_{3}|\leq$
$\displaystyle\frac{\overline{f}M^{n+2}}{4(n+1)!}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}(\tau+s)^{n+1}\text{d}\tau\text{d}s\text{d}z$
$\displaystyle\leq$
$\displaystyle\frac{\overline{f}M^{n+2}}{4(n+1)!}\frac{1}{n+2}\int_{\eta}^{\xi}\int_{0}^{\eta}(z+\eta)^{n+2}\text{d}s\text{d}z$
$\displaystyle=$
$\displaystyle\frac{\overline{f}M^{n+2}}{4(n+1)!}\frac{\eta}{n+2}\int_{\eta}^{\xi}(z+\eta)^{n+2}\text{d}s\text{d}z$
$\displaystyle\leq$
$\displaystyle\frac{\overline{f}M^{n+2}}{4(n+1)!}\frac{1}{n+2}\int_{\eta}^{\xi}(z+\eta)^{n+2}\text{d}z$
$\displaystyle=$
$\displaystyle\frac{\overline{f}M^{n+2}}{4(n+1)!}\frac{1}{(n+2)(n+3)}\left((\xi+\eta)^{n+3}-(2\eta)^{n+3}\right),$
$\displaystyle|I_{4}|\leq$
$\displaystyle\frac{\overline{f}M^{n+2}}{2(n+1)!}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}(\tau+s)^{n+1}\text{d}\tau\text{d}s\text{d}z$
$\displaystyle\leq$
$\displaystyle\frac{\overline{f}M^{n+2}}{2(n+1)!}\frac{1}{n+2}\int_{0}^{\eta}\int_{0}^{z}(2z)^{n+2}\text{d}s\text{d}z$
$\displaystyle\leq$
$\displaystyle\frac{\overline{f}M^{n+2}}{2(n+1)!}\frac{2}{n+2}\int_{0}^{\eta}(2z)^{n+2}\text{d}z$
$\displaystyle=$
$\displaystyle\frac{\overline{f}M^{n+2}}{2(n+1)!}\frac{1}{(n+2)(n+3)}(2\eta)^{n+3}.$

Then we get

$\displaystyle|\Delta G_{n+1}(\xi,\eta)|\leq$
$\displaystyle\left(|I_{1}|+|I_{2}|\right)+\left(|I_{3}|+|I_{4}|\right)$
$\displaystyle\leq$
$\displaystyle\frac{{{\lambda_{1}}}}{4(n+3)}\frac{M^{n+2}}{(n+2)!}(\xi+\eta)^{n+3}{+}\frac{{\overline{f}}}{2(n+3)}\frac{M^{n+2}}{(n+2)!}(\xi+\eta)^{n+3}$
$\displaystyle\leq$
$\displaystyle\left(\frac{{{\lambda_{1}}}}{2(n+3)}+\frac{{\overline{f}}}{n+3}\right)\frac{M^{n+2}}{(n+2)!}(\xi+\eta)^{n+2}$
$\displaystyle\leq$
$\displaystyle\frac{M^{n+3}}{(n+2)!}\left(\xi+\eta\right)^{n+2},$

which implies that (44) holds for $n+1$. Therefore, (44) holds true for
$n\in\mathbb{N}_{0}$.

We conclude that the series $\sum_{n=0}^{\infty}\Delta G_{n}(\xi,\eta)$
converges absolutely and uniformly in $\eta\in[0,1]$ and
$\xi\in[\eta,2-\eta]$, and $G(\xi,\eta):=\lim_{n\to\infty}G_{n}(\xi,\eta)$ is
a solution of (40). Moreover, $G(\xi,\eta)$ is continuous in $\xi$ and $\eta$.
Furthermore, in view of (4) with $f\in C^{1}\left([0,1]\times[0,1]\right)$ and
$c_{1}\in C^{1}\left([0,1]\right)$, we deduce that
$G_{\xi\xi},G_{\eta\eta},G_{\xi\eta}$ exist and are Hölder continuous with an
arbitrary exponent $\beta^{\prime}\in(0,1)$. In particular,
$G_{\xi\xi},G_{\eta\eta},G_{\xi\eta}$ are Hölder continuous with the exponent
$\beta$. Therefore, the problem (8) admits a solution
$k\in\mathcal{H}^{2+\beta}\left(D\right)$.

Regarding the uniqueness of solution of (8), which is equivalent to the
uniqueness of solution of the integral equation (4), it suffices to show that
any twice continuously differentiable solution of (4) can be approximated by
the sequence $\\{G_{n}(\xi,\eta)\\}$ constructed via (42). More precisely, for
any twice continuously differentiable solution $\overline{G}(\xi,\eta)$ of (4)
, we intend to show that the estimate

$\displaystyle|G_{n}(\xi,\eta)-\overline{G}(\xi,\eta)|\leq\frac{LM^{n}}{n!}\left(\xi+\eta\right)^{n},$
(47)

holds true for all $n\in\mathbb{N}_{0}$, where

$\displaystyle
L:=\max_{\eta\in[0,1],\xi\in[\eta,2-\eta]}|\Phi_{\overline{G}}(\xi,\eta)|.$

This is because (47) implies the uniform convergence of
$\\{G_{n}(\xi,\eta)\\}$ to $\overline{G}(\xi,\eta)$, while the uniqueness of
$G(\xi,\eta):=\lim_{n\to\infty}G_{n}(\xi,\eta)$ leads to
$\overline{G}(\xi,\eta)\equiv G(\xi,\eta)$. So the solution of (4) is unique,
and therefore the solution of (4) is also unique.

Now we prove (47) by induction. First, since $\overline{G}(\xi,\eta)$ is a
solution (4), using the definition of $\Phi_{\overline{G}}$, we obtain

$\displaystyle|G_{0}(\xi,\eta)-\overline{G}(\xi,\eta)|=$
$\displaystyle|G_{0}(\xi,\eta)-\left(G_{0}(\xi,\eta)+\Phi_{\overline{G}}(\xi,\eta)\right)|=|\Phi_{\overline{G}}(\xi,\eta)|\leq
L,$

which shows that (47) holds true for $n=0$.

Suppose that (47) holds true for a general $n\in\mathbb{N}$. Then by this
assumption and the linearity of $\Phi_{H}(\xi,\eta)$ w.r.t. $H$, we have

$\displaystyle|G_{n+1}(\xi,\eta)-\overline{G}(\xi,\eta)|=\left|\Phi_{G_{n}}(\xi,\eta)-\Phi_{\overline{G}}(\xi,\eta)\right|=\left|\Phi_{G_{n}-\overline{G}}(\xi,\eta)\right|\leq
I_{5}+I_{6}+I_{7}+I_{8},$

where

$\displaystyle I_{5}:=$ $\displaystyle
L\frac{{\lambda_{1}}M^{n}}{4n!}\int_{\eta}^{\xi}\int_{0}^{\eta}(\tau+s)^{n}\text{d}s\text{d}\tau,$
$\displaystyle I_{6}:=$ $\displaystyle
L\frac{{\lambda_{1}}M^{n}}{2n!}\int_{0}^{\eta}\int_{0}^{\tau}(\tau+s)^{n}\text{d}s\text{d}\tau,$
$\displaystyle I_{7}:=$ $\displaystyle
L\frac{\overline{f}M^{n}}{4n!}\int_{\eta}^{\xi}\int_{0}^{\eta}\int_{z}^{z+\eta-s}(\tau+s)^{n}\text{d}\tau\text{d}s\text{d}z,$
$\displaystyle I_{8}:=$ $\displaystyle
L\frac{\overline{f}M^{n}}{2n!}\int_{0}^{\eta}\int_{0}^{z}\int_{z}^{2z-s}(\tau+s)^{n}\text{d}\tau\text{d}s\text{d}z.$

Analogous to the estimates of $I_{1},I_{2},I_{3}$ and $I_{4}$, it holds that

$\displaystyle I_{5}\leq$ $\displaystyle
L\frac{{\lambda_{1}}M^{n}}{4n!}\frac{1}{(n+1)(n+2)}\left((\xi+\eta)^{n+2}-(2\eta)^{n+2}\right),$
$\displaystyle I_{6}\leq$ $\displaystyle
L\frac{{\lambda_{1}}M^{n}}{4n!}\frac{1}{(n+1)(n+2)}(2\eta)^{n+2},$
$\displaystyle I_{7}\leq$ $\displaystyle
L\frac{\overline{f}M^{n}}{4n!}\frac{1}{(n+1)(n+2)}\left((\xi+\eta)^{n+2}-(2\eta)^{n+2}\right),$
$\displaystyle I_{8}\leq$ $\displaystyle
L\frac{\overline{f}M^{n}}{2n!}\frac{1}{(n+1)(n+2)}(2\eta)^{n+2}.$

Then we obain

$\displaystyle|G_{n+1}(\xi,\eta)-\overline{G}(\xi,\eta)|\leq$ $\displaystyle
L\left(\frac{{{\lambda_{1}}}}{2(n+2)}+\frac{{\overline{f}}}{n+2}\right)\frac{M^{n}}{(n+1)!}(\xi+\eta)^{n+1}\leq{\frac{LM^{n+1}}{(n+1)!}\left(\xi+\eta\right)^{n+1}},$

which implies that (47) holds true for $n+1$. Thus, (47) holds true for all
$n\in\mathbb{N}_{0}$. We conclude that the solution of (8) is unique.

Finally, by changing the variables and using the method of successive
approximation as above, we may prove the well-posedness and regularity of the
kernel function $l$ for (17).$\hfill\blacksquare$

######  Remark 4.1.

For the stabilization problem of linear parabolic PDEs with time-independent
coefficients, the “uniqueness of kernel functions” was claimed in [12] and
proved by estimating

$\displaystyle|\Delta
G(\xi,\eta)|:=|G^{\prime}(\xi,\eta)-G^{\prime\prime}(\xi,\eta)|\leq\frac{C}{n!}(\xi+\eta)^{n}$
(48)

with some constant $C>0$, where $G^{\prime},G^{\prime\prime}$ are two kernel
functions; see (39) and (40) of [12].

It is worth noting that a gap exists in the proof of [12]. On one hand, if
$G^{\prime}$ and $G^{\prime\prime}$ are given by (see (39) of [12])

$\displaystyle
G^{\prime}(\xi,\eta)=\sum_{n=0}^{\infty}G_{n}(\xi,\eta),~{}G^{\prime\prime}(\xi,\eta)=\sum_{n=0}^{\infty}G_{n}(\xi,\eta),$
(49)

as indicated in [12], the estimate (48) can be obtained in the same way as in
(38) of [12]. Therefore the equality of $G^{\prime}=G^{\prime\prime}$ can be
concluded by letting $n\rightarrow\infty$. However, since the limit of
$\sum_{i=0}^{n}G_{i}(\xi,\eta)$ is unique, which is known by the limit theory,
due to (49), the equality of $G^{\prime}=G^{\prime\prime}$ holds true
trivially. So the proof presented in [12] is essentially for the uniqueness of
the limit of $\\{G_{n}\\}$ rather than the uniqueness of solution.

On the other hand, if $G^{\prime}$ and $G^{\prime\prime}$ are not given by
(49), without imposing more conditions, the estimate (48) can not be obtained
in the same way as in (38) of [12].

In order to address the above-mentioned issue, as presented in this paper, a
feasible idea of proving the uniqueness of kernel functions is to prove that
all solutions of integral equations (e.g., (4) or (40)) can be approximated by
the sequence $\\{G_{n}\\}$ constructed. That is to say, $G^{\prime}$ and
$G^{\prime\prime}$ in [12] have to be with a form of (49). Then the uniqueness
of solution is determined by the uniqueness of the limit of $\\{G_{n}\\}$.
Note that, by using this method, the estimate considered in this paper (see
(47)) is different from (48). In particular, (47) implies (48).

######  Remark 4.2.

It is worth noting that for general functions $f$ and $c$ it is impossible to
obtain an analytic solution $G$ of the integral equation (4) by using the
iterating formula(43) (or (42)). In some special cases, the solution $G$ of
(4) can be given by a refined iteration formula. For example, consider
$f\equiv 0$ and $c_{1}(x)=rx^{2}$ with a nonnegative constant $r$. The
integral equation (4) becomes

$\displaystyle
G(\xi,\eta)={\frac{\lambda_{0}}{4}}\left(\xi+\eta\right)+{{\frac{{1}}{4}}\int_{\eta}^{\xi}\int_{0}^{\eta}(\lambda_{0}-r\tau
s)G(\tau,s)\text{d}s\text{d}\tau}+{{\frac{{1}}{2}}\int_{0}^{\eta}\int_{0}^{\tau}(\lambda_{0}-r\tau
s)G(\tau,s)\text{d}s\text{d}\tau},$ (50)

whose solution is uniquely determined by

$\displaystyle G(\xi,\eta)=$
$\displaystyle\frac{\lambda_{0}}{4}(\xi+\eta)+\frac{\lambda_{0}}{4}\sum_{n=1}^{\infty}\sum_{i=0}^{n}\left(\frac{1}{4}\right)^{n}\lambda_{0}^{i}A_{n+1-i}^{n}T_{2n-i}(\xi,\eta),$
(51)

where

$\displaystyle{A_{n+1-i}^{n}}:=$ $\displaystyle\left\\{\begin{aligned}
&(-r)^{n}\Pi_{m=1}^{n}C_{2m},&&i=0,\\\
&(A_{n-i}^{n-1}+A_{n+1-i}^{n-1})C_{2n-i},&&0<i<n,\\\
&\Pi_{m=1}^{n}C_{m},&&i=n,\end{aligned}\right.$ $\displaystyle C_{n}:=$
$\displaystyle\frac{1}{n(n+1)},$ $\displaystyle T_{2n-i}(\xi,\eta):=$
$\displaystyle\xi^{2n+1-i}\eta^{2n-i}+\xi^{2n-i}\eta^{2n+1-i}.$

The proof is provided in Appendix.

##  5 Well-posedness and Stability of the Original System

In this section, we prove the well-posedness and exponential stability in
different norms for the original system, i.e., system (4).

#### Proof of Theorem 2.1.

The assertion of Theorem 2.1 (i) follows immediately from the inverse
transformation (16), Proposition 4.1 and Proposition 3.1 (i).

For the proof of Theorem 2.1 (ii), we define first the following qualities:

$\displaystyle\alpha_{1}:=$ $\displaystyle\max\limits_{(x,y)\in
D}|k(x,y)|,~{}\beta_{1}:=\max\limits_{(x,y)\in D}|l(x,y)|,$
$\displaystyle\alpha_{2}:=$
$\displaystyle\max\limits_{x\in[0,1]}|k(x,x)|,~{}\beta_{2}:=\max\limits_{x\in[0,1]}|l(x,x)|,$
$\displaystyle\alpha_{3}:=$ $\displaystyle\max\limits_{(x,y)\in
D}|k_{x}(x,y)|,~{}\beta_{3}:=\max\limits_{(x,y)\in D}|l_{x}(x,y)|.$

For $p\in[1,+\infty)$, we infer from (16) and the Cauchy-Schwartz’ inequality
that

$\displaystyle\|w[t]\|_{p}^{p}\leq 2^{p-1}(1+\beta_{1}^{p})\|u[t]\|_{p}^{p}.$
(52)

Note that

$\displaystyle u_{0}(x)=w_{0}(x)+\int_{0}^{x}k(x,y)w_{0}(y)\text{d}y,$

which implies

$\displaystyle\|u_{0}\|_{p}^{p}\leq$ $\displaystyle
2^{p-1}(1+\alpha_{1}^{p})\|w_{0}\|_{p}^{p},$ (53)
$\displaystyle\|u_{0x}\|_{p}^{p}\leq$ $\displaystyle
3^{p-1}(\alpha_{2}^{p}+\alpha_{3}^{p})\|w_{0}\|_{p}^{p}+3^{p-1}\|w_{0x}\|_{p}^{p}.$
(54)

By (20), (52) and (53), we obtain the exponential stability of system (4) in
$L^{p}$-norm for $p\in[1,+\infty)$:

$\displaystyle\|w[t]\|_{p}\leq
C_{1}e^{-\underline{\lambda}t}\|w_{0}\|_{p},~{}\forall t\in{\mathbb{R}_{>0}},$
(55)

where
$C_{1}:=\left(4^{p-1}(1+\alpha_{1}^{p})(1+\beta_{1}^{p})\right)^{\frac{1}{p}}$.

Now differentiating two-hand sides of (16) w.r.t. $x$, taking $L^{p}$-norm,
and using the Cauchy-Schwartz’ inequality, we have

$\displaystyle\|w_{x}[t]\|_{p}^{p}\leq
3^{p-1}\|u_{x}[t]\|_{p}^{p}+3^{p-1}\left(\beta_{2}^{p}+\beta_{3}^{p})\right\|u[t]\|_{p}^{p},$

which along with (21), (54) and (55) implies the exponential stability of
system (4) in $W^{1,p}$-norm for $p\in[1,+\infty)$:

$\displaystyle\|w[t]\|_{{1,p}}\leq
C_{2}e^{-\underline{\lambda}t}\|w_{0}\|_{{1,p}},$ (56)

where
$C_{2}:=\max\left\\{9^{\frac{p-1}{p}}\gamma_{1}^{\frac{1}{p}},\gamma_{2}^{\frac{1}{p}}\right\\}$
with

$\displaystyle\gamma_{1}:=$
$\displaystyle\max\\{1,\beta_{2}^{p}+\beta_{3}^{p}\\},$
$\displaystyle\gamma_{2}:=$ $\displaystyle
C_{1}^{p}+9^{p-1}\gamma_{1}\left(1+\alpha_{1}^{p}+\alpha_{2}^{p}+\alpha_{3}^{p}\right).$

Furthermore, letting $p\rightarrow+\infty$ in (55) and (56), we obtain the
exponential stability of system (4) in $L^{\infty}$-norm and
$W^{1,\infty}$-norm, respectively:

$\displaystyle\|w[t]\|_{\infty}\leq
C_{3}e^{-\underline{\lambda}t}\|w_{0}\|_{\infty},~{}\forall
t\in{\mathbb{R}_{>0}},$ $\displaystyle\|w[t]\|_{{1,\infty}}\leq
C_{4}e^{-\underline{\lambda}t}\|w_{0}\|_{{1,\infty}},~{}\forall
t\in{\mathbb{R}_{>0}},$

where $C_{3}:=4{\gamma_{3}}$ and
$C_{4}:=\max\\{9{\gamma_{4}},C_{1}+9{\gamma_{4}}(1+\alpha_{1}+\alpha_{2}+\alpha_{3})\\}$
with

$\displaystyle{\gamma_{3}}:=$ $\displaystyle\left\\{\begin{aligned}
&1,&&0<\alpha_{1}\leq 1,0<\beta_{1}\leq 1,\\\ &\beta_{1},&&0<\alpha_{1}\leq
1,\beta_{1}>1,\\\ &\alpha_{1},&&\alpha_{1}>1,0<\beta_{1}\leq 1,\\\
&\alpha_{1}\beta_{1},&&\alpha_{1}>1,\beta_{1}>1,\end{aligned}\right.$
$\displaystyle{\gamma_{4}}:=$ $\displaystyle\max\\{1,\beta_{2}+\beta_{3}\\}.$

Therefore, the assertion of Theorem 2.1 (ii) holds true.

Let $w_{1}$ and $w_{2}$ be the solution of system (4) with initial data
$w_{01}\in W_{0}$ and $w_{02}\in W_{0}$, respectively. Using the linearity of
system (4) and Theorem 2.1 (ii), the estimates (10) and (11) are satisfied
with $w:=w_{1}-w_{2}$, which imply the assertion of Theorem 2.1 (iii). The
proof is complete. $\hfill\blacksquare$

##  6 Conclusion

In this paper, a combinatorial method, i.e., the method of backstepping and
the technique of ALFs, was applied to address the exponential stabilization
and continuous dependence of solutions on initial data for a class of $1$-D
space-time-varying linear parabolic PDEs without imposing a Gevrey-like
condition on the time-varying coefficient. Compared to the results of the
existing literature, the kernel functions obtained via the proposed method are
independent of the time variable, therefore the associated calculations have
been extensively simplified. In addition, exponential stability and continuous
dependence of solutions on initial data could be established in different
norms ($L^{p}$ and $W^{1,p}$ norms) by using the technique of ALFs, which was
used to deal with singularities in the case of $p\in[1,2)$.

It is worth noting that, with a Gevrey-like condition, the proposed method is
still suitable to address the exponential stabilization and continuous
dependence of solutions on initial data in different norms for $1$-D linear
parabolic PDEs having a general form, e.g., $c(x,t)$ need not to be divided
into two terms as (5). However, without involving a Gevrey-like condition, it
is still challenging to apply the method to obtain time-independent kernel
functions and establish the stability estimates in different norms for
parabolic PDEs having a general form, which will be studied in our future
work.

##  Appendix A Appendix

### A.1. The Bellman-Gronwall-Peano’s inequality

The following lemma used in this paper is concerned with the Bellman-Gronwall-
Peano’s inequality, which can be found in, e.g., [28, pp. 94].

######  Lemma A.1.

Let $T\in{\mathbb{R}_{>0}}$, if the function $z$ is a nonnegative, absolutely
continuous function on $[0,T]$, and satisfies the inequality

$\displaystyle\frac{\text{d}z}{\text{d}t}\leq q(t)z(t)+h(t)\ a.e.\ in\ [0,T],$

where $q$, $h\in L^{1}(0,T)$, then

$\displaystyle z(t)\leq
e^{\int_{0}^{t}q(s)\text{d}s}z(0)+\int_{0}^{t}e^{\int_{s}^{t}q(r)\text{d}r}h(s)\text{d}s,~{}\forall
t\in[0,T].$

### A.2. Proof of (52) in Remark 4.2

We apply a new approximation to obtain (52), which is the solution of the
integral equation (50).

Let $G_{0}(\xi,\eta):=\frac{\lambda_{0}}{4}\left(\xi+\eta\right)$, and

$\displaystyle G_{n+1}(\xi,\eta):=$
$\displaystyle\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}(\lambda_{0}-r\tau
s)G_{n}(\tau,s)\text{d}s\text{d}\tau+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{\tau}(\lambda_{0}-r\tau
s)G_{n}(\tau,s)\text{d}s\text{d}\tau.$

According to the proof of Proposition 4.1, the solution of (50) exists and is
unique. Therefore, as long as the series $\sum_{n=0}^{\infty}G_{n}(\xi,\eta)$
converges, $G(\xi,\eta):=\sum_{n=0}^{\infty}G_{n}(\xi,\eta)$ is the unique
solution of (50).

Now we apply the induction to prove the convergence of
$\sum_{n=0}^{\infty}G_{n}(\xi,\eta)$ and the fact that

$\displaystyle
G_{n}(\xi,\eta)=\frac{\lambda_{0}}{4}\left(\frac{1}{4}\right)^{n}\sum_{i=0}^{n}\lambda_{0}^{i}A_{n+1-i}^{n}T_{2n-i}(\xi,\eta).$
(57)

Regarding the convergence of $\sum_{n=0}^{\infty}G_{n}(\xi,\eta)$, it suffices
to show that

$\displaystyle|G_{n}(\xi,\eta)|\leq\frac{M_{1}^{n+1}}{(n+1)!}\left(\xi^{n+1}\eta^{n}+\xi^{n}\eta^{n+1}\right),\forall
n\in\mathbb{N}_{0},$ (58)

where $M_{1}:=r+|\lambda_{0}|$. Firstly, the following result can be
established:

$\displaystyle|G_{0}(\xi,\eta)|\leq$ $\displaystyle
M_{1}\left(\xi+\eta\right),$

which shows that (58) holds true for $n=0$. The next step is to prove that if
(58) holds true for a general $n\in\mathbb{N}$, then (58) also holds true for
$n+1$. Indeed, by direct computations, we have

$\displaystyle|G_{n+1}(\xi,\eta)|\leq$
$\displaystyle\frac{1}{4}\int_{\eta}^{\xi}\int_{0}^{\eta}(|\lambda_{0}|+r\tau
s)|G_{n}(\tau,s)|\text{d}s\text{d}\tau+\frac{1}{2}\int_{0}^{\eta}\int_{0}^{\tau}(|\lambda_{0}|+r\tau
s)|G_{n}(\tau,s)|\text{d}s\text{d}\tau$ $\displaystyle\leq$
$\displaystyle\frac{1}{4}\frac{M_{1}^{n+1}}{(n+1)!}\frac{1}{n+2}\left(\frac{|\lambda_{0}|}{n+1}+\frac{r\xi\eta}{n+3}\right)\left(\xi^{n+2}\eta^{n+1}+\xi^{n+1}\eta^{n+2}\right)$
$\displaystyle\leq$
$\displaystyle\frac{M_{1}^{n+2}}{(n+2)!}\left(\xi^{n+2}\eta^{n+1}+\xi^{n+1}\eta^{n+2}\right),$

which implies that (58) holds for $n+1$. Thus, (58) holds true for all
$n\in\mathbb{N}_{0}$. Therefore, $\sum_{n=0}^{\infty}G_{n}(\xi,\eta)$
converges, and $G:=\sum_{n=0}^{\infty}G_{n}(\xi,\eta)$ is the solution of
(50).

We need to prove (57). Note that

$\displaystyle\int_{\eta}^{\xi}\int_{0}^{\eta}\left(\tau^{n+1}s^{n}+\tau^{n}s^{n+1}\right)\text{d}s\text{d}\tau+2\int_{0}^{\eta}\int_{0}^{\tau}\left(\tau^{n+1}s^{n}+\tau^{n}s^{n+1}\right)\text{d}s\text{d}\tau=\frac{1}{(n+1)(n+2)}\xi^{n+1}\eta^{n+1}(\xi+\eta).$
(59)

For any function $\nu$, define $K$ via

$\displaystyle K(\nu(\tau,s)):=$
$\displaystyle\int_{\eta}^{\xi}\int_{0}^{\eta}\nu(\tau,s)\text{d}s\text{d}\tau+2\int_{0}^{\eta}\int_{0}^{\tau}\nu(\tau,s)\text{d}s\text{d}\tau.$

By (59), we have

$\displaystyle K(T_{n}(\tau,s))=C_{n+1}T_{n+1}(\xi,\eta),$ (60)

where

$\displaystyle T_{n}(\tau,s):=\tau^{n+1}s^{n}+\tau^{n}s^{n+1},$ $\displaystyle
C_{n+1}:=\frac{1}{(n+1)(n+2)}.$

Then, it holds that

$\displaystyle
K(\lambda_{0}T_{n}(\tau,s))=\lambda_{0}C_{n+1}T_{n+1}(\xi,\eta),$ (61)
$\displaystyle K(\tau sT_{n}(\tau,s))=C_{n+2}T_{n+2}(\xi,\eta).$ (62)

By (60), (61) and (62), we obtain

$\displaystyle G_{1}(\xi,\eta)=$
$\displaystyle\frac{\lambda_{0}}{4}\frac{1}{4}K((\lambda_{0}-r\tau
s)T_{0}(\tau,s))=\frac{\lambda_{0}}{4}\frac{1}{4}\left(\lambda_{0}C_{1}T_{1}(\xi,\eta)-rC_{2}T_{2}(\xi,\eta)\right),$

which implies that (57) holds true for $n=1$.

Supposing that (57) holds true for $1,2,\ldots,n$, it follows that

$\displaystyle G_{n+1}(\xi,\eta)=$
$\displaystyle\frac{1}{4}K\left((\lambda_{0}-r\tau
s)\frac{\lambda_{0}}{4}\left(\frac{1}{4}\right)^{n}\sum_{i=0}^{n}\lambda_{0}^{i}A_{n+1-i}^{n}T_{2n-i}(\xi,\eta)\right)$
$\displaystyle=$
$\displaystyle\frac{\lambda_{0}}{4}\left(\frac{1}{4}\right)^{n+1}\left(\sum_{i=0}^{n}\lambda_{0}^{i+1}A_{n+1-i}^{n}C_{2n+1-i}T_{2n+1-i}(\xi,\eta)\right.\left.-\sum_{i=0}^{n}r\lambda_{0}^{i}A_{n+1-i}^{n}C_{2n+2-i}T_{2n+2-i}(\xi,\eta)\right)$
$\displaystyle=$
$\displaystyle\frac{\lambda_{0}}{4}\left(\frac{1}{4}\right)^{n+1}\left(\sum_{i=0}^{n-1}\lambda_{0}^{i+1}A_{n+1-i}^{n}C_{2n+1-i}T_{2n+1-i}(\xi,\eta)\right.$
$\displaystyle+\lambda_{0}^{n+1}A_{1}^{n}C_{n+1}T_{n+1}(\xi,\eta)-rA_{n+1}^{n}C_{2n+2}T_{2n+2}(\xi,\eta)\left.+\sum_{i=1}^{n}\lambda_{0}^{i}A_{n+1-i}^{n}C_{2n+2-i}T_{2n+2-i}(\xi,\eta)\right)$
$\displaystyle=$
$\displaystyle\frac{\lambda_{0}}{4}\left(\frac{1}{4}\right)^{n+1}\bigg{(}-rA_{n+1}^{n}C_{2n+2}T_{2n+2}(\xi,\eta)+\sum_{i=1}^{n}\lambda_{0}^{i}\left(A_{n+1-i}^{n}+A_{n+2-i}^{n}\right)C_{2n+2-i}T_{2n+2-i}(\xi,\eta)$
$\displaystyle+\lambda_{0}^{n+1}A_{1}^{n}C_{n+1}T_{n+1}(\xi,\eta)\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{\lambda_{0}}{4}\left(\frac{1}{4}\right)^{n+1}\sum_{i=0}^{n+1}\lambda_{0}^{i}A_{n+2-i}^{n+1}T_{2n+2-i}(\xi,\eta),$

which implies that (57) holds true for $n+1$. Therefore, by induction, (57)
holds true for all $n\geq 1$. The proof is complete.

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