# Stability of Abstract Thermoelastic Systems with Cattaneo’s Law and Inertial
Term ††thanks: CHXD was supported by the China Scholarship Council, ZJH was
supported by the National Natural Science Foundation of China (grants No.
62073236). QZ was supported by the National Natural Science Foundation of
China (grants No. 12271035, 12131008) and Beijing Municipal Natural Science
Foundation (grant No. 1232018).

Chenxi Deng , Zhong-Jie Han, Zhaobin Kuang, Qiong Zhang School of Mathematics
and Statistics, Beijing Institute of Technology, Beijing 100081, China (email:
3120215730@bit.edu.cn)School of Mathematics, BIIT Lab, Tianjin University,
Tianjin 300354, China (email<EMAIL_ADDRESS>Computer Science Department,
Stanford University, Stanford 94305, U.S.A.(email<EMAIL_ADDRESS>School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI,
Beijing Institute of Technology, Beijing 100081, China (email:
zhangqiong@bit.edu.cn)Corresponding author.

###### Abstract

This paper investigates the polynomial stability of solutions to abstract
thermoelastic systems with Cattaneo’s law on Hilbert space. The study includes
two cases: one with an inertial term and the other without it. For the case
with an inertial term, we partition the region of parameters for non-
exponential stability into three sub-regions and obtain distinct polynomial
decay rates for each sub-region. Meanwhile, for the case without an inertial
term, we explore two areas and obtain dissimilar polynomial decay rates in
these areas. Moreover, we demonstrate that all the given polynomial decay
rates are sharp. Finally, we give some examples to explain our conclusion.

AMS subject classification. 35Q74, 74F05

Key words: thermoelastic system; Cattaneo’s law; inertial term; polynomial
stability.

##  1 Introduction

Let $H$ be a complex Hilbert space with inner product $(\cdot,\,\cdot)$ and
the induced norm $\|\cdot\|$, $A$ be a self-adjoint, positive definite
operator with compact resolvent on $H$. We consider the following abstract
coupled thermoelastic system:

$\left\\{\begin{array}[]{l}u_{tt}+mA^{\gamma}u_{tt}+\sigma
Au-A^{\alpha}\theta=0,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\theta_{t}-A^{\frac{\beta}{2}}q+A^{\alpha}u_{t}=0,\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle\tau
q_{t}+q+A^{\frac{\beta}{2}}\theta=0,\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle
u(0)=u_{0},\;\;u_{t}(0)=u_{1},\;\;\theta(0)=\theta_{0},\;\;q(0)=q_{0},\end{array}\right.$
(1.1)

where the constants $m\geq 0,\;\sigma>0,\;\tau>0$ correspond to the inertial
term, wave speed and relaxation parameter, respectively. The parameters
$(\alpha,\;\beta,\;\gamma)\in E:=[0,1]\times[0,1]\times(0,1]$ represent the
coupling order, damping order, and inertial order, respectively. The system
with $\gamma=0$, corresponding to the system without inertial term, can be
contained in the case $m=0$, which will be investigated in Section 4.

If $\tau=0$, the heat conduction in the system follows the classical Fourier’s
law. However, if $\tau>0$, the system follows the Cattaneo’s law, which is
used to describe the finite speed of heat propagation in a medium. This
eliminates the paradox of infinite speed of heat transfer in Fourier’s law and
describes the wave-like motion of heat, also known as the second sound in
physics.

The objective of this work is to investigate the long-time behavior of the
coupled system (1.1) with Cattaneo’s law. In particular, we demonstrate the
impact of thermoelastic parameters $\alpha,\;\beta$ and the inertial term
(determined by $m$) on the stability of this abstract coupled system (1.1).

The natural energy of system (1.1) is

$E(t)=\frac{1}{2}\big{(}\|u_{t}\|^{2}+m\|A^{\frac{\gamma}{2}}u_{t}\|^{2}+\sigma\|A^{\frac{1}{2}}u\|^{2}+\|\theta\|^{2}+\tau\|q\|^{2}\big{)}.$

A direct computation gives

${d\over dt}E(t)=-\|q\|^{2}\leq 0.$

Hence, the system is dissipative.

There is a significant body of work that studies the asymptotic behavior of
thermoelastic systems (1.1) under various heat conduction mechanisms,
including Fourier’s law ($\tau=0$) and Cattaneo’s law ($\tau\neq 0$), with
($m\neq 0$) or without ($m=0$) inertial terms. In [25], Russell developed an
elastic system with indirect damping that encompasses thermoelastic string,
beam, and plate equations. It is precisely the system (1.1) with
$\tau=m=0,~{}\sigma=1$. Later, Rivera and Racke studied the regularity and
exponential stability of the system in [22], where they also provided specific
examples of the system (1.1). Ammar-Khodja _et al._ [1] identified the regions
of exponential stability for the system (1.1) with $\tau=m=0$. Hao and Liu
[11] investigated the polynomial stability of the system outside the regions
of exponential stability in [1] and presented optimal polynomial decay rates.
Further details on the case $\tau=m=0$ can be found in [3, 12, 14, 16, 17] and
the references therein.

Regarding the case $\tau=0,~{}m\neq 0$, Dell’Oro _et al._ [8] investigated the
stability of the system (1.1) for the special case $\beta=\gamma=\frac{1}{2}$,
$0\leq\alpha\leq\frac{3}{4}$, which, to the best of the authors’ knowledge,
was the first result that considered the inertial term. In [9], the authors
provided a comprehensive overview of the regions of exponential stability and
polynomial stability for parameters $(\alpha,\beta)\in[0,1]\times[0,1]$,
$\gamma\in(0,1]$. The authors in [15] pointed out that the polynomial decay
rates in [9] for the system with Fourier’s law are sharp.

The study of thermoelastic systems with Fourier’s law is relatively complete,
but other heat conduction mechanisms, such as Cattaneo’s law and Vernotte’s
law, are also interesting to study, see [6, 7, 20, 26] for more details. Liu
_et al._ studied the thermoelastic model with Cattaneo’s law in [9]. They
claimed that “the change from Fourier’s law to Cattaneo’s law leads to a loss
of exponential stability in most coupled systems.” Later, Han, Kuang and Zhang
[10] showed that if the system has the same wave speed, that is,
$\sigma\tau=1$, the region of exponential stability shown in [9] can be
larger. Moreover, they characterized the polynomial decay rates of the system
(1.1) with two special choices of $\alpha$.

In this paper, we aim to address the question posed in the article by Han _et
al._ [10] regarding the stability of the system (1.1) on Hilbert space with
Cattaneo’s law. While Han _et al._ focused on exponentially stable regions of
the system, we investigate the polynomial stability of the system both with
and without an inertial term outside those exponentially stable regions. For
the system with an inertial term, we divide the non-exponentially stable
region into three parts and get different polynomial decay rates for each
part; whereas in the case without an inertial term, we only need to consider
two regions and also obtain different polynomial decay rates on these regions.
Furthermore, we prove that the polynomial decay rates above are all optimal.
Our main analysis method is the frequency domain method developed by Borichev
and Tomilov [5], along with interpolation inequalities and detailed spectral
analysis to support our proofs.

The paper is structured as follows. In Section 2, we describe the problem and
show our main results. We demonstrate the proof of polynomial stability of the
system with and without an inertial term in Section 3 and Section 4,
respectively. Then we show the decay rate in each area is optimal in Section
5. In Section 6, the polynomial stability of some Rayleigh beam equations with
thermal damping is given using our results. Finally, we give a conclusion of
our work in Section 7.

##  2 Preliminaries and Main Results

In this section, we state some preliminaries and the main results of this
paper. To describe system (1.1) as an abstract first-order evolution equation,
we define a Hilbert space

$\mathcal{H}:=\mathcal{D}(A^{\frac{1}{2}})\times\mathcal{D}(A^{\frac{\gamma}{2}})\times
H\times H,$

which is equipped with the inner product

$\langle
U_{1},U_{2}\rangle_{\mathcal{H}}=\sigma(A^{\frac{1}{2}}u_{1},A^{\frac{1}{2}}u_{2})+m(A^{\frac{\gamma}{2}}v_{1},A^{\frac{\gamma}{2}}v_{2})+(v_{1},v_{2})+(\theta_{1},\theta_{2})+\tau(q_{1},q_{2}),$

where $U_{i}=(u_{i},v_{i},\theta_{i},q_{i})\in\mathcal{H},\;i=1,2$.

Let $t>0$, introduce a new variable $v(t):=u_{t}(t)$, define
$U(t):=(u(t),v(t),\theta(t),q(t))^{\top}$,
$U_{0}:=(u_{0},u_{1},\theta_{0},q_{0})^{\top}\in\mathcal{H}$, then the system
(1.1) can be written as an evolution equation in $\mathcal{H}$:

$\left\\{\begin{array}[]{l}\displaystyle{d\over dt}U(t)=\mathcal{A}U(t),\quad
t>0,\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
U(0)=U_{0},\end{array}\right.$ (2.1)

where the operator $\mathcal{A}:{\cal D}({\cal
A})\subset\mathcal{H}\to\mathcal{H}$ is defined by

$\mathcal{A}\left[\begin{array}[]{c}u\\\ v\\\ \theta\\\
q\end{array}\right]=\left(\begin{array}[]{c}v\\\ -(I+mA^{\gamma})^{-1}(\sigma
Au-A^{\alpha}\theta)\\\ -A^{\alpha}v+A^{\frac{\beta}{2}}q\\\
\displaystyle{1\over\tau}(-q-A^{\frac{\beta}{2}}\theta)\end{array}\right),$

with domain

$\mathcal{D}(\mathcal{A})=\left\\{(u,v,\theta,q)\in\mathcal{H}\,\left|\,v\in\mathcal{D}(A^{1\over
2}),\;\sigma
Au-A^{\alpha}\theta\in\mathcal{D}(A^{-\frac{\gamma}{2}}),\;-A^{\alpha}v+A^{\frac{\beta}{2}}q\in
H,\;q+A^{\frac{\beta}{2}}\theta\in H\right.\right\\}.$

It is clear that

$\mathcal{D}(\mathcal{A})\subset\left\\{(u,v,\theta,q)\in\mathcal{D}(\mathcal{A})\,\left|\,v\in\mathcal{D}(A^{\alpha}),\;q\in\mathcal{D}(A^{\frac{\beta}{2}})\right.\right\\}\quad\;\mbox{when
}\;0\leq\alpha\leq{1\over 2},$

and

$\mathcal{D}(\mathcal{A})\subset\left\\{(u,v,\theta,q)\in\mathcal{D}(\mathcal{A})\,\left|\,u\in\mathcal{D}(A^{1-\alpha})\right.\right\\}\quad\;\mbox{when
}\;\frac{1}{2}<\alpha\leq 1.$

The well-posedness of the system (2.1) is stated as follows.

######  Lemma 2.1.

Let $\mathcal{A}$ and $\mathcal{H}$ be defined as above and
$(\alpha,\beta,\gamma)\in{[0,1]\times[0,1]\times(0,1]}$. If $\beta\geq
2\alpha-1$, then $0\in\rho(\mathcal{A})$ and $\mathcal{A}$ generates a $C_{0}$
semigroup $e^{{\cal A}t}$ of contractions on $\mathcal{H}$.

###### Proof.

It is easy to see that $\mathcal{A}$ is a densely defined operator on
$\mathcal{H}$ and

${\cal R}e\langle\mathcal{A}U,U\rangle_{\mathcal{H}}=-\|q\|^{2}\leq
0,\;\forall\;U\in\mathcal{D}(\mathcal{A}),$

then $\mathcal{A}$ is dissipative in $\mathcal{H}$.

We claim that $\mathcal{A}$ is a bijection. On the one hand, for any given
$G:=(g_{1},g_{2},g_{3},g_{4})^{\top}\in\mathcal{H}$, let
$U=(u,v,\theta,q)^{\top}$, where

$\displaystyle v=g_{1},\quad\theta=-A^{-\frac{\beta}{2}}(\tau
g_{4}+A^{-\frac{\beta}{2}}(g_{3}+A^{\alpha}g_{1})),\quad
q=A^{-\frac{\beta}{2}}(g_{3}+A^{\alpha}g_{1}),$

and

$u=\sigma^{-1}A^{-1}(A^{\alpha}\theta-(I+mA^{\gamma})g_{2}).$

Then $U\in\mathcal{D}(\mathcal{A})$ and $\mathcal{A}U=G$. On the other hand,
it is clear that ${\cal A}$ is an injective. Thus $\mathcal{A}^{-1}$ exists.

Now we show $\mathcal{A}^{-1}$ is bounded. Note that
$\mathcal{D}(A^{\frac{1}{2}})\subset\mathcal{D}(A^{\frac{\gamma}{2}})$, we
have

$\|v\|_{\mathcal{D}(A^{\frac{\gamma}{2}})}\leq\|g_{1}\|_{\mathcal{D}(A^{\frac{1}{2}})}\leq
C\|G\|_{\mathcal{H}}.$ (2.2)

Recalling $\beta\geq 2\alpha-1$, we can also get

$\|q\|\leq C(\|g_{3}\|+\|A^{\alpha-\frac{\beta}{2}}g_{1}\|)\leq
C(\|g_{3}\|+\|g_{1}\|_{\mathcal{D}(A^{\frac{1}{2}})})\leq
C\|G\|_{\mathcal{H}},$ (2.3) $\|\theta\|\leq C(\|g_{4}\|+\|q\|)\leq
C\|G\|_{\mathcal{H}},$ (2.4)

and

$\begin{array}[]{lcl}&\leq&C\big{(}\|A^{-\frac{1}{2}+\alpha-\frac{\beta}{2}}g_{4}\|+\|A^{-\frac{1}{2}+\alpha-\beta}g_{3}\|+\|A^{2\alpha-\beta-\frac{1}{2}}g_{1}\|+\|A^{-\frac{1}{2}}(I+mA^{\gamma})g_{2}\|\big{)}\cr&\leq&C\|G\|_{\mathcal{H}}\quad(\mbox{Using
the assumption}\;\beta\geq 2\alpha-1).\end{array}$ (2.5)

Consequently, according to (2.2)-(2.5), one has that $\mathcal{A}^{-1}$ is
bounded. By the Lumer-Phillips Theorem [23], $\mathcal{A}$ generates a $C_{0}$
semigroup $e^{{\cal A}t}$ of contractions on $\mathcal{H}$. The proof is
completed. ∎

Our interest is the stability properties of system (1.1), especially the
polynomial stability when $\alpha,\;\beta,\;\gamma$ satisfy certain
conditions. Let us recall the corresponding definition.

######  Definition 2.1.

Let $\mathcal{A}\,:\,\mathcal{D}(\mathcal{A})\subset\mathcal{H}\to\mathcal{H}$
generate a $C_{0}$ semigroup $e^{\mathcal{A}t}$ of contractions on a Hilbert
space $\mathcal{H}$.

  1. (i)

$e^{\mathcal{A}t}$ is said to be strongly stable if
$\underset{t\to\infty}{\lim}\|e^{\mathcal{A}t}x_{0}\|_{\mathcal{H}}=0,\quad\forall\;x_{0}\in\mathcal{H};$

  2. (ii)

$e^{\mathcal{A}t}$ is said to be exponentially stable with decay rate
$\omega>0$, if there exists a constant $C>0$ such that

$\|e^{\mathcal{A}t}\|_{{\cal L}(\mathcal{H})}\leq Ce^{-\omega
t},\quad\forall\;t\geq 0;$

  3. (iii)

$e^{\mathcal{A}t}$ is said to be polynomially stable of order $k>0$ if there
exists a constant $C>0$ such that

$\|e^{\mathcal{A}t}\mathcal{A}^{-1}\|_{{\cal L}(\mathcal{H})}\leq
Ct^{-k},\quad\forall\;t>0.$

Following the result in [2, 13], the strong stability of the semigroup
$e^{t\mathcal{A}}$ is equivalent to $i{\mathbb{R}}\subset\rho(\mathcal{A})$
since $0\in\rho(\mathcal{A})$ and $\mathcal{A}^{-1}$ is compact ([10]). The
proof of the following lemma will be given in Appendix.

######  Lemma 2.2.

Assume that $(\alpha,\beta,\gamma)\in E$ satisfies $\beta\geq 2\alpha-1$. Then
$i{\mathbb{R}}\subset\rho(\mathcal{A})$ for $m\geq 0$, and consequently, the
semigroup $e^{t\mathcal{A}}$ is strongly stable.

We further discuss the explicit decay rates of the system with
$(\alpha,\beta,\gamma)\in E$. To begin with, we divide the parameter space $E$
into the following regions according to the computation in Section 5.

######  Definition 2.2.

The parameter space
$E=\left\\{(\alpha,\beta,\gamma)\mid\alpha\in[0,1],\beta\in[0,1],\gamma\in(0,1]\right\\}$
can be partitioned into the following non-overlapping regions

$\displaystyle T_{1}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;\frac{1}{2}<\alpha<\frac{\beta+1}{2},\;0<\beta\leq
1,\;0<\gamma<2\alpha-\beta\right\\},$ $\displaystyle T_{2}=$
$\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;0\leq\alpha<\frac{1}{2},\;0<\beta<1,\;0<\gamma<1-\beta\right\\},$
$\displaystyle T_{3}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;0\leq\alpha<\frac{\beta+\gamma}{2},\;1-\gamma<\beta\leq
1,\;0<\gamma\leq 1\right\\},$ $\displaystyle T_{4}=$
$\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;\frac{\beta+1}{2}<\alpha\leq 1,\;0\leq\beta<1,\;0<\gamma\leq
1\right\\},$ $\displaystyle F_{12}=$
$\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;\alpha=\frac{1}{2},\;0<\beta<1-\gamma,\;0<\gamma<1\right\\},$
$\displaystyle F_{13}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;\alpha=\frac{\beta+\gamma}{2},\;1-\gamma<\beta\leq
1,\;0<\gamma<1\right\\},$ $\displaystyle F_{14}=$
$\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;\alpha=\frac{\beta+1}{2},\;0<\beta\leq 1,\;0<\gamma<1\right\\},$
$\displaystyle F_{2}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;0\leq\alpha<\frac{1}{2},\;\beta=0,\;0<\gamma<1\right\\},$
$\displaystyle F_{23}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;0\leq\alpha<\frac{1}{2},\;\beta=1-\gamma,\;0<\gamma<1\right\\},$
$\displaystyle L_{123}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;\alpha=\frac{1}{2},\;\ \beta=1-\gamma,\;\ 0<\gamma<1\right\\},$
$\displaystyle L_{124}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;\alpha=\frac{1}{2},\ \beta=0,\ 0<\gamma<1\right\\},\;$
$\displaystyle L_{2}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;0\leq\alpha<\frac{1}{2},\;\beta=0,\;\gamma=1\right\\},\;$
$\displaystyle L_{34}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;\alpha=\frac{\beta+1}{2},0<\beta\leq 1,\;\gamma=1\right\\},$
$\displaystyle P_{234}=$ $\displaystyle\left\\{(\alpha,\beta,\gamma)\in
E\;\Big{|}\;\alpha=\frac{1}{2},\;\beta=0,\;\gamma=1\right\\}.$

######  Remark 2.1.

From the definition, it is clear that planes
$F_{12},\;F_{13},\;F_{14},\;F_{23}$ are (parts) of the intersections of
$T_{i}$ and $T_{j}$, lines $L_{123},\;L_{124}$ are intersections of
$T_{i},\;T_{j},\;T_{k}$. Furthermore, plane $F_{2}$ is a part of the boundary
of $T_{2}$, $L_{2}$ is a special part of the boundary of $F_{2}$, $L_{34}$ is
a part of the boundary of $F_{14}$, $P_{234}$ is a point. The regions in
Definition 2.2 are plotted in Figure 2.

It is known from [10, Theorem 2.1] that on the plane $F_{13}$, the lines
$L_{123}$, $L_{34}$ and the point $P_{234}$, the semigroup is exponentially
stable. On the region $T_{4},$ $0$ is a spectrum of $\mathcal{A}$. Moreover,
the asymptotic behavior on $L_{2}$ is not clear now (see computation in
Section 5). In this paper, we ignore these cases.

Figure 1: Regions of polynomial stability when $m>0$.

(a) $\gamma=\frac{1}{2}$

(b) $\gamma=\frac{1}{4}$

(c) $\gamma=\frac{1}{2}$

(d) $\gamma=\frac{3}{4}$

Figure 2: Partition of $E$ according to Definition 2.2 when $\gamma$ is
$\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, respectively.

According to the computation in Section 5, we shall analyze the polynomial
stability of system (1.1) with $m\not=0$ when parameters
$(\alpha,\,\beta,\,\gamma)$ belong to the following three regions:

$\begin{array}[]{l}\displaystyle V_{1}:=T_{1}\cup F_{12}\cup F_{14}\cup
L_{124},\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
V_{2}:=T_{2}\cup F_{2},\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr\displaystyle
V_{3}:=T_{3}\cup F_{23}.\end{array}$ (2.6)

The regions $V_{i},\;i=1,2,3$ are shown in Figure 1. The first main result of
this paper read as follows.

######  Theorem 2.1.

Let $m>0$ and $V_{i},i=1,2,3$ be defined as in (2.6). Then semigroup
$e^{\mathcal{A}t}$ associated with (2.1) has the following stability
properties:

  1. (i)

It is polynomially stable of order
$k_{1}=\displaystyle\frac{2\alpha-\gamma}{2(2\alpha-\beta-\gamma)}$ in
$V_{1}$;

  2. (ii)

It is polynomially stable of order
$k_{2}=\displaystyle\frac{1-\gamma}{2(-2\alpha-\beta-\gamma+2)}$ in $V_{2}$;

  3. (iii)

It is polynomially stable of order
$k_{3}=\displaystyle\frac{1-\gamma}{2(-2\alpha+\beta+\gamma)}$ in $V_{3}$.

We also consider the case when there is no inertial term in system (2.1).
Similarly, we partition the parameter space $E^{*}:=[0,1]^{2}$ as follows.

######  Definition 2.3.

The parameter space
$E^{*}=\left\\{(\alpha,\beta)\mid\alpha\in[0,1],\beta\in[0,1]\right\\}$ can be
partitioned into the following non-overlapping regions

$\displaystyle T_{1}^{*}=$ $\displaystyle\left\\{(\alpha,\beta)\in
E\Big{|}\frac{1}{2}<\alpha<\frac{\beta+1}{2},0<\beta\leq 1\right\\},$
$\displaystyle T_{2}^{*}=$ $\displaystyle\left\\{(\alpha,\beta)\in
E\Big{|}0\leq\alpha<\frac{1}{2},0<\beta<1\right\\},$ $\displaystyle
T_{4}^{*}=$ $\displaystyle\left\\{(\alpha,\beta)\in
E\Big{|}\frac{\beta+1}{2}<\alpha\leq 1,0\leq\beta<1\right\\},$ $\displaystyle
F^{*}_{12}=$ $\displaystyle\left\\{(\alpha,\beta)\in
E\Big{|}\alpha=\frac{1}{2},0<\beta<1\right\\},$ $\displaystyle F^{*}_{14}=$
$\displaystyle\left\\{(\alpha,\beta)\in
E\Big{|}\alpha=\frac{\beta+1}{2},0<\beta\leq 1\right\\},$ $\displaystyle
F^{*}_{2}=$ $\displaystyle\left\\{(\alpha,\beta)\in
E\Big{|}0\leq\alpha<\frac{1}{2},\beta=0\right\\},$ $\displaystyle
L^{*}_{124}=$ $\displaystyle\left\\{(\alpha,\beta)\in
E\Big{|}\alpha=\frac{1}{2},\beta=0\right\\},$ $\displaystyle L^{*}_{23}=$
$\displaystyle\left\\{(\alpha,\beta)\in
E\Big{|}0\leq\alpha<\frac{1}{2},\beta=1\right\\},$ $\displaystyle
P^{*}_{123}=$ $\displaystyle\left\\{(\alpha,\beta)\in
E\Big{|}\alpha=\frac{1}{2},\beta=1\right\\}.$

We plot the regions given in Definition 2.3 in Figure 4. The lines
$F^{*}_{12},F^{*}_{14}$ denote the boundary of $T^{*}_{1}$, $T^{*}_{2}$ and
$T^{*}_{1}$, $T^{*}_{4}$, respectively. The line $F^{*}_{2}$ and $L_{23}$ are
the boundary of $T^{*}_{2}$. Point $L^{*}_{124}$ is the intersection of
$T_{i}^{*},i=1,2,4$. The region $T^{*}_{4}$ is the area where $0$ is a
spectrum of $\mathcal{A}$. It is known from [10, Theorem 2.2] that the system
is exponentially stable on the point $P_{123}^{*}$. We consider the decay rate
of the solution when parameters belonging to the remaining area
$E^{*}\backslash(T_{4}^{*}\cup P^{*}_{123})$ and partition the region into the
following two parts according to the computation in Section 5.

$\begin{array}[]{l}\displaystyle V_{1}^{*}:=T_{1}^{*}\cup F^{*}_{12}\cup
F^{*}_{14}\cup L^{*}_{124},\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle V_{2}^{*}:=T_{2}^{*}\cup F^{*}_{2}\cup
L^{*}_{23}.\end{array}$ (2.7)

Figure 3: Partition of $E^{*}$ according to Definition 2.3

Figure 4: Partition of $V_{i}^{*},i=1,2$

The regions of $V_{i}^{*}$ are shown in Figure 4. The second main result is as
follows.

######  Theorem 2.2.

Let $m=0$ and $V_{i}^{*},i=1,2$ be defined as in(2.7). Then semigroup
$e^{\mathcal{A}t}$ associated with (2.1) satisfies

  1. (i)

It is polynomially stable of order $\displaystyle
k_{1}^{*}=\frac{\alpha}{2\alpha-\beta}$ in $V_{1}^{*}$;

  2. (ii)

It is polynomially stable of order $\displaystyle
k_{2}^{*}=\frac{1}{2(2-2\alpha-\beta)}$ in $V_{2}^{*}$.

Furthermore, through a detailed and creative spectral analysis, we can show
the sharpness of the decay rates obtained in the above two theorems.

######  Theorem 2.3.

The orders of polynomial decay in Theorem 2.1 and Theorem 2.2 are all optimal.

In Theorem 2.1 and 2.2, we provide a complete polynomial stability analysis
for system (1.1). When $m>0$, i.e., the system (1.1) is with an inertial term,
we divide the parameter region into three parts, where the semigroup decays
polynomially at different rates. In the case where there is no inertial term
(i.e., $m=0$), we also establish a relationship between the parameters
$\alpha,\;\beta$ and the order of polynomial stability. Our results complement
the stability analysis of the system previously discussed in [10]. On the
other hand, it is known that the decay rates, as obtained in Theorem 2.2, are
slower when compared to the findings presented in [12]. The works in [12]
focused on a thermoelastic system with Fourier’s law while neglecting the
inertial term. This observation reveals that the thermal damping of Cattaneo’s
type is weaker than the Fourier’s law.

Our subsequent findings on polynomial stability reply on the following lemmas,
which provide necessary and sufficient conditions for the polynomial stability
of $C_{0}$ semigroup (see [5, 18] or [4, 24] for more general case).

######  Lemma 2.3.

Assume that $\mathcal{A}$ generates a $C_{0}$ contraction semigroup
$e^{\mathcal{A}t}$ on a Hilbert space $\mathcal{H}$ and satisfies
$i\mathbb{R}\subset\rho(\mathcal{A})$. Then, $e^{\mathcal{A}t}$ is
polynomially stable with order $k$, if and only if

$\limsup_{|s|\to\infty}\limits s^{-{1\over
k}}\|(is-\mathcal{A})^{-1}\|_{\mathcal{L}(\mathcal{H})}<\infty.$ (2.8)

The following interpolation theorem will play a crucial role in our proof.

######  Lemma 2.4.

Let $A\,:\,{\cal D}(A)\subset H$ be self-adjoint and positive definite,
$r,\;p\;,q\in\mathbb{R}$. Then

$\displaystyle\|A^{p}x\|\leq\|A^{q}x\|^{p-r\over q-r}\|A^{r}x\|^{q-p\over
q-r},\quad\forall\;r\leq p\leq q,\;x\in{\cal D}(A^{q}).$ (2.9)

###### Proof.

The case $0\leq r\leq p\leq q$ can be found in [19].

If $r\leq 0$ and $0\leq p\leq q$, we let $y\in D(A^{q-r}),\;x=A^{-r}y.$ It
follows from (2.9) that

$\|A^{p}x\|=\|A^{p-r}y\|\leq\|A^{q-r}y\|^{{p-r\over q-r}}\|y\|^{q-p\over
q-r}=\|A^{q}x\|^{p-r\over q-r}\|A^{r}x\|^{q-p\over q-r}.$

The other cases can be proved by the same argument.

∎

##  3 Polynomial Stability of the System with an Inertial Term (Proof of
Theorem 2.1)

This section is devoted to analyzing the polynomial stability of system (1.1)
with $m>0$. Due to Lemma 2.2 and 2.3, it is sufficient to show that there
exists a constant $r>0$ such that

$\inf\big{\\{}\lambda^{k}\|i\,\lambda U-{\cal A}U\|_{\cal
H}\,\big{|}\quad\|U\|_{\cal H}=1,\>\lambda\in{\mathbb{R}}\big{\\}}\geqslant r$
(3.1)

holds for some $k>0$.

Suppose by contradiction that (3.1) fails. Then there exists a sequence
$\\{\lambda_{n},U_{n}\\}_{n=1}^{\infty}$, where
$\lambda_{n}\in{\mathbb{R}},\;U_{n}:=(u_{n},\,{v}_{n},\,\theta_{n},\,q_{n})\in
D({\cal A})$ such that

$\|U_{n}\|_{\cal
H}=\big{\|}(u_{n},\,{v}_{n},\,\theta_{n},\,q_{n})\big{\|}_{\cal H}=1,$ (3.2)

and as $n\to\infty$,

$\displaystyle\lambda_{n}\to\infty,$
$\displaystyle\displaystyle\lambda_{n}^{k}\big{(}i\,\lambda_{n}U_{n}-{\cal
A}U_{n}\big{)}=o(1)\quad\;\mbox{ in }\;{\cal H},$ (3.3)

i.e.,

$\displaystyle\lambda_{n}^{k}(i\lambda_{n}A^{\frac{1}{2}}u_{n}-A^{\frac{1}{2}}v_{n})=o(1)\quad\;\mbox{
in }\;H,$ (3.4)
$\displaystyle\lambda_{n}^{k}A^{-\frac{\gamma}{2}}\left(i\lambda_{n}v_{n}+i\lambda_{n}mA^{\gamma}v_{n}+\sigma
Au_{n}-A^{\alpha}\theta_{n}\right)=o(1)\quad\;\mbox{ in }\;H,$ (3.5)
$\displaystyle\lambda_{n}^{k}(i\lambda_{n}\theta_{n}-A^{\frac{\beta}{2}}q_{n}+A^{\alpha}v_{n})=o(1)\quad\;\mbox{
in }\;H,$ (3.6) $\displaystyle\lambda_{n}^{k}(i\lambda_{n}\tau
q_{n}+q_{n}+A^{\frac{\beta}{2}}\theta_{n})=o(1)\quad\;\mbox{ in }\;H.$ (3.7)

######  Lemma 3.1.

For $m\geq 0$, one has

$\displaystyle\|q_{n}\|=\lambda_{n}^{-\frac{k}{2}}o(1),$ (3.10)
$\displaystyle\|A^{\frac{\beta}{2}}\theta_{n}\|=\lambda_{n}^{1-\frac{k}{2}}o(1),$
$\displaystyle\|A^{\frac{1}{2}}u_{n}\|,\;\|v_{n}\|,\;\|\theta_{n}\|,\;\|\lambda_{n}^{-1}A^{\frac{1}{2}}v_{n}\|=O(1).$

Furthermore, when $m>0,$ it holds

$\|A^{\gamma\over 2}v_{n}\|,\;\;\|\lambda_{n}A^{\gamma\over 2}u_{n}\|=O(1).$
(3.11)

###### Proof.

We can obtain (3.10) by the dissipativeness of $\mathcal{A}$ and (3.3). Then,
combining (3.7) and (3.10) yields (LABEL:v1_8). (LABEL:+211) and (3.11) are
clear from (3.2) and (3.4). ∎

Now, we proceed to show (i)–(iii) in Theorem 2.1, respectively.

(i)  Assume $(\alpha,\,\beta,\,\gamma)\in V_{1}$ and let
$k=\frac{2(2\alpha-\beta-\gamma)}{2\alpha-\gamma}$.

From (3.6) and the assumption $-\alpha+\frac{\gamma}{2}<-{\beta\over 2}\leq
0$, we have that

$i\lambda_{n}A^{-\alpha+\frac{\gamma}{2}}\theta_{n}-A^{-\alpha+\frac{\gamma}{2}+\frac{\beta}{2}}q_{n}+A^{\frac{\gamma}{2}}v_{n}=o(1).$

Combining these with (3.10), (3.11) yields

$\|A^{-\alpha+\frac{\gamma}{2}}\theta_{n}\|=\lambda_{n}^{-1}O(1).$ (3.12)

By using Lemma 2.4, (LABEL:v1_8), (3.12) and the choice of $k$ given above,
one has

$\left\|\theta_{n}\right\|\leq\|A^{\frac{\beta}{2}}\theta_{n}\|^{\frac{2\alpha-\gamma}{2\alpha+\beta-\gamma}}\|A^{-\alpha+\frac{\gamma}{2}}\theta_{n}\|^{\frac{\beta}{2\alpha+\beta-\gamma}}=o(1).$
(3.13)

Note that $A^{-\alpha+\frac{\gamma}{2}}\theta_{n}$ and
$A^{-\alpha}(I+mA^{\gamma})v_{n}$ are bounded due to
$-\alpha+\frac{\gamma}{2}<0$ and (3.11). Then, taking the inner product of
(3.5) with $\lambda_{n}^{-k}A^{-\alpha+\frac{\gamma}{2}}\theta_{n}$, (3.6)
with $\lambda_{n}^{-k}A^{-\alpha}(I+mA^{\gamma})v_{n}$, respectively, we have

$(\theta_{n},i\lambda_{n}A^{-\alpha}(I+{m}A^{\gamma})v_{n})+\sigma(A^{1-\alpha}\theta_{n},u_{n})-(\theta_{n},\theta_{n})=o(1),$
$(i\lambda_{n}\theta_{n},A^{-\alpha}(I+mA^{\gamma})v_{n})-(A^{\frac{\beta}{2}-\alpha}(I+{m}A^{\gamma})q_{n},v_{n})+((I+mA^{\gamma})v_{n},v_{n})=o(1).$

Adding the above two equations, we get

$\sigma(A^{1-\alpha}\theta_{n},\,u_{n})-(A^{\frac{\beta}{2}-\alpha}(I+mA^{\gamma})q_{n},\,v_{n})-\|\theta_{n}\|^{2}+\|v_{n}\|^{2}+m\|A^{\frac{\gamma}{2}}v_{n}\|^{2}=o(1).$
(3.14)

Combining $\alpha>{1\over 2},\;\alpha>{\beta+\gamma\over 2}$ and (3.10),
(LABEL:+211), (3.11), (3.13), one has

$(A^{{1\over 2}-\alpha}\theta_{n},\,A^{1\over
2}u_{n}),\;\;(A^{\frac{\beta+\gamma}{2}-\alpha}q_{n},\,A^{-{\gamma\over
2}}(I+mA^{\gamma})v_{n})=o(1).$ (3.15)

Substituting (3.15) into (3.14), along with (3.13), yields

$\|v_{n}\|,~{}~{}\|A^{\frac{\gamma}{2}}v_{n}\|=o(1).$ (3.16)

Now, taking the inner product of (3.5) with
${\lambda_{n}^{-k}}A^{\frac{\gamma}{2}}u_{n}$ on $H$, we get

$(i\lambda_{n}v_{n},u_{n})+(i\lambda_{n}{m}A^{\gamma}v_{n},u_{n})+\sigma(Au_{n},u_{n})-(A^{\alpha}\theta_{n},u_{n})\to
0,$ (3.17)

which together with (3.4), one can conclude that

$-\|v_{n}\|^{2}-m\|A^{\frac{\gamma}{2}}v_{n}\|^{2}+\sigma\|A^{\frac{1}{2}}u_{n}\|^{2}-(A^{\alpha}\theta_{n},u_{n})=o(1).$
(3.18)

Thanks to $\alpha-{\beta\over 2}\leq{1\over 2}$, $\eqref{s1+}$ and
(LABEL:v1_8), we get

$(A^{\alpha}\theta_{n},\,u_{n})=(A^{\frac{\beta}{2}}\theta_{n},\,(i\lambda_{n})^{-1}A^{\alpha-\frac{\beta}{2}}v_{n}+\lambda_{n}^{-1-k}o(1))=(A^{\frac{\beta}{2}}\theta_{n},\,(i\lambda_{n})^{-1}A^{\alpha-\frac{\beta}{2}}v_{n})+{\lambda_{n}^{-\frac{3k}{2}}}o(1).$
(3.19)

By $\eqref{s3+}$, we know

$A^{\alpha-\frac{\beta}{2}}v_{n}=-i\lambda_{n}A^{-\frac{\beta}{2}}\theta_{n}+q_{n}+\lambda_{n}^{-k}o(1).$
(3.20)

Thus, by substituting (3.20) into (3.19) and using (3.10), (LABEL:v1_8),
(3.13), we conclude that

$(A^{\alpha}\theta_{n},\,u_{n})=o(1).$ (3.21)

Combining (3.16), (3.18) and (3.21) yields

$\|A^{\frac{1}{2}}u_{n}\|={o(1)}.$ (3.22)

In summary, by (3.10), (3.13), (3.16) and (3.22), we obtain that
$\|U_{n}\|_{\mathcal{H}}=o(1)$, which contradicts the assumption (3.2). The
proof of (i) is completed.

(ii) Assume $(\alpha,\;\beta,\;\gamma)\in V_{2}$, and let
$k=\frac{2(-2\alpha-\beta-\gamma+2)}{1-\gamma}$.

We first show that for $\alpha<{1\over 2}$,

$\|A^{-\frac{1}{2}}(I+mA^{\gamma})v_{n}\|=\lambda_{n}^{-1}O(1).$ (3.23)

In fact, it follows from (3.5) that

$\|\lambda_{n}A^{-\frac{1}{2}}(I+mA^{\gamma})v_{n}\|\leq\sigma\|A^{1\over
2}u_{n}\|+\|A^{-{1\over 2}+\alpha}\theta_{n}\|+\lambda_{n}^{-k}o(1).$ (3.24)

Thus, (3.23) follows from (LABEL:+211) and (3.24).

Taking the inner product of (3.6) with $\lambda_{n}^{-1-k}\theta_{n}$ on $H$
yields

$i\|\theta_{n}\|^{2}-\lambda_{n}^{-1}(A^{\frac{\beta}{2}}q_{n},\theta_{n})+\lambda_{n}^{-1}(A^{\alpha}v_{n},\theta_{n})=\lambda_{n}^{-1-k}o(1).$
(3.25)

From (3.10) and (LABEL:v1_8), it is easy to see that

$|\lambda_{n}^{-1}(A^{\frac{\beta}{2}}q_{n},\theta_{n})|\leq\|q_{n}\|\|\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n}\|=\lambda_{n}^{-k}o(1).$
(3.26)

To deal with the third term of (3.25), we use the Cauchy-Schwarz inequality
and (LABEL:v1_8) to get

$|\lambda_{n}^{-1}(A^{\alpha}v_{n},\theta_{n})|\leq\|A^{\alpha-\frac{\beta}{2}}v_{n}\|\|\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n}\|={\lambda_{n}}^{-\frac{k}{2}}\|A^{\alpha-\frac{\beta}{2}}v_{n}\|o(1).$
(3.27)

Furthermore, by Lemma 2.4, (LABEL:+211), (3.11), (3.23) and $\beta+\gamma<1$,
we deduce that

$\|A^{\alpha-\frac{\beta}{2}}v_{n}\|\leq\left\\{\begin{array}[]{ll}O(1)&\;\mbox{when}\;\;\displaystyle
0\leq\alpha\leq\frac{\gamma}{2},\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle{\|A^{\gamma-\frac{1}{2}}v_{n}\|^{1-\frac{2\alpha-\beta-2\gamma+1}{2(1-\gamma)}}\|A^{\frac{1}{2}}v_{n}\|^{\frac{2\alpha-\beta-2\gamma+1}{2(1-\gamma)}}=\lambda_{n}^{\frac{2\alpha-\beta-\gamma}{1-\gamma}}O(1)}&\;\mbox{when}\;\;\displaystyle\frac{\gamma}{2}<\alpha<{1\over
2}.\par\end{array}\right.$ (3.28)

Substituting (3.26)-(3.28) into (3.25) and noticing the choice of $k$, we have

$\|\theta_{n}\|=\left\\{\begin{array}[]{ll}\lambda_{n}^{-{k\over
4}}o(1)&\;\mbox{when}\;\;\displaystyle 0\leq\alpha\leq\frac{\gamma}{2},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\lambda_{n}^{\frac{2\alpha-1}{1-\gamma}}o(1)&\;\mbox{when}\;\;\displaystyle\frac{\gamma}{2}<\alpha<\frac{1}{2}.\end{array}\right.$
(3.29)

Consequently, one has

$\|\theta_{n}\|=o(1).$ (3.30)

Now we proceed to prove that

$\|v_{n}\|,\;\;\|A^{\frac{\gamma}{2}}v_{n}\|=o(1).$ (3.31)

Based on the estimation (3.29), we shall prove (3.31) by considering the
following two cases, respectively (Case ii-A and B).

Case ii-A. Assume $\alpha\leq\frac{\gamma}{2}$.

Note that $\|A^{\frac{\beta-1}{2}}(I+mA^{\gamma})v_{n}\|\leq
C\|A^{\frac{\beta+\gamma-1}{2}}A^{\gamma\over 2}v_{n}\|=O(1)$ due to
$\beta+\gamma<1$ and (3.11). Then, we can take the inner product of (3.6) with
${\lambda_{n}^{-k}}A^{\frac{\beta-1}{2}}(I+mA^{\gamma})v_{n}$ to get

$\begin{array}[]{l}-(A^{\frac{\beta}{2}}\theta_{n},i\lambda_{n}A^{-\frac{1}{2}}(I+mA^{\gamma})v_{n})-(q_{n},A^{\frac{2\beta-1}{2}}(I+mA^{\gamma})v_{n})\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle+\|A^{\frac{2\alpha+\beta-1}{4}}v_{n}\|^{2}+m\|A^{\frac{2\alpha+2\gamma+\beta-1}{4}}v_{n}\|^{2}=\lambda_{n}^{-k}o(1).\end{array}$
(3.32)

By (LABEL:+211) and $\beta+\gamma<1$, we have

$\|A^{\frac{2\beta-1}{2}}(I+mA^{\gamma})v_{n}\|\leq
C\|A^{\beta+\gamma-1}A^{{1\over 2}}v_{n}\|=\lambda_{n}O(1).$

Then, substituting the above and (3.10), (LABEL:v1_8), (3.23) into (3.32), we
have

$\|A^{\frac{2\alpha+\beta-1}{4}}v_{n}\|,~{}~{}\|A^{\frac{2\alpha+2\gamma+\beta-1}{4}}v_{n}\|=\lambda_{n}^{\frac{1}{2}-\frac{k}{4}}o(1)=\lambda_{n}^{\frac{2\alpha+\beta-1}{2(1-\gamma)}}o(1).$
(3.33)

Thanks to $\alpha\leq\frac{\gamma}{2}$, we easily obtain
$\frac{2\alpha+2\gamma+\beta-1}{4}<\frac{\gamma}{2}<\frac{1}{2}$, and thus, by
interpolation, one gets

$\|A^{\frac{\gamma}{2}}v_{n}\|\leq\|A^{\frac{2\alpha+2\gamma+\beta-1}{4}}v_{n}\|^{1-\frac{1-\beta-2\alpha}{3-\beta-2\alpha-2\gamma}}\|A^{\frac{1}{2}}v_{n}\|^{\frac{1-\beta-2\alpha}{3-\beta-2\alpha-2\gamma}}.$
(3.34)

Therefore, we obtain (3.31) by combining (LABEL:+211), (3.33) and (3.34).

Case ii-B. Assume $\frac{\gamma}{2}<\alpha<{1\over 2}.$

Using Lemma 2.4, (3.11) and (3.23) yields

$\|A^{-\alpha}(I+mA^{\gamma})v_{n}\|\leq\|A^{-\frac{1}{2}}(I+mA^{\gamma})v_{n}\|^{\frac{2\alpha-\gamma}{1-\gamma}}\|A^{-\frac{\gamma}{2}}(I+mA^{\gamma})v_{n}\|^{1-\frac{2\alpha-\gamma}{1-\gamma}}=\lambda_{n}^{\frac{-2\alpha+\gamma}{1-\gamma}}O(1).$
(3.35)

Taking the inner product of (3.6) with
$\lambda_{n}^{-k}A^{-\alpha}(I+mA^{\gamma})v_{n}$, we get

$(i\lambda_{n}\theta_{n},A^{-\alpha}(I+mA^{\gamma})v_{n})-(q_{n},A^{\frac{\beta}{2}-\alpha}(I+mA^{\gamma})v_{n})+((I+mA^{\gamma})v_{n},v_{n})=o(1).$
(3.36)

Combining (3.35) and (3.29), one has

$(i\lambda_{n}\theta_{n},A^{-\alpha}(I+mA^{\gamma})v_{n})=o(1).$ (3.37)

To estimate the second term of (3.36), we notice that if
$\frac{\gamma}{2}<\alpha\leq\frac{\beta+\gamma}{2}$,

$\|A^{-\alpha+\frac{\beta}{2}+\gamma}v_{n}\|\leq\|A^{\frac{\gamma}{2}}v_{n}\|^{1-\frac{-2\alpha+\beta+\gamma}{1-\gamma}}\|A^{\frac{1}{2}}v_{n}\|^{\frac{-2\alpha+\beta+\gamma}{1-\gamma}}=\lambda_{n}^{\frac{-2\alpha+\beta+\gamma}{1-\gamma}}O(1),$
(3.38)

where we have used Lemma 2.4, 3.1 and $\beta+\gamma<1$. Therefore, from
(3.10), (LABEL:+211) and (3.38), we can deduce that

$|(q_{n},A^{\frac{\beta}{2}-\alpha}(I+mA^{\gamma})v_{n})|\leq\left\\{\begin{array}[]{ll}C\|q_{n}\|\|A^{-\alpha+\frac{\beta}{2}+\gamma}v_{n}\|=\lambda_{n}^{2(\beta+\gamma-1)\over
1-\gamma}o(1),&\displaystyle{\gamma\over
2}<\alpha\leq\frac{\beta+\gamma}{2},\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\|A^{-\alpha+\frac{\beta+\gamma}{2}}q_{n}\|\|A^{-\frac{\gamma}{2}}(I+mA^{\gamma})v_{n}\|=\lambda_{n}^{-{k\over
2}}o(1),&\displaystyle\frac{\beta+\gamma}{2}<\alpha<{1\over
2}.\end{array}\right.$ (3.39)

Therefore, we get (3.31) by substituting (3.37) and (3.39) into (3.36).

Finally, similar to (i), we can show that the fourth term in (3.18) tends to
zero. Indeed, we have
$|(A^{\alpha}\theta_{n},u_{n})|\leq\|A^{\alpha-\frac{1}{2}}\theta_{n}\|\|A^{\frac{1}{2}}u_{n}\|=o(1)$
thanks to (LABEL:+211), (3.30) and $\alpha<\frac{1}{2}$. Thus, by (3.18),
along with (3.31), we get $\|A^{\frac{1}{2}}u_{n}\|=o(1)$. Therefore,
combining this with (3.30), (3.31), one can arrive at the contradiction
$\|U_{n}\|_{\mathcal{H}}=o(1)$.

(iii) Assume $(\alpha,\beta,\gamma)\in V_{3}$, and let
$k=\displaystyle\frac{2(-2\alpha+\beta+\gamma)}{1-\gamma}$.

Taking the inner product of (3.6) with $\lambda_{n}^{-k}\theta_{n}$ yields

$i\|\theta_{n}\|^{2}-\lambda_{n}^{-1}(A^{\frac{\beta}{2}}q_{n},\theta_{n})+\lambda_{n}^{-1}(A^{\alpha}v_{n},\theta_{n})=\lambda_{n}^{-1-k}o(1).$
(3.40)

From Lemma 3.1, it is easy to get

$|(q_{n},\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n})|\leq\|q_{n}\|\|\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n}\|={\lambda_{n}^{-k}}o(1),$
(3.41)

and thanks to $\alpha<\frac{\beta+\gamma}{2}$, we have

$|(A^{\alpha}v_{n},\lambda_{n}^{-1}\theta_{n})|\leq\|A^{\alpha-\frac{\beta}{2}}v_{n}\|\|\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n}\|={\lambda_{n}^{-\frac{k}{2}}}o(1).$
(3.42)

Thus, from (3.40)-(3.42), we obtain

$\|\theta_{n}\|={\lambda_{n}}^{-\frac{k}{4}}o(1).$ (3.43)

Now, let us divide $V_{3}$ into four regions (see the 2-D picture Figure 5),
and we will prove $\|v_{n}\|,$ $\|A^{\frac{\gamma}{2}}v_{n}\|,$
$\|A^{\frac{1}{2}}u_{n}\|=o(1)$ in each region, respectively (Case iii-A, B,
C, D).

Figure 5: Subregions of $V_{3}$.

Case iii-A. Let $(\alpha,\beta,\gamma)\in V_{3}$ and
$0\leq\alpha\leq\frac{\gamma}{2}$.

Since $\beta+\gamma\geq 1$ and (3.5), (LABEL:+211), (3.43), it is easy to get

$\|A^{-\frac{\beta+\gamma}{2}}\left(I+mA^{\gamma}\right)v_{n}\|\leq\lambda_{n}^{-1}(\sigma\|A^{1-\frac{\beta+\gamma}{2}}u_{n}\|+\|A^{\alpha-\frac{\beta+\gamma}{2}}\theta_{n}\|+\lambda_{n}^{-k}o(1))=\lambda_{n}^{-1}O(1).$

Consequently, we have

$\|A^{{1\over 2}-\beta-\gamma}\left(I+mA^{\gamma}\right)v_{n}\|=\|A^{{1\over
2}-\frac{\beta+\gamma}{2}}A^{-\frac{\beta+\gamma}{2}}\left(I+mA^{\gamma}\right)v_{n}\|=\lambda_{n}^{-1}O(1).$
(3.44)

Taking the inner product of (3.6) with
$\lambda_{n}^{-k}A^{\frac{1}{2}-\frac{\beta}{2}-\gamma}(I+mA^{\gamma})v_{n}$,
we have

$\begin{array}[]{l}i\lambda_{n}(A^{\frac{\beta}{2}}\theta_{n},A^{\frac{1}{2}-\beta-\gamma}(I+mA^{\gamma})v_{n})-(q_{n},A^{{1\over
2}-\gamma}(I+mA^{\gamma})v_{n})\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle+\|A^{{1\over
2}(\alpha-\frac{\beta}{2}-\gamma+\frac{1}{2})}v_{n}\|^{2}+m\|A^{{1\over
2}(\alpha-\frac{\beta}{2}+\frac{1}{2})}v_{n}\|^{2}=\lambda_{n}^{-k}o(1).\end{array}$
(3.45)

We deduce from (3.10)-(LABEL:+211) and (3.44) that

$i\lambda_{n}(A^{\frac{\beta}{2}}\theta_{n},A^{\frac{1}{2}-\beta-\gamma}(I+mA^{\gamma})v_{n}),\;\;(q_{n},A^{{1\over
2}-\gamma}(I+mA^{\gamma})v_{n})=\lambda_{n}^{1-\frac{k}{2}}o(1).$ (3.46)

Combining (3.45)-(3.46) yields

$\|A^{{1\over
2}(\alpha-\frac{\beta}{2}-\gamma+\frac{1}{2})}v_{n}\|,\quad\|A^{{1\over
2}(\alpha-\frac{\beta}{2}+\frac{1}{2})}v_{n}\|=\lambda_{n}^{\frac{1}{2}-\frac{k}{4}}o(1).$
(3.47)

Note that ${1\over
2}(\alpha-\frac{\beta}{2}+\frac{1}{2})\leq\frac{\gamma}{2}<\frac{1}{2}$ due to
$\alpha\leq{\gamma\over 2}$ and $\beta+\gamma\geq 1.$ Then using Lemma 2.4,
along with (LABEL:+211) and (3.47), one has

$\|A^{\frac{\gamma}{2}}v_{n}\|\leq\|A^{{1\over
2}(\alpha-\frac{\beta}{2}+\frac{1}{2})}v_{n}\|^{1-\frac{\beta+2\gamma-2\alpha-1}{1-2\alpha+\beta}}\|A^{\frac{1}{2}}v_{n}\|^{\frac{\beta+2\gamma-2\alpha-1}{1-2\alpha+\beta}}=o(1).$

Thus, (3.31) is proved. It is clear that
$|(A^{\alpha}\theta_{n},u_{n})|\leq\|\theta_{n}\|\|A^{\alpha}u_{n}\|=o(1)$ by
$\alpha\leq{\gamma\over 2}\leq{1\over 2}$, (3.2) and (3.43). Therefore, by
(3.18) along with (3.31), we obtain $\|A^{1\over 2}u_{n}\|=o(1)$, and
consequently, $\|U_{n}\|_{\cal H}=o(1)$ by (3.10), (3.18), (3.31) and (3.43).

Case iii-B. Let $(\alpha,\beta,\gamma)\in
V_{3},\;\frac{\gamma}{2}<\alpha\leq\frac{1}{2}$ and
$-\alpha+\frac{\beta}{2}+\gamma\leq\frac{1}{2}$.

Similar to (i), one has that (3.14) still holds since $\alpha>{\gamma\over
2}.$ Let us estimate the first two terms of (3.14). First, note that
$0<1-\alpha-\frac{\beta}{2}\leq\frac{1}{2}$ since
$\frac{\gamma}{2}<\alpha\leq\frac{1}{2}$ and $\gamma+\beta\geq 1.$ From
(LABEL:+211), (3.11), we can deduce that

$\|A^{1-\alpha-\frac{\beta}{2}}u_{n}\|\leq\left\\{\begin{array}[]{ll}\lambda_{n}^{-1}O(1),&0<1-\alpha-\frac{\beta}{2}\leq{\gamma\over
2},\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\|A^{\frac{\gamma}{2}}u_{n}\|^{1-\frac{2-2\alpha-\beta-\gamma}{1-\gamma}}\|A^{\frac{1}{2}}u_{n}\|^{\frac{2-2\alpha-\beta-\gamma}{1-\gamma}}=\lambda_{n}^{\frac{1-2\alpha-\beta}{1-\gamma}}O(1),&{\gamma\over
2}<1-\alpha-\frac{\beta}{2}\leq{1\over 2}.\end{array}\right.$ (3.48)

Consequently, by (LABEL:v1_8) and (3.48),

$|(A^{{\beta\over
2}}\theta_{n},A^{1-\alpha-\frac{\beta}{2}}u_{n})|\leq\|A^{{\beta\over
2}}\theta_{n}\|\|A^{1-\alpha-\frac{\beta}{2}}u_{n}\|=\left\\{\begin{array}[]{ll}\lambda_{n}^{-{k\over
2}}o(1),&0<1-\alpha-\frac{\beta}{2}\leq{\gamma\over 2},\\\ \vskip 6.0pt plus
2.0pt minus
2.0pt\cr\displaystyle\lambda_{n}^{\frac{2(1-\beta-\gamma)}{1-\gamma}}o(1),&{\gamma\over
2}<1-\alpha-\frac{\beta}{2}\leq{1\over 2}.\end{array}\right.$ (3.49)

Note that $\beta+\gamma\geq 1$, it turns out that

$|(A^{1-\alpha}\theta_{n},u_{n})|=o(1).$ (3.50)

By $\frac{\gamma}{2}<-\alpha+\frac{\beta}{2}+\gamma\leq\frac{1}{2}$, we have
the following estimation by Lemma 2.4, (LABEL:+211), (3.11),

$\|A^{-\alpha+\frac{\beta}{2}+\gamma}v_{n}\|\leq\|A^{\frac{\gamma}{2}}v_{n}\|^{1-\frac{-2\alpha+\beta+\gamma}{1-\gamma}}\|A^{\frac{1}{2}}v_{n}\|^{\frac{-2\alpha+\beta+\gamma}{1-\gamma}}=\lambda_{n}^{\frac{-2\alpha+\beta+\gamma}{1-\gamma}}O(1).$

This combined with (3.10) yields

$|(q_{n},A^{-\alpha+\frac{\beta}{2}}(I+mA^{\gamma})v_{n})|\leq
C\|q_{n}\|\|A^{-\alpha+\frac{\beta}{2}+\gamma}v_{n}\|=o(1).$ (3.51)

Substituting (3.43), (3.50), (3.51) into (3.14), we get
$\|v_{n}\|,\|A^{\gamma\over 2}v_{n}\|=o(1),$ i.e., (3.31) holds. Then, similar
to the discussion in Case iii-A, we can get
$|(A^{\alpha}\theta_{n},u_{n})|\leq\|\theta_{n}\|\|A^{\alpha}u_{n}\|=o(1)$ due
to $\alpha\leq{1\over 2}$ and (3.43), which together with (3.18) and (3.31)
implies $\|A^{1\over 2}u_{n}\|=o(1)$. Finally, combining this with (3.43), we
get the contradiction $\|U_{n}\|_{\cal H}=o(1).$

Case iii-C. Let $(\alpha,\beta,\gamma)\in
V_{3},\;\frac{\gamma}{2}<\alpha\leq\frac{1}{2},$ and
$-\alpha+\frac{\beta}{2}+\gamma>\frac{1}{2}$.

It follows from (3.6) that

$\|A^{\frac{\beta}{2}}q_{n}\|\leq\|\lambda_{n}\theta_{n}\|+\|A^{\alpha}v_{n}\|+\lambda_{n}^{-k}o(1).$
(3.52)

From (LABEL:+211), (3.11), by interpolation (2.9), we get

$\|A^{\alpha}v_{n}\|\leq\|A^{\gamma\over
2}v_{n}\|^{1-\frac{2\alpha-\gamma}{1-\gamma}}\|A^{1\over
2}v_{n}\|^{\frac{2\alpha-\gamma}{1-\gamma}}=\lambda_{n}^{\frac{2\alpha-\gamma}{1-\gamma}}O(1).$
(3.53)

We shall show that

$\|\lambda_{n}\theta_{n}\|=\left\\{\begin{array}[]{ll}\displaystyle\lambda_{n}^{1-\frac{k}{4}}o(1),&\mbox{if}~{}~{}~{}1-\frac{k}{4}\leq{\frac{2\alpha-\gamma}{1-\gamma}},\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\lambda_{n}^{\frac{2\alpha-\gamma}{1-\gamma}}o(1),&\mbox{if}~{}~{}~{}1-\frac{k}{4}>{\frac{2\alpha-\gamma}{1-\gamma}}.\end{array}\right.$
(3.54)

Therefore, combining (3.52)-(3.54), we see

$\|A^{\frac{\beta}{2}}q_{n}\|=\lambda_{n}^{\frac{2\alpha-\gamma}{1-\gamma}}O(1).$
(3.55)

Thanks to $0<-\alpha+\frac{\beta}{2}+\frac{\gamma}{2}<\frac{\beta}{2}$, Lemma
2.4, (3.10), (3.11) and (3.55), we obtain

$\begin{array}[]{ll}|(q_{n},A^{-\alpha+\frac{\beta}{2}}(I+mA^{\gamma})v_{n})|&\leq
C\|A^{-\alpha+\frac{\beta}{2}+\frac{\gamma}{2}}q_{n}\|\|A^{\frac{\gamma}{2}}v_{n}\|\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr&\leq
C\|A^{-\alpha+\frac{\beta}{2}+\frac{\gamma}{2}}q_{n}\|\\\ \displaystyle&\leq
C\|q_{n}\|^{\frac{2\alpha-\gamma}{\beta}}\|A^{\frac{\beta}{2}}q_{n}\|^{1-\frac{2\alpha-\gamma}{\beta}}=o(1).\end{array}$

Then, using the same argument after (3.51), one can also arrive at the
contradiction $\|U_{n}\|_{\cal H}=o(1).$

Now we are the place to prove (3.54). By (3.43), it is obvious that

$\|\lambda_{n}\theta_{n}\|=\lambda_{n}^{1-\frac{k}{4}}o(1)=\lambda_{n}^{\frac{2+2\alpha-\beta-3\gamma}{2(1-\gamma)}}o(1).$
(3.56)

On the other hand, if $1-\frac{k}{4}>{\frac{2\alpha-\gamma}{1-\gamma}}$, i.e.,
$\alpha+\frac{\beta+\gamma}{2}<1$, one can get a better decay rate than
(3.56). In fact, recall that $\frac{1}{2}-\beta-\frac{\gamma}{2}\leq 0$ due to
$\beta+\gamma\geq 1$. Then, it follows from (3.5) that

$i\lambda_{n}A^{\frac{1}{2}-\beta}(mI+A^{-\gamma})v_{n}+\sigma
A^{\frac{3}{2}-\beta-\gamma}u_{n}-A^{\frac{1}{2}+\alpha-\beta-\gamma}\theta_{n}=o(1).$
(3.57)

By (LABEL:+211), (3.43), (3.57), $\frac{3}{2}-\beta-\gamma\leq\frac{1}{2}$ and
$\frac{1}{2}+\alpha-\beta-\gamma\leq 0$, one has

$\|A^{\frac{1}{2}-\beta}v_{n}\|=\lambda_{n}^{-1}O(1).$ (3.58)

Note that
$\frac{1}{2}-\beta\leq\alpha-\frac{\beta}{2}<-\beta-\frac{\gamma}{2}+1$ where
we use $\alpha+\frac{\beta+\gamma}{2}<1$ and $\beta+\gamma\geq 1$. Combining
(3.11), (3.58) and interpolation, we have

$\|A^{\alpha-\frac{\beta}{2}}v_{n}\|\leq\|A^{\frac{1}{2}-\beta}v_{n}\|^{\frac{2-2\alpha-\beta-\gamma}{1-\gamma}}\|A^{-\beta-\frac{\gamma}{2}+1}v_{n}\|^{1-\frac{2-2\alpha-\beta-\gamma}{1-\gamma}}=\lambda_{n}^{-\frac{2-2\alpha-\beta-\gamma}{1-\gamma}}O(1),$
(3.59)

where $\|A^{-\beta-\frac{\gamma}{2}+1}v_{n}\|$ is bounded since
$-\beta-\frac{\gamma}{2}+1\leq\frac{\gamma}{2}$. Using (LABEL:v1_8) and (3.59)
yields

$|\lambda_{n}^{-1}(A^{\alpha}v_{n},\theta_{n})|\leq\|A^{\alpha-\frac{\beta}{2}}v_{n}\|\|\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n}\|=\lambda_{n}^{\frac{4\alpha-2}{1-\gamma}}o(1).$
(3.60)

Note that $-k\leq\frac{4\alpha-2}{1-\gamma}$ due to $\beta+\gamma\geq 1$.
Thus, substituting (3.41) and (3.60) into (3.40) yields
$\|\theta_{n}\|=\lambda_{n}^{\frac{2\alpha-1}{1-\gamma}}o(1)$. Consequently,
the second part of (3.54) is obtained.

Case iii-D. Let $\frac{1}{2}<\alpha\leq\frac{\beta+\gamma}{2}$.

Note that by (3.5), (LABEL:+211) and (3.43), along with $\alpha>\frac{1}{2}$,
we see that

$\|\lambda_{n}A^{-\alpha}(I+mA^{\gamma})v_{n}\|=O(1).$ (3.61)

Thus, taking the inner product of (3.6) with
$\lambda_{n}^{-k}A^{-\alpha}(I+mA^{\gamma})v_{n}$ in $H$, we have

$(i\lambda_{n}\theta_{n},A^{-\alpha}(I+mA^{\gamma})v_{n})-(A^{\frac{\beta+\gamma}{2}-\alpha}q_{n},A^{-\frac{\gamma}{2}}(I+mA^{\gamma})v_{n})+\|v_{n}\|^{2}+m\|A^{\frac{\gamma}{2}}v_{n}\|^{2}=o(1).$
(3.62)

Recalling that $\alpha>\frac{1}{2}$ and (3.6), one has

$\|i\lambda_{n}A^{\frac{1}{2}-\alpha}\theta_{n}-A^{\frac{1}{2}-\alpha+\frac{\beta}{2}}q_{n}+A^{\frac{1}{2}}v_{n}\|=\lambda_{n}^{-k}o(1).$

Thus, we deduce from (LABEL:+211) and (3.43) that

$\|A^{\frac{\beta+1-2\alpha}{2}}q_{n}\|=\lambda_{n}O(1).$ (3.63)

Thus, by interpolation (2.9), along with (3.10) and (3.63), we get

$\|A^{\frac{\beta+\gamma}{2}-\alpha}q_{n}\|\leq\|q_{n}\|^{1-{\frac{\beta+\gamma-2\alpha}{\beta+1-2\alpha}}}\|A^{\frac{\beta+1-2\alpha}{2}}q_{n}\|^{\frac{\beta+\gamma-2\alpha}{\beta+1-2\alpha}}=o(1).$
(3.64)

Note that
$\|A^{-\frac{\gamma}{2}}(I+mA^{\gamma})v_{n}\|\leq\|A^{-\frac{\gamma}{2}}v_{n}\|+\|mA^{\frac{\gamma}{2}}v_{n}\|=O(1)$.
Substituting (3.43), (3.61) and (3.64) into (3.62) yields

$\|v_{n}\|,\;\|A^{\frac{\gamma}{2}}v_{n}\|=o(1).$ (3.65)

Finally, due to $0<\alpha-\frac{\beta}{2}<\frac{\gamma}{2}$ and (3.4),
(LABEL:v1_8), it is clear that

$|(A^{\alpha}\theta_{n},u_{n})|\leq\|\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n}\|\|\lambda_{n}A^{\alpha-\frac{\beta}{2}}u_{n}\|\leq\|\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n}\|\|A^{\alpha-\frac{\beta}{2}}v_{n}\|=\lambda_{n}^{-\frac{k}{2}}o(1).$
(3.66)

Then, by (3.18), along with (3.65) and (3.66), we obtain

$\|A^{\frac{1}{2}}u_{n}\|=o(1).$ (3.67)

In summary, by (3.10), (3.43), (3.65) and (3.67), we have arrived at the
contradiction $\|U_{n}\|_{\mathcal{H}}=o(1)$.

##  4 Polynomial Stability of the System without an Inertial Term (Proof of
Theorem 2.2)

This section is devoted to considering the stability of the system (1.1)
without an inertial term, i.e., $m=0$. Note that by Lemma 2.2, we have known
that the corresponding semigroup $e^{t{\cal A}}$ is strongly stable. We shall
further estimate the polynomial decay rates of the solutions to the system
when parameters $(\alpha,\;\beta)\in V_{i}^{*},\;i=1,2,$ respectively.

Similar to the argument given in Section 3, we still employ the proof by
contradiction to show Theorem 2.2. Specifically, suppose (3.1) fails. Then,
there at least exists a sequence
$\\{\lambda_{n},U_{n}\\}_{n=1}^{\infty}\subset{\mathbb{R}}\times D({\cal A})$
such that (3.2)-(3.3) hold with $m=0$, that is

$\displaystyle\lambda_{n}^{k}(i\lambda_{n}A^{\frac{1}{2}}u_{n}-A^{\frac{1}{2}}v_{n})=o(1)\quad\;\mbox{
in }\;H,$ (4.1) $\displaystyle\lambda_{n}^{k}\left(i\lambda_{n}v_{n}+\sigma
Au_{n}-A^{\alpha}\theta_{n}\right)=o(1)\quad\;\mbox{ in }\;H,$ (4.2)
$\displaystyle\lambda_{n}^{k}(i\lambda_{n}\theta_{n}-A^{\frac{\beta}{2}}q_{n}+A^{\alpha}v_{n})=o(1)\quad\;\mbox{
in }\;H,$ (4.3) $\displaystyle\lambda_{n}^{k}(i\lambda_{n}\tau
q_{n}+q_{n}+A^{\frac{\beta}{2}}\theta_{n})=o(1)\quad\;\mbox{ in }\;H.$ (4.4)

Consequently, (3.10)-(LABEL:+211) remain true. Now, we proceed to show
(i)-(ii) in Theorem 2.2, respectively.

(i) Let $(\alpha,\;\beta)\in V_{1}^{*}$(see (2.7)) and
$k=\frac{2\alpha-\beta}{\alpha}$.

Thanks to $-\alpha+\frac{\beta}{2}\leq 0$ and (3.2), we can deduce from (4.3)
that

$\|\lambda_{n}A^{-\alpha}\theta_{n}\|=O(1).$ (4.5)

Then, using (LABEL:v1_8), (4.5) and the interpolation inequality (2.9) in
Lemma 2.4, we obtain

$\left\|\theta_{n}\right\|\leq\left\|A^{-\alpha}\theta_{n}\right\|^{\frac{\beta}{2\alpha+\beta}}\|A^{\frac{\beta}{2}}\theta_{n}\|^{\frac{2\alpha}{2\alpha+\beta}}=o(1).$
(4.6)

We take the inner product of (4.3) with $\lambda_{n}^{-k}A^{-\alpha}v_{n}$ to
get

$(i\lambda_{n}\theta_{n},A^{-\alpha}v_{n})-(A^{\frac{\beta}{2}-\alpha}q_{n},v_{n})+\|v_{n}\|^{2}=o(1).$
(4.7)

By (4.2), it is clear that $\|i\lambda_{n}A^{-\alpha}v_{n}+\sigma
A^{1-\alpha}u_{n}-\theta_{n}\|=\lambda_{n}^{-k}o(1)$. Combining this with
(3.2), along with $\alpha\geq\frac{1}{2}$, yields

$\|A^{-\alpha}v_{n}\|=\lambda_{n}^{-1}O(1).$ (4.8)

Substituting (3.2), (3.10), (4.6), (4.8) into (4.7), along with
$\frac{\beta}{2}-\alpha\leq 0$, we obtain

$\|v_{n}\|=o(1).$ (4.9)

Furthermore, similar to (3.17)-(3.18), we can deduce from (4.1) and (4.2) that

$-\|v_{n}\|^{2}+\sigma\|A^{\frac{1}{2}}u_{n}\|^{2}-i(\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n},A^{\alpha-\frac{\beta}{2}}v_{n})=o(1).$
(4.10)

It follows from (3.10), (LABEL:v1_8), (4.3) and (4.6) that

$(\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n},\,A^{\alpha-\frac{\beta}{2}}v_{n})=(\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n},\,q_{n}-i\lambda_{n}A^{-\frac{\beta}{2}}\theta_{n})+\lambda_{n}^{-1-k}o(1)=o(1).$

Combining this with (4.9) and (4.10) yields

$\|A^{\frac{1}{2}}u_{n}\|=o(1).$ (4.11)

Consequently, we arrive at the contradiction $\|U_{n}\|_{\mathcal{H}}=o(1)$ by
(3.10), (4.6), (4.9) and (4.11).

(ii) Let $(\alpha,\;\beta)\in V_{2}^{*}$ and $k=2(2-2\alpha-\beta)$.

Taking the inner product of (4.3) with $\lambda_{n}^{-k-1}\theta_{n}$, one has

$(i\lambda_{n}\theta_{n},\lambda_{n}^{-1}\theta_{n})-(A^{\frac{\beta}{2}}q_{n},\lambda_{n}^{-1}\theta_{n})+(A^{\alpha}v_{n},\lambda_{n}^{-1}\theta_{n})=\lambda_{n}^{-1-k}o(1).$
(4.12)

By (3.10) and (LABEL:v1_8), we get

$|(A^{\frac{\beta}{2}}q_{n},\lambda_{n}^{-1}\theta_{n})|\leq\|q_{n}\|\|\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n}\|=\lambda^{-k}o(1).$
(4.13)

Moreover, by (4.2) and (LABEL:+211), we have

$\displaystyle\|\lambda_{n}A^{-\frac{1}{2}}v_{n}\|=O(1).$ (4.14)

Recalling $0\leq\alpha<{1\over 2}$, we deduce from (2.9), (LABEL:+211) and
(4.14) that

$\|A^{\alpha-\frac{\beta}{2}}v_{n}\|\leq\|A^{-\frac{1}{2}}v_{n}\|^{{1\over
2}-\alpha+{\beta\over 2}}\|A^{\frac{1}{2}}v_{n}\|^{{1\over
2}+\alpha-{\beta\over 2}}=\lambda_{n}^{2\alpha-\beta}O(1).$

This implies that

$|(A^{\alpha}v_{n},\lambda_{n}^{-1}\theta_{n})|\leq\|A^{\alpha-\frac{\beta}{2}}v_{n}\|\|\lambda_{n}^{-1}A^{\frac{\beta}{2}}\theta_{n}\|=\lambda^{4\alpha-2}o(1).$
(4.15)

Thus, substituting (4.13) and (4.15) into (4.12), along with $-k\leq
4\alpha-2$, yields

$\|\theta_{n}\|=\lambda^{2\alpha-1}o(1).$ (4.16)

We claim that

$\|v_{n}\|=o(1).$ (4.17)

Indeed, note that (4.7) still holds for this case. Then, in order to show
(4.17), it suffices to prove that the first and second terms in (4.7) are both
$o(1)$.

From (3.2) and (4.14), it is easy to get

$\|A^{-\alpha}v_{n}\|\leq\|A^{-\frac{1}{2}}v_{n}\|^{2\alpha}\|v_{n}\|^{1-2\alpha}=\lambda_{n}^{-2\alpha}O(1).$
(4.18)

Combining this with (4.16) gives that the first term in (4.7) satisfies

$|(i\lambda_{n}\theta_{n},A^{-\alpha}v_{n})|\leq\|\theta_{n}\|\|\lambda_{n}A^{-\alpha}v_{n}\|=o(1).$
(4.19)

We proceed to show the second term in (4.7) is also $o(1)$. In fact, if
$0\leq\alpha\leq\frac{\beta}{2}$, by interpolation inequality (2.9), along
with (3.2) and (4.1), we have

$\|A^{\frac{\beta}{2}-\alpha}v_{n}\|\leq\|A^{\frac{1}{2}}v_{n}\|^{\beta-2\alpha}\|v_{n}\|^{1-\beta+2\alpha}=\lambda_{n}^{\beta-2\alpha}O(1).$
(4.20)

Thus, by (3.10) and (4.20), we get

$|(q_{n},A^{\frac{\beta}{2}-\alpha}v_{n})|\leq\|q_{n}\|\|A^{\frac{\beta}{2}-\alpha}v_{n}\|=\lambda_{n}^{2\beta-2}o(1)=o(1).$
(4.21)

If $\frac{\beta}{2}<\alpha<\frac{1}{2}$, it is clear from (3.2) and (3.10)
that

$|(q_{n},A^{\frac{\beta}{2}-\alpha}v_{n})|\leq\|q_{n}\|\|A^{\frac{\beta}{2}-\alpha}v_{n}\|=\lambda_{n}^{-{k\over
2}}o(1).$ (4.22)

By (4.21) and (4.22), we have proved that the second term in (4.7) is $o(1)$,
and hence, (4.17) holds.

Moreover, it is clear that
$|(\theta_{n},A^{\alpha}u_{n})|\leq\|\theta_{n}\|\|A^{\alpha}u_{n}\|=\lambda_{n}^{2\alpha-1}o(1)$
by (3.2), (4.16) and $\alpha<{1\over 2}$. This along with (4.10) and (4.17),
yields

$\|A^{\frac{1}{2}}u_{n}\|=o(1).$ (4.23)

In summary, by (3.10), (4.16), (4.17) and (4.23), we have arrived at the
contradiction $\|U_{n}\|_{\mathcal{H}}=o(1)$. The desired result follows.

##  5 Proof of Optimality of Decay Rates (Theorem 2.3)

In this section, we are going to prove Theorem 2.3, which shows the orders of
polynomial decay achieved in Theorem 2.1 and Theorem 2.2 are optimal. To this
end, we analyze the eigenvalues $\lambda$ of the operator $\mathcal{A}$ when
$m\neq 0$ (with an inertial term) and when $m=0$ (without an inertial term).
In addition, we assume that the system has different wave speeds, that is,
$\sigma\tau\neq 1$. In what follows, we first describe the characteristic
equation associated with $\mathcal{A}$ (Section 5.1). We then describe the
solutions to the characteristic equation in an asymptotic setting in Section
5.2 when the system is with an inertial term (Table 1) and without an inertial
term (Table 2), respectively. Finally, we show that these eigenvalues indicate
the optimality of the polynomial decay rates described in Theorem 2.1 and
Theorem 2.2 (Section 5.3).

###  5.1 Characteristic Equations

We describe the characteristic equations associated with the operator
$\mathcal{A}$ when the system is with an inertial term ($m\neq 0$) and without
an inertial term ($m=0$). For this purpose, recall that $A$ is a self-adjoint,
positive-definite operator with compact resolvent. Thus, there exists a
sequence of eigenvalues $\\{\mu_{n}\\}_{n\geq 1}$ of $A$ such that

$0<\mu_{1}\leq\mu_{2}\leq\cdots\leq\mu_{n}\leq\cdots,~{}~{}\qquad\lim_{n\to\infty}\mu_{n}=\infty.$

As such, the eigenvalues $\lambda$ of operator ${\cal A}$ satisfies the
following quartic equation:

$f(\lambda,\mu_{n})\equiv(m\tau\cdot\mu_{n}^{\gamma}+\tau)\lambda^{4}+(m\cdot\mu_{n}^{\gamma}+1)\lambda^{3}+(\tau\cdot\mu_{n}^{2\alpha}+m\cdot\mu_{n}^{\beta+\gamma}+\sigma\tau\cdot\mu_{n}+\mu_{n}^{\beta})\lambda^{2}+(\mu_{n}^{2\alpha}+\sigma\cdot\mu_{n})\lambda+\sigma\cdot\mu_{n}^{1+\beta}=0.$
(5.1)

Because the system has different wave speeds, we take $\sigma=2$ and $\tau=1$
so that $\sigma\tau=2\neq 1$ without loss of generality. In addition, by
taking $m=1$ and $m=0$, we can derive the characteristic equation of the
system from (5.1) when it is with and without an inertial term respectively as
follows:

$(\mu_{n}^{\gamma}+1)\lambda_{n}^{4}+(\mu_{n}^{\gamma}+1)\lambda_{n}^{3}+(\mu_{n}^{2\alpha}+\mu_{n}^{\beta+\gamma}+2\mu_{n}+\mu_{n}^{\beta})\lambda_{n}^{2}+(\mu_{n}^{2\alpha}+2\mu_{n})\lambda_{n}+2\mu_{n}^{1+\beta}=0.$
(5.2)
$\lambda_{n}^{4}+\lambda_{n}^{3}+(\mu_{n}^{2\alpha}+2\mu_{n}+\mu_{n}^{\beta})\lambda_{n}^{2}+(\mu_{n}^{2\alpha}+2\mu_{n})\lambda_{n}+2\mu_{n}^{1+\beta}=0.$
(5.3)

In (5.2) and (5.3), $\lambda_{n}$’s are sequences of eigenvalues of
$\mathcal{A}$ associated with $\mu_{n}\rightarrow\infty$ as
$n\rightarrow\infty$. We seek to solve (5.2) and (5.3) for $\lambda_{n}$ in an
asymptotic setting, that is, in the setting as $\mu_{n}\rightarrow\infty$.
Several existing works [10, 12, 15] have described the applications of a
procedure that can also be used to solve (5.2) and (5.3). In this paper, we
apply this procedure to solve (5.2) and (5.3) and omit the discussion of its
details. We refer interested readers to the aforementioned works for details
of this procedure. The solutions to (5.2) and (5.3) are summarized in Table 1
and 2, respectively.

###  5.2 Eigenvalues of the System

First, we consider the solutions to (5.2), which are eigenvalues of
$\mathcal{A}$ with an inertial term, in each regions given in Definition 2.2.
It should be noticed that the partition in Definition 2.2 is similar to that
reported in [10], where the exponential stability was proved when parameters
$(\alpha,\beta,\gamma)\in F_{13},\;L_{123},\;L_{34}$. More precisely, the
solutions to (5.2) are given in Table 1. As a result, one can conclude that
([21]) in region $V_{1}$, although the coefficients for the real and imaginary
parts of the eigenvalues on $T_{1},F_{12},F_{14},$ and $L_{124}$ are
different, there always exists a sequence of eigenvalues on each region
satisfying ${\cal R}e\lambda_{n}=|{\cal I}m\lambda_{n}|^{-{1\over k_{1}}},$
where $k_{1}$ is defined in Theorem 2.1. Therefore, the decay properties of
the corresponding semigroup are uniform in $V_{1}$.

By a similar argument, we can obtain that the long-time behavior of the system
(1.1) is the same when parameters are in $V_{2},\;V_{3}$ or
$V_{1}^{*},\;V_{2}^{*}$, respectively. Note that to characterize the long-time
behavior in $V_{1}^{*}$ and $V_{2}^{*}$, we have used the solutions to (5.3)
given in Table 2.

Table 1: Solutions to (5.2). In $P_{234}$, $j_{1}\approx-24.0858$,
$j_{2}\approx-5.52312$, and $j_{3}\approx 0.608892$ are the three roots of
$j^{3}+29j^{2}+115j-81=0$.

Region |  $\lambda_{n,1}$ and $\lambda_{n,2}$ |  $\lambda_{n,3}$ and $\lambda_{n,4}$
---|---|---
$T_{1}$ | $-\frac{1}{2}\mu_{n}^{-2\alpha+\beta+\gamma}(1+o(1))\pm i\mu_{n}^{\alpha-\frac{\gamma}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\sqrt{2}\mu_{n}^{-\alpha+\frac{\beta}{2}+\frac{1}{2}}(1+o(1))$
$T_{2}$ | $-\frac{1}{8}\mu_{n}^{2\alpha+\beta+\gamma-2}(1+o(1))\pm i\sqrt{2}\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\mu_{n}^{\frac{\beta}{2}}(1+o(1))$
$T_{3}$ | $-\frac{1}{2}(1+o(1))\pm i\mu_{n}^{\frac{\beta}{2}}(1+o(1))$ | $-\frac{1}{2}\mu_{n}^{2\alpha-\beta-\gamma}(1+o(1))\pm i\sqrt{2}\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$
$F_{12}$ | $-\frac{1}{18}\mu_{n}^{\beta+\gamma-1}(1+o(1))\pm i\sqrt{3}\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\frac{\sqrt{6}}{3}\mu_{n}^{\frac{\beta}{2}}(1+o(1))$
$F_{13}$ | $-\frac{1}{4}(1+o(1))\pm i\sqrt{2}\mu_{n}^{\beta/2}(1+o(1))$ | $-\frac{1}{4}(1+o(1))\pm i\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$
$F_{14}$ | $-\frac{1}{2}\mu_{n}^{\gamma-1}(1+o(1))\pm i\mu_{n}^{\frac{\beta}{2}-\frac{\gamma}{2}+\frac{1}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm\frac{i\sqrt{7}}{2}(1+o(1))$
$F_{2}$ | $-\frac{1}{8}\mu_{n}^{2\alpha+\gamma-2}(1+o(1))\pm i\sqrt{2}\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm\frac{i\sqrt{3}}{2}(1+o(1))$
$F_{23}$ | $-\frac{1}{2}\mu_{n}^{2\alpha-1}(1+o(1))\pm i\sqrt{2}\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$
$L_{123}$ | $(-\frac{1}{4}+\frac{\sqrt{2}}{8})(1+o(1))\pm i\sqrt{2+\sqrt{2}}\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$ | $(-\frac{1}{4}-\frac{\sqrt{2}}{8})(1+o(1))\pm i\sqrt{2-\sqrt{2}}\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$
$L_{124}$ | $-\frac{1}{18}\mu_{n}^{\gamma-1}(1+o(1))\pm i\sqrt{3}\mu_{n}^{\frac{1}{2}-\frac{\gamma}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm\frac{i\sqrt{15}}{6}(1+o(1))$
$L_{2}$ | $-\frac{1}{6}\mu_{n}^{2\alpha-1}(1+o(1))\pm i\sqrt{2}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm\frac{i\sqrt{3}}{2}(1+o(1))$
$L_{34}$ | $-\frac{1}{4}(1+o(1))\pm i\sqrt{2}\mu_{n}^{\frac{\beta}{2}}(1+o(1))$ | $-\frac{1}{4}(1+o(1))\pm\frac{i\sqrt{15}}{4}(1+o(1))$
$P_{234}$ | $-\frac{1}{4}(1-\sqrt{j_{3}})(1+o(1))\pm i\frac{\sqrt{-j_{1}}+\sqrt{-j_{2}}}{4}(1+o(1))$ | $-\frac{1}{4}(1+\sqrt{j_{3}})(1+o(1))\pm i\frac{\sqrt{-j_{1}}-\sqrt{-j_{2}}}{4}(1+o(1))$

Table 2: Solutions to (5.3)

Region |  $\lambda_{n,1}$ and $\lambda_{n,2}$ |  $\lambda_{n,3}$ and $\lambda_{n,4}$
---|---|---
$T_{1}^{*}$ | $-\frac{1}{2}\mu_{n}^{\beta-2\alpha}(1+o(1))\pm i\mu_{n}^{\alpha}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\sqrt{2}\mu_{n}^{-\alpha+\frac{\beta}{2}+\frac{1}{2}}(1+o(1))$
$T_{2}^{*}$ | $-\frac{1}{8}\mu_{n}^{2\alpha+\beta-2}(1+o(1))\pm i\sqrt{2}\mu_{n}^{\frac{1}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\mu_{n}^{\beta/2}(1+o(1))$
$F^{*}_{12}$ | $-\frac{1}{18}\mu_{n}^{\beta-1}(1+o(1))\pm i\sqrt{3}\mu_{n}^{\frac{1}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm\frac{i\sqrt{6}}{3}\mu_{n}^{\frac{\beta}{2}}(1+o(1))$
$F^{*}_{14}$ | $-\frac{1}{2}\mu_{n}^{-1}(1+o(1))\pm i\mu_{n}^{\frac{\beta}{2}+\frac{1}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\frac{\sqrt{7}}{2}(1+o(1))$
$F^{*}_{2}$ | $-\frac{1}{8}\mu_{n}^{2\alpha-2}(1+o(1))\pm i\mu_{n}^{\frac{1}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\frac{\sqrt{3}}{2}(1+o(1))$
$L^{*}_{124}$ | $-\frac{1}{18}\mu_{n}^{-1}(1+o(1))\pm i\sqrt{3}\mu_{n}^{\frac{1}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\frac{\sqrt{15}}{6}(1+o(1))$
$L^{*}_{23}$ | $-\frac{1}{2}\mu^{2\alpha-1}(1+o(1))\pm i\sqrt{2}\mu_{n}^{\frac{1}{2}}(1+o(1))$ | $-\frac{1}{2}(1+o(1))\pm i\mu_{n}^{\frac{1}{2}}(1+o(1))$
$P^{*}_{123}$ | $\left(-\frac{1}{4}+\frac{1}{4\sqrt{2}}\right)(1+o(1))\pm i\sqrt{2+\sqrt{2}}\mu^{\frac{1}{2}}(1+o(1))$ | $\left(-\frac{1}{4}-\frac{1}{4\sqrt{2}}\right)(1+o(1))\pm i\sqrt{2-\sqrt{2}}\mu^{\frac{1}{2}}(1+o(1))$

###  5.3 Optimality of the Polynomial Decay Rates

With the eigenvalues of $\mathcal{A}$ as reported in Table 1 and Table 2, we
can show that the polynomial decay rates given in Theorem 2.1 and Theorem 2.2
are optimal.

We first show the rates achieved in Theorem 2.1 are optimal. From Table 1, we
can deduce that there exists a sequence of eigenvalues satisfying

${\cal R}e\lambda_{n}=|{\cal I}m\lambda_{n}|^{-{1\over k_{i}}},\quad\;\mbox{
when }\;(\alpha,\;\beta,\;\gamma)\in V_{i},\;i=1,2,3,$ (5.4)

where $k_{i}$ and $V_{i}$ are defined in Theorem 2.1. Then by the same
argument as given in [12, Corollary 4.7], we can obtain

$\varlimsup\limits_{\lambda\in{\mathbb{R}},|\lambda|\to\infty}|\lambda|^{-{1\over
k_{i}}}\|(i\lambda I-{\cal A})^{-1}\|_{{\cal L}({\cal H})}\geq
C>0,\quad\;\mbox{ when }\;(\alpha,\;\beta,\;\gamma)\in V_{i},\;i=1,2,3.$ (5.5)

Therefore, the obtained decay rates in Theorem 2.1 are optimal.

Following the similar discussion as given above, we can also derive that the
obtained decay rates given in Theorem 2.2 are also optimal based on the
expression of eigenvalues in Table 2.

##  6 Examples

This section is devoted to presenting some examples. From these examples, we
can find that the decay rates of the semigroups associated with the coupled
systems can be easily identified based on our main results Theorem 2.1 and
Theorem 2.2.

Example 1.  Assume $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$ with
boundary $\partial\Omega=\Gamma_{0}\cup\Gamma_{1}$. $\Gamma_{0}$ and
$\Gamma_{1}$ are smooth. Let $A=\Delta^{2}$ be the bi-Laplace operator on
$\Omega$ with domain $D(A)=\\{u\in
H^{4}(\Omega)\,|\,u|_{\Gamma_{0}}=\partial_{\nu}u|_{\Gamma_{0}}=\Delta
u|_{\Gamma_{1}}=\partial_{\nu}\Delta u|_{\Gamma_{1}}=0\\},\;H=L^{2}(\Omega),$
where $\nu$ is the outward unit normal vector to the boundary. If $m>0$ and
$\gamma={1\over 2},\;\alpha=1,\;\beta=1$, the system (1.1) is as follows.

$\left\\{\begin{array}[]{ll}u_{tt}-m\Delta
u_{tt}+\sigma\Delta^{2}u-\Delta^{2}\theta=0,&x\in\Omega,t>0,\\\
\theta_{t}-\Delta q+\Delta^{2}u_{t}=0,&x\in\Omega,t>0,\\\ \tau
q_{t}+q+\Delta\theta=0,&x\in\Omega,t>0,\\\
u=\partial_{\nu}u=\theta=\partial_{\nu}\theta=q=\partial_{\nu}q=0,&x\in\Gamma_{0},t>0,\\\
\Delta u=\partial_{\nu}\Delta u=0,&x\in\Gamma_{1},t>0,\\\
u(0)=u_{0},\;u_{t}(0)=u_{1},\;\theta(0)=\theta_{0},\;q(0)=q_{0},&x\in\Omega.\end{array}\right.$
(6.1)

The model (6.1) describes the vibration of Rayleigh beam with thermal damping
of Cattaneo type. By Theorem 2.1, we obtain that $(\alpha,\;\beta,\;\gamma)\in
V_{1}$ and the semigroup associated with system (6.1) decays polynomially with
optimal order $t^{-\frac{3}{2}}$.

Moreover, it was known from [10] that the energy of system (1.1) decays
exponentially when $\gamma={1\over 2},\;\alpha={3\over 4},\;\beta=1$, which
corresponds to the Rayleigh beam with Cattaneo thermal conduction and weaker
coupling. Comparing this with the long-time behavior of system (6.1), one can
conclude that the coupling order should coordinate with the damping order to
get a better decay rate.

Example 2.  Consider the following one-dimensional thermo-elastic Rayleigh
beam system with Cattaneo’s law.

$\left\\{\begin{array}[]{l}u_{tt}-mu_{ttxx}+\sigma u_{xxxx}-\theta=0,\quad
x\in(0,1),\;t>0,\\\ \theta_{t}-q_{x}+u_{t}=0,\quad x\in(0,1),\;t>0,\\\ \tau
q_{t}+q+\theta_{x}=0,\quad x\in(0,1),\;t>0,\\\
u(0)=u_{x}(0)=u_{xx}(1)=u_{xxx}(1)=\theta(0)=\theta(1)=q(0)=q(1)=0,\quad
t>0,\\\ u(0)=u_{0},\;u_{t}(0)=u_{1},\;\theta(0)=\theta_{0},\;q(0)=q_{0},\quad
x\in(0,1).\end{array}\right.$ (6.2)

To apply the abstract results in Theorem 2.1 and 2.2 to this equation, we let
operator $A=\partial_{xxxx}$ on $H=L^{2}(0,1)$ with domain $D(A)=\\{u\in
H^{4}(0,1)\,|\,u(0)=u_{x}(0)=u_{xx}(1)=u_{xxx}(1)=0\\}.$ Note that system
(6.2) is corresponding to the abstract system (1.1) with
$(\alpha,\,\beta,\,\gamma)=(0,\;{1\over 2},\;{1\over 2})$ when $m>0.$ Thus, we
easily see that the semigroup corresponding to (6.2) is polynomially stable
with order $t^{-{1\over 4}}$ by choosing parameters
$(\alpha,\,\beta,\,\gamma)=(0,\;{1\over 2},\;{1\over 2})\in V_{3}$ in Theorem
2.1. While for the case $m=0$, the decay rate of the semigroup is $t^{-{1\over
3}}$ due to Theorem 2.2 with parameters $(\alpha,\,\beta)=(0,\;{1\over 2})\in
V_{2}^{*}$.

##  7 Conclusions

In this work, we focus on investigating the polynomial stability of abstract
thermoelastic systems with Cattaneo’s law and mainly obtained the following
results:

For the system with an inertial term ($m>0$), based on system parameter
choices $(\alpha,\beta,\gamma)$, we classified the “non-exponential stability”
area into three subregions, and obtained explicit polynomial decay rates
dependent on these parameters for each subregion.

For the system without an inertial term ($m=0$), we also divided the “non-
exponential stability” region into two subregions, and estimated its
polynomial decay rates at each subregion, respectively. Upon comparing the
stability outcomes between systems with and without an inertial term, it is
evident that this term plays a substantial role in determining stability for
such coupled PDEs’ systems.

Furthermore, we conducted a detailed asymptotic spectral analysis on the
system operator ($m>0$ or $m=0$), which led us to reveal the eigenvalues
asymptotics of the system operator. The optimality of the above decay rates
was verified, based on these results, respectively.

##  Appendix A Appendix

Proof of Lemma 2.2. Define

$\widetilde{\lambda}:=\sup\big{\\{}R>0\,|\,[-iR,\,iR]\subset\rho(\mathcal{A})\big{\\}}.$

It is sufficient to show $\widetilde{\lambda}=\infty$ since
$0\in\rho({\mathcal{A}})$ (see [10]). By contradiction, if
$0<\widetilde{\lambda}<\infty$, there exists a sequence
$\lambda_{n}\to\widetilde{\lambda}$ and a sequence
$Z_{n}=(u_{n},v_{n},\theta_{n},q_{n})\in\mathcal{D}(\mathcal{A})$ with
$\|Z_{n}\|_{\mathcal{H}}=1$ such that

$\|(i\lambda_{n}I-\mathcal{A})Z_{n}\|_{\mathcal{H}}=o(1),\quad n\to\infty,$
(8.1)

i.e.,

$\displaystyle
i\lambda_{n}A^{\frac{1}{2}}u_{n}-A^{\frac{1}{2}}v_{n}:=f_{1n}=o(1)\quad\mbox{in
}\;H,$ (8.2) $\displaystyle
A^{-\frac{\gamma}{2}}\left(i\lambda_{n}v_{n}+i\lambda_{n}mA^{\gamma}v_{n}+A^{\alpha}(\sigma
A^{1-\alpha}u_{n}-\theta_{n})\right):=f_{2n}=o(1)\quad\mbox{in }\;H,$ (8.3)
$\displaystyle
i\lambda_{n}\theta_{n}+A^{\alpha}v_{n}-A^{\frac{\beta}{2}}q_{n}:=f_{3n}=o(1)\quad\mbox{in
}\;H,$ (8.4) $\displaystyle i\lambda_{n}\tau
q_{n}+q_{n}+A^{\frac{\beta}{2}}\theta_{n}:=f_{4n}=o(1)\quad\mbox{in }\;H.$
(8.5)

It follows from ${\|Z_{n}\|_{\mathcal{H}}}=1$ that

$\|A^{\frac{1}{2}}u_{n}\|,\;\|A^{\frac{\gamma}{2}}v_{n}\|,\;\|v_{n}\|,\;\|\theta_{n}\|,\;\|q_{n}\|\;\;\mbox{are
bounded.}$ (8.6)

Moreover, (8.2), (8.6) and the fact $0<\widetilde{\lambda}<\infty$ imply that
$\|A^{\frac{1}{2}}v_{n}\|$ is bounded. By the dissipativeness of
$\mathcal{A}$, we get

${\cal
R}e\langle(i\lambda_{n}I-\mathcal{A})Z_{n},Z_{n}\rangle_{\mathcal{H}}=-{\cal
R}e\langle\mathcal{A}{Z_{n}},{Z_{n}}\rangle_{\mathcal{H}}=\|q_{n}\|^{2}=o(1).$
(8.7)

Thus, by (8.5) and (8.7), we get

$\|A^{\frac{\beta}{2}}\theta_{n}\|=o(1),$ (8.8)

and hence,

$\|\theta_{n}\|=o(1).$ (8.9)

By taking the inner product of (8.4) with $A^{-\alpha}v_{n}$ on $H$, we obtain

$(i\lambda_{n}\theta_{n},A^{-\alpha}v_{n})+(A^{\alpha}v_{n},A^{-\alpha}v_{n})-(A^{\frac{\beta}{2}}q_{n},A^{-\alpha}v_{n})=o(1).$
(8.10)

It is clear that the first term in (8.10) converges to $0$ due to the
boundedness of $\|A^{-\alpha}v_{n}\|$ and (8.9). Moreover, due to the
boundedness of $\|A^{\frac{1}{2}}v_{n}\|$ and (8.7), one has

$|(A^{\frac{\beta}{2}}q_{n},A^{-\alpha}v_{n})|=|(q_{n},A^{\frac{\beta}{2}-\alpha}v_{n})|\leq\|q_{n}\|\|A^{\frac{\beta}{2}-\alpha}v_{n}\|\leq\|q_{n}\|\|A^{\frac{1}{2}}v_{n}\|=o(1).$

Therefore, it follows from (8.10) that

$\|v_{n}\|=o(1).$ (8.11)

By the interpolation inequality in Lemma 2.4, along with (8.11) and the
boundedness of $\|A^{\frac{1}{2}}v_{n}\|$, we have

${\|A^{\frac{\gamma}{2}}v_{n}\|\leq\|v_{n}\|^{1-\gamma}\|A^{\frac{1}{2}}v_{n}\|^{\gamma}=o(1).}$
(8.12)

Note that $\|A^{\frac{\gamma}{2}}u_{n}\|$ is bounded from (8.6) and
$0\leq\gamma\leq 1$. Thus, taking the inner product of (8.3) with
$A^{\frac{\gamma}{2}}u_{n}$ on $H$, substituting (8.2) into it, we get

$-(v_{n},v_{n})-m(A^{\frac{\gamma}{2}}v_{n},A^{\frac{\gamma}{2}}v_{n})+\sigma(A^{\frac{1}{2}}u_{n},A^{\frac{1}{2}}u_{n})-(A^{\alpha}\theta_{n},u_{n})=o(1),\quad
m\geq 0.$ (8.13)

One has that $\|A^{-\frac{\beta}{2}+\alpha}u_{n}\|$ is bounded thanks to (8.6)
and the fact that $\beta\geq 2\alpha-1$. Recalling (8.8), we have

$(A^{\alpha}\theta_{n},u_{n})=(A^{\frac{\beta}{2}}\theta_{n},A^{-\frac{\beta}{2}+\alpha}u_{n})\leq\|A^{\frac{\beta}{2}}\theta_{n}\|\|A^{-\frac{\beta}{2}+\alpha}u_{n}\|=o(1).$
(8.14)

Hence, substituting (8.11), (8.12) and (8.14) into (8.13) yields

$\|A^{\frac{1}{2}}u_{n}\|=o(1).$ (8.15)

In summary, by (8.7), (8.9), (8.11), (8.12) and (8.15), we arrive at
$\|Z_{n}\|_{\mathcal{H}}=\|(u_{n},v_{n},\theta_{n},q_{n})\|_{\mathcal{H}}=o(1)$,
which contradicts the assumption that $\|Z_{n}\|_{\mathcal{H}}=1$. The desired
result follows.

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