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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 73, pp. 1--26.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/73\hfil Thermoelastic Bresse system]
{Decay rates for solutions to thermoelastic Bresse systems of types I and III}

\author[F. A. Gallego, J. E. Mu\~{n}oz Rivera \hfil EJDE-2017/73\hfilneg]
{Fernando A. Gallego, Jaime E. Mu\~{n}oz Rivera}

\address{Fernando A. Gallego \newline
Centre de Robotique (CAOR), MINES Paristech,
PSL Research University, 60 boulevard Saint-Michel,
75272 Paris Cedex 06, France}
\email{ferangares@gmail.com}

\address{Jaime E. Mu\~{n}oz Rivera \newline
Laborat\'orio de Computu\c{c}\~ao Cient\'ifica, LNCC,
Petr\'opolis, 25651-070, RJ, Brazil. \newline
Instituto de Matem\'atica,
Universidade Federal do Rio de Janeiro,
UFRJ, P.O. Box 68530, 21945-970,
Rio de Janeiro, RJ, Brazil}
\email{rivera@lncc.br, rivera@im.ufrj.br}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted February 17, 2016. Published March 15, 2017.}
\subjclass[2010]{35B35, 35L55, 93D20}
\keywords{Decay rate; heat conduction; Bresse system; thermoelasticity}

\begin{abstract}
 In this article, we study the energy decay for the thermoelastic Bresse system
 in the whole line with two  dissipative mechanisms, given by heat conduction
 (Types I and III). We prove that the decay rate of the solutions are very slow.
 More precisely, we show that the solutions decay with the rate of
 $(1+t)^{-1/8}$ in the $L^2$-norm, whenever the initial data belongs to
 $L^1(\mathbb{R}) \cap H^{s}(\mathbb{R})$ for a suitable $s$.
 The wave speeds of propagation have influence on the decay rate with respect
 to the regularity of the initial data. This phenomenon is known as
 \textit{regularity-loss}. The main tool used to prove our results is the
 energy method in the Fourier space.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

In this article, we consider two Cauchy problems related to the Bresse model
 with two  dissipative mechanisms corresponding to the heat conduction 
coupled to the system. The first of them  is the Bresse system with 
thermoelasticity of Type I:
\begin{equation}\label{ee1}
\begin{gathered}
\rho_1\varphi _{tt}-k ( \varphi_x - \psi -l\omega)_x
 -k_0l( \omega_x - l\varphi )+l\gamma \theta_1 = 0 \quad
 \text{in $\mathbb{R} \times (0,\infty)$},  \\
\rho_2\psi_{tt}-b\psi_{xx}-k( \varphi_x - \psi -l\omega)+\gamma \theta_{2x} =0
 \quad \text{in $\mathbb{R} \times (0,\infty)$},  \\
\rho_1\omega_{tt}-k_0(\omega_x - l \varphi )_x- kl( \varphi_x - \psi -l\omega)
 +\gamma \theta_{1x} =0 \quad \text{in $\mathbb{R} \times (0,\infty)$},\\
\theta_{1t} -k_1\theta_{1xx}+m_1(\omega_x - l \varphi )_t=0 \quad
 \text{in $\mathbb{R} \times (0,\infty)$}, \\
\theta_{2t} -k_2\theta_{2xx}+m_2\psi_{xt}=0  \quad \text{in $\mathbb{R} \times (0,\infty)$},
\end{gathered}
\end{equation}
with the initial data
\begin{equation*}
(\varphi, \varphi_t, \psi, \psi_t,\omega, \omega_t,\theta_1,\theta_2)(x,0)
=(\varphi_0, \varphi_1, \psi_0, \psi_1,\omega_0, \omega_1,\theta_{10},\theta_{20})(x).
\end{equation*}
The second one, is the Bresse system with thermoelasticity of Type III:
\begin{equation}\label{ee2}
\begin{gathered}
\rho_1\varphi _{tt}-k ( \varphi_x - \psi -l\omega)_x -k_0l( \omega_x - l\varphi )
 +l\gamma \theta_{1t} = 0 \quad \text{in $\mathbb{R} \times (0,\infty)$},  \\
\rho_2\psi_{tt}-b\psi_{xx}-k( \varphi_x - \psi -l\omega)+\gamma \theta_{2xt} =0
\quad  \text{in $\mathbb{R} \times (0,\infty)$},  \\
\rho_1\omega_{tt}-k_0(\omega_x - l \varphi )_x- kl( \varphi_x - \psi -l\omega)
 +\gamma \theta_{1xt} =0 \quad \text{in $\mathbb{R} \times (0,\infty)$},\\
\theta_{1tt} -k_1\theta_{1xx}-\alpha_1\theta_{1xxt}+m_1(\omega_x - l \varphi )_t=0
\quad  \text{in $\mathbb{R} \times (0,\infty)$}, \\
\theta_{2tt} -k_2\theta_{2xx}-\alpha_2\theta_{2xxt}+m_2\psi_{xt}=0  \quad
  \text{in $\mathbb{R} \times (0,\infty)$},
\end{gathered}
\end{equation}
 with the initial data
\begin{equation*}
(\varphi, \varphi_t, \psi, \psi_t,\omega, \omega_t,\theta_1,\theta_2, 
\theta_{1t},\theta_{2t})(x,0)
=(\varphi_0, \varphi_1, \psi_0, \psi_1,\omega_0, \omega_1,\theta_{10},
\theta_{20},\theta_{11},\theta_{21})(x),
\end{equation*}
where $\alpha_1$, $\alpha_2$, $\rho_1,\rho_2, \gamma, b, k, k_0,k_1,k_2,l$ $m_1$ 
and $m_2$ are positive constants.

Terms $k_0(\omega_x-l\varphi)$, $k(\varphi-\psi-l\omega)$ and $b\psi_x$  
denote the axial force, the shear force and the bending moment, where 
$\omega$, $\varphi$  and $\psi$ are the longitudinal, vertical and shear 
angle displacements, respectively. Furthermore, $\rho_1 = \rho A$, $\rho_2 = \rho I$, 
$k_0 = EA$, $k = k'GA$, $b = EI$ and $l = R^{-1}$, where $\rho$ denotes the density, 
$E$ is the elastic modulus, $G$ is the shear modulus, $k'$ is the shear factor, 
$A$ is the cross-sectional area, $I$ is the second moment of area of the 
cross-section and $R$ is the radius of curvature of the beam. Here, we assume that
all the above coefficients are positive constants.  In what concerns of 
the Thermoelastic of type III, we refer the work of Green and Naghdi  
\cite{green1991re, green1992undamped}. They re-examined the classical model 
of thermoelasticity and introduced the so-called model of thermoelasticity 
of type III, which the constitutive assumption on the heat flux vector is 
different from Fourier's law. They  developed a model of thermoelasticity 
that includes temperature gradient and thermal displacement gradient among 
the constitutive variables and proposed a heat conduction law as
\begin{equation}\label{t1}
q(x,t)=-(\kappa  \theta_x (x,t)+\kappa^* v_x(x,t)),
\end{equation}
where $v_t = \theta$ and $v$ is the thermal displacement gradient,
$\kappa$ and $\kappa^*$ are constants.
Combining \eqref{t1} with the energy balance law
\begin{equation}\label{t2}
\rho\theta_{t} + \varrho  \, div\, q = 0,
\end{equation}
lead to the equation
\begin{equation*}
\rho \theta_{tt} - \varrho \kappa  \theta_{xx} -\varrho \kappa^*  \theta_{xx} = 0,
\end{equation*}
which permits propagation of thermal waves at finite speed. The common feature 
of these theories, is that all of them lead to hyperbolic differential 
equations and model heat flow as thermal waves traveling at finite speed. 
More information about mathematical modeling can be found in 
\cite{chandrasekharaiah1998hyperbolic, green1992undamped, lagnese1993}.


The main purpose of this article is to investigate the asymptotic behavior 
of the solutions to the Cauchy problems  \eqref{ee1} and \eqref{ee2} posed
 on $\mathbb{R}$. To the best of our knowledge, the stability of the Bresse model 
does not have any physical explanation when it is considered in the real line. 
Be that  as it may, from  mathematical point of view, a considerable number
 of stability issues concerning the Bresse model in a whole space, have 
received considerable attention in the last years
 \cite{ghoul2016,  said2015asymptotic, 2015bressetypeiii, said2014bresse, 
soufyane2014effect}. This has been due to \textit{the regularity-loss phenomenon} 
that usually appears in the pure Cauchy problems (for instance, see 
\cite{djou2014, duan2001, haramotokawashima2008, hosono2006, idekawashima2008, 
ueda2011} and references therein). Roughly speaking, the decay rate of the 
solution is of the \textit{regularity-loss type}, when it is obtained only
 by assuming some additional order regularity on the initial date. Thus,  
based on this refinement  of the initial data, we investigate the relationship 
between damping terms, the wave speeds of propagation and their influence 
on the decay rate of the vector solutions $V_1$ and $V_2$ 
(see \eqref{vectorsolution1}-\eqref{vectorsolution2} below) of the systems 
\eqref{ee1} and \eqref{ee2}, respectively.
Thus, our main result reads as follows.

\begin{theorem}\label{teo2}
Let $s$ be a non-negative integer, suppose that 
$V^0_j \in H^s(\mathbb{R})\cap L^1(\mathbb{R})$ for $j=1,2$. Then, 
the vector solutions $V_j$ of thermoelastic Bresse problems  \eqref{ee1} and 
\eqref{ee2}, respectively, satisfy the following decay estimates:
\begin{enumerate}
\item If $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$, then
\begin{equation}\label{e32}
\|\partial^k_xV_j(t)\|_2\leq C_1(1+t)^{-\frac{1}{8}-\frac{k}{4}}\|V_j^0\|_1
+ C_2(1+t)^{-\frac{l}{4}}\|\partial_x^{k+l}V_j^0\|_2,  
\end{equation}
for $j=1,2$ and $t\geq 0$.

\item If $\frac{\rho_1}{\rho_2} \neq\frac{k}{b}$ or $k\neq k_0$, then
\begin{equation}\label{e32'}
\|\partial^k_xV_j(t)\|_2\leq C_1(1+t)^{-\frac{1}{8}-\frac{k}{4}}\|V_j^0\|_1 
+ C_2(1+t)^{-\frac{l}{6}}\|\partial_x^{k+l}V_j^0\|_2, 
\end{equation}
for $j=1,2$,  and $t\geq 0$.
\end{enumerate}
where $k+l \leq s$, $C_1,C_2$ are two positive constants.
\end{theorem}

Our proof is based on some estimates for the Fourier image of the solution  
as well as a suitable linear combination of series of energy estimates. 
The key idea is to construct functionals to capture the dissipation of all 
the components of the vector solution. These functional allows to build an
 appropriate Lyapunov functionals equivalent to the energy, which gives the 
dissipation of all the components in the vector $\hat{V}^0_1(\xi,t)$ and 
$\hat{V}^0_2(\xi,t)$ (See \eqref{eq45} below). Finally, we rely on the 
Plancherel theorem and some asymptotic inequalities to show the desired decay 
estimates.

The decay rate $(1 + t)^{-1/8}$  can be obtained only under the regularity 
$V_0 \in  H^s(\mathbb{R})$. This regularity loss comes to analyze the Fourier 
image of the solution. Indeed, for $\hat{V}(\xi, t)$, we have 
(see \eqref{eq27'}, \eqref{eq27} and \eqref{eq45} below) that
\begin{equation}
| \hat{V}(\xi, t)|^2 \leq C e^{-\beta s(\xi)t}| \hat{V}(\xi, 0)|^2,
\end{equation}
where
\begin{equation*}
s(\xi) = \begin{cases}
C_1\frac{\xi^4}{(1+\xi^8)}, & \text{if $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and 
$k=k_0$}, \\[4pt]
C_2\frac{\xi^4}{(1+\xi^2)(1+\xi^8)}, & \text{if $\frac{\rho_1}{\rho_2} 
\neq\frac{k}{b}$ or $k\neq k_0$.}
\end{cases}
\end{equation*}

As we will see, the decay estimate \eqref{e32}-\eqref{e32'} depends in a 
critical way on the properties of the function $s(\xi)$. Obviously,
the function $s(\xi)$ behaves like $\xi^4$ in the low frequency region
 $(|\xi| \leq 1)$ and like $\xi^{-4}$ near infinity whenever
 $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$. Otherwise, if the wave speeds 
of propagation are different the function $s(\xi)$ behaves also like $\xi^4$ 
in the low frequency region and like $\xi^{-6}$ near infinity, which means 
that the dissipation in the high frequency region is very weak and produces 
the regularity loss phenomenom. It has been known recently that this regularity 
loss leads to some difficulties in the nonlinear problems, 
see \cite{haramotokawashima2008, idekawashima2008} for more details.

There are many works on the global existence and asymptotic stability of solutions 
to the initial boundary value problem for the Bresse system with dissipation. 
In this direction, we refer the IBVP associated to \eqref{ee1} considered by 
Liu and Rao in \cite{liu2009energy}. They proved that the exponential decay 
 exists only when the velocities of the wave propagation are the same. 
If the wave speeds are different, they showed that the energy  decays polynomially 
to zero with the rate $t^{-\frac{1}{2}}$ and $t^{-\frac{1}{4}}$, provided that
 the boundary conditions are Dirichlet-Neumann-Neumann
\[
\omega_x(x,t)=\varphi(x,t)=\psi_x(x,t)=\theta_1(x,t)=\theta_2(x,t)=0, \quad 
\text{for $x=0,l$},
\]
and Dirichlet-Dirichlet-Dirichlet type,
\[
\omega(x,t)=\varphi(x,t)=\psi(x,t)=\theta_1(x,t)=\theta_2(x,t)=0, \quad 
\text{for $x=0,l$.}
\]
An improvement of the above results was made by Fatori and Mu\~{n}oz Rivera 
in \cite{fatori2010rates}. They showed that, in general, the Thermoelastic Bresse 
system of Type I is not exponentially stable, but there exists polynomial 
stability with rates that depend on the wave propagation and the regularity 
of the initial data.

As far as we know, there exist just a few results related to the stability 
of the pure Cauchy problem to the Bresse model. The decay rate of the solution 
of the IVP for Bresse system in the whole line has been first studied by 
Said-Houari and Soufyane in \cite{soufyane2014effect}. They considered the system
\begin{equation}\label{eee1}
\begin{gathered}
\varphi _{tt}- ( \varphi_x - \psi -l\omega)_x -k_0^2l( \omega_x - l\varphi ) = 0
\quad\text{in $\mathbb{R} \times (0,\infty)$},  \\
\psi_{tt}-a^2\psi_{xx}-k( \varphi_x - \psi -l\omega)\gamma_1 \psi_t =0
\quad\text{in $\mathbb{R} \times (0,\infty)$}, \\
\omega_{tt}-k_0^2(\omega_x - l \varphi )_x- l( \varphi_x - \psi -l\omega)
 +\gamma_2 \omega_t =0\quad \text{in $\mathbb{R} \times (0,\infty)$},
\end{gathered}
\end{equation}
and investigated the relationship between the frictional damping terms, 
the wave speeds of propagation and their influence on the decay rate of the solution.
 In addition, they showed that the $L^2$-norm of the solution decays with the rate 
$(1 + t)^{-1/4}$. Later on, the same authors in \cite{said2014bresse}, proved 
that the vector solution $V$ of the Bresse system damped by heat conduction:
\begin{equation}\label{eee2}
\begin{gathered}
\varphi _{tt}-k ( \varphi_x - \psi -l\omega)_x -k_0^2l( \omega_x - l\varphi ) = 0
\quad\text{in $\mathbb{R} \times (0,\infty)$},  \\
\psi_{tt}-a^2\psi_{xx}-k( \varphi_x - \psi -l\omega)+m \theta_x =0
\quad\text{ in $\mathbb{R} \times (0,\infty)$}, \\
\omega_{tt}-k_0^2(\omega_x - l \varphi )_x- l( \varphi_x - \psi -l\omega)
 +\gamma \omega_t =0 \quad\text{in $\mathbb{R} \times (0,\infty)$}, \\
\theta_{t} -k_1\theta_{xx}+m\psi_{xt}=0\quad \text{in $\mathbb{R} \times (0,\infty)$},
\end{gathered}
\end{equation}
decays with the rate
\begin{equation}\label{eee3}
\|\partial_x^kV(t)\|_{L^2} \leq C(1+t)^{-\frac{1}{12}
-\frac{k}{6}}\|V_0\|_{L^1}+C(1+t)^{-\frac{l}{2}}\|\partial_x^{k+l}V_0\|_{L^2},
\end{equation}
for $a=1$, and
\begin{equation}\label{eee4}
\|\partial_x^kV(t)\|_{L^2} \leq C(1+t)^{-\frac{1}{12}-\frac{k}{6}}
\|V_0\|_{L^1}+C(1+t)^{-\frac{l}{4}}\|\partial_x^{k+l}V_0\|_{L^2},
\end{equation}
for $a\neq 1$, $k=1,2,\dots,s-l$.
More recently, Said-Houari and Hamadouche  \cite{said2015asymptotic} studied
 the decay properties of the Bresse-Cattaneo system
\begin{equation}\label{eee5}
\begin{gathered}
\varphi _{tt}-( \varphi_x - \psi -l\omega)_x -k_0^2l( \omega_x - l\varphi ) = 0
\quad\text{in $\mathbb{R} \times (0,\infty)$},  \\
\psi_{tt}-a^2\psi_{xx}-( \varphi_x - \psi -l\omega)+m \theta_x =0
\quad\text{in $\mathbb{R} \times (0,\infty)$}, \\
\omega_{tt}-k_0^2(\omega_x - l \varphi )_x- l( \varphi_x - \psi -l\omega)
+\gamma \omega_t =0 \quad\text{in $\mathbb{R} \times (0,\infty)$}, \\
\theta_{t} +q_x+m\psi_{xt}=0 \quad \text{in $\mathbb{R} \times (0,\infty)$}, \\
\tau_q q_t  +\beta q+\theta_{x}=0\quad \text{in $\mathbb{R} \times (0,\infty)$},
\end{gathered}
\end{equation}
obtaining the same decay rate as the one of the solution for the Bresse-Fourier
 model \eqref{eee2}.  This fact has been also seen in \cite{said2013damping}, 
where the authors investigated the Timoshenko-Cattaneo and Timoshenko-Fourier 
models and showed the same behavior for the solutions of both systems. 
Finally, concerning to the Termoelasticity type III (in one-dimensional space), 
Said-Houari and Hamadouche in \cite{2015bressetypeiii} have been recently 
analyzed the system
\begin{gather*}
\varphi _{tt}-( \varphi_x - \psi -l\omega)_x -k_0^2l( \omega_x - l\varphi ) = 0
\quad\text{in $\mathbb{R} \times (0,\infty)$},  \\
\psi_{tt}-a^2\psi_{xx}-( \varphi_x - \psi -l\omega)+m \theta_{tx} =0
\quad\text{in $\mathbb{R} \times (0,\infty)$}, \\
\omega_{tt}-k_0^2(\omega_x - l \varphi )_x- l( \varphi_x - \psi -l\omega)
+\gamma \omega_t =0\quad\text{in $\mathbb{R} \times (0,\infty)$}, \\
\theta_{tt} -k_1\theta_{xx}+\beta\psi_{tx}-k_2\theta_{txx}=0
\quad\text{in $\mathbb{R} \times (0,\infty)$}.
\end{gather*}
They proved that the solution decay with the rate
\begin{equation*}
\|\partial_x^kV(t)\|_{L^2} \leq C(1+t)^{-\frac{1}{12}-\frac{k}{6}}
\|V_0\|_{L^1}+C(1+t)^{-\frac{l}{2}}\|\partial_x^{k+l}V_0\|_{L^2},
\end{equation*}
for $a=1$, and
\begin{equation*}
\|\partial_x^kV(t)\|_{L^2} \leq C(1+t)^{-\frac{1}{12}
-\frac{k}{6}}\|V_0\|_{L^1}+C(1+t)^{-\frac{l}{8}}\|\partial_x^{k+l}V_0\|_{L^2},
\end{equation*}
for $a\neq 1$, $k=1,2,\dots,s-l$.


This paper is organized as follows:
In Section 2, we analyze the ODE system generated by the Fourier transform 
applies to the Cauchy problem, obtaining  appropriate decay estimates for 
the Fourier image of the solution.
Section 3 is dedicated to proof our main result.


\section{Energy method in the Fourier space}

 In this section, we establish  decay rates for the Fourier image of the 
solutions of thermoelastic Bresse systems.  To obtain the estimates of the 
Fourier image is actually the hardest and technical part. These estimates 
will play to a crucial role in proving the Theorems \ref{teo1} and \ref{teo1'}, 
below.

\subsection{Thermoelastic Bresse system of type I}
Taking Fourier Transform in \eqref{ee1}, we obtain the  ODE system
\begin{gather}
\rho_1\hat{\varphi} _{tt}-ik\xi ( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  ) 
-k_0l( i\xi \hat{\omega } - l \hat{\varphi} ) 
+l\gamma \hat{\theta}_1 = 0 \quad \text{in $\mathbb{R} \times (0,\infty)$} \label{e1}\\
\rho_2\hat{\psi}_{tt}+b\xi ^{2}\hat{\psi }- k( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  )+i\gamma \xi  \hat {\theta}_2=0 
\quad \text{in $\mathbb{R} \times (0,\infty)$}\label{e2}\\
\rho_1\hat{\omega}_{tt}-ik_0\xi ( i\xi \hat{\omega } - l \hat{\varphi} )
- kl( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )
 + i\gamma \xi \hat{\theta}_1=0 \quad \text{in $\mathbb{R} \times (0,\infty)$}\label{e3}\\
\hat{\theta}_{1t} +k_1\xi ^{2}\hat{\theta}_1+m_1 ( i\xi \hat{\omega } 
 - l \hat{\varphi} )_t=0 \quad \text{in $\mathbb{R} \times (0,\infty)$}\label{e4} \\
\hat{\theta}_{2t} +k_2\xi ^{2}\hat{\theta}_2+im_2 \xi \hat{\psi}_{t} =0 
\quad \text{in $\mathbb{R} \times (0,\infty)$}\label{e5}
\end{gather}
The energy functional associated to the above system is defined by
\begin{equation}\label{energytypeII}
\begin{aligned}
\hat{E}( \xi,t  )
&=\rho_1|\hat{\varphi}_{t} |^{2}+\rho_2|\hat{\psi}_{t}|^{2}
 +\rho_1 |\hat{\omega}_{t}|^{2}+\frac{\gamma}{m_1}|\hat{\theta}_1 |^{2}
 +\frac{\gamma}{m_2}|\hat{\theta}_2 |^{2} \\
&\quad +b| \xi|^{2}|\hat{\psi}|^{2} 
 +k|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^{2}
  + k_0|i\xi \hat{\omega } - l \hat{\varphi}|^{2}.
\end{aligned}
\end{equation}

\begin{lemma}\label{lem1}
Consider the energy functional $\hat{E}$ associated to the system 
\eqref{e1}-\eqref{e5}. Then
\begin{equation} \label{e7}
\frac{d}{dt}\hat{E}(\xi,t)=-2\gamma\xi^2(\frac{k_1}{m_1}|\hat{\theta}_1|^2
+\frac{k_2}{m_2}|\hat{\theta}_2|^2).
\end{equation}
\end{lemma}

\begin{proof}
Multiplying \eqref{e1} by $\overline{\hat{\varphi}}_t$,  \eqref{e2}  by 
 $\overline{\hat{\psi}}_t$,  \eqref{e3}  by $\overline{\hat{\omega}}_t$,  
\eqref{e4} by $\frac{\gamma}{m_1}\overline{\hat{\theta}_1}$, and \eqref{e5}
 by $\frac{\gamma}{m_2}\overline{\hat{\theta}_2}$, adding and  taking real part, 
\eqref{e7} follows.
\end{proof}

We show that the decay rate of the solution will depend on the wave speeds of 
propagation. More precisely, we analyze two cases: First, we suppose that
\[
\frac{\rho_1}{\rho_2}=\frac{k}{b} \quad \text{and} \quad k=k_0.
\]
Otherwise, we consider the case when the wave speeds of propagation are 
different ($\frac{\rho_1}{\rho_2}\neq\frac{k}{b}$ or $k\neq k_0$). 
The proof of our main results in this section (Theorems \ref{teo1} and
 \ref{teo1'} below) are based on the following lemmas:

\begin{lemma}\label{lem2}
The functional
\begin{equation*}
J_1(\xi,t)=\operatorname{Re}(i\rho_2\xi \hat{\psi}_t\overline{\hat{\theta}_2}),
\end{equation*}
satisfies
\begin{equation}\label{e9}
\begin{aligned}
&\frac{d}{dt}J_1(\xi,t) + \frac{m_2\rho_2}{2}\xi^2|\hat{\psi}_t|^2 \\
&\leq b|\xi|^3|\hat{\psi}||\hat{\theta}_2| 
 + k|\xi||\hat{\theta}_2| | i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }| 
 +  C_1(1+\xi^2)\xi^2|\hat{\theta}_2|^2,
\end{aligned}
\end{equation}
where $C_1$ is a positive constant.
\end{lemma}

\begin{proof}
Multiplying \eqref{e5} by $-i\rho_2\xi \overline{\hat{\psi}_t}$ and taking 
real part, we obtain
\[
\frac{d}{dt}\operatorname{Re}(-i\rho_2\xi \overline{\hat{\psi}_t}\hat{\theta}_{2})
+\operatorname{Re}(i\rho_2\xi \overline{\hat{\psi}_{tt}}\hat{\theta}_{2}) 
-\operatorname{Re}(ik_2\rho_2\xi^3 \overline{\hat{\psi}_t}\hat{\theta}_2)
+m_2\rho_2 \xi^2 |\hat{\psi}_{t}|^2 =0.
\]
By \eqref{e2}, we have
\begin{align*}
&\frac{d Re}{dt}(-i\rho_2\xi \overline{\hat{\psi}_t}\hat{\theta}_{2})
 +m_2\rho_2 \xi^2 |\hat{\psi}_{t}|^2 \\
&\leq k_2\rho_2|\xi|^3 |\hat{\psi}_t||\hat{\theta}_2|
 +b|\xi|^3 |\hat{\psi}| |\hat{\theta}_{2}| 
+ k|\xi||\hat{\theta}_{2}|| i\xi \hat{\varphi} - \hat{\psi}
  -l \hat{\omega } |+\gamma \xi^2 |\hat {\theta}_2|^2
\end{align*}
Applying Young's inequality, we obtain \eqref{e9}.
\end{proof}

\begin{lemma}\label{lem3'}
The functional
\begin{equation*}
T_1(\xi,t)=\operatorname{Re}\big(-\rho_1\hat{\varphi}_t \overline{(i\xi \hat{\omega }
 - l \hat{\varphi})} - \frac{\rho_1}{m_1}\hat{\varphi}_t\overline{\hat{\theta}_1}\big)
\end{equation*}
satisfies
\begin{equation}\label{e10}
\begin{aligned}
&\frac{d}{dt}T_1(\xi,t) +\frac{k_0l}{2}|i\xi \hat{\omega } 
 - l \hat{\varphi}|^2 \\
&\leq \frac{\rho_1k_1}{m_1}|\xi|^2|\hat{\varphi}_t||\hat{\theta}_1|
 -\operatorname{Re}(ik\xi(i\xi\hat{\varphi}-\hat{\psi}
 -l\hat{\omega})\overline{(i\xi \hat{\omega } - l \hat{\varphi})})  \\
&\quad +\frac{k}{m_1}|\xi||\hat{\theta}_1||i\xi\hat{\varphi}
 -\hat{\psi}-l\hat{\omega}|+C_2 |\hat{\theta}_1|^2,
\end{aligned}
\end{equation}
where $C_2$ is a positive constant.
\end{lemma}

\begin{proof}
Multiplying \eqref{e1} by $-\overline{(i\xi \hat{\omega } - l \hat{\varphi})}$ 
and taking real part, we have
\begin{align*}
&\frac{d}{dt} \operatorname{Re}(-\rho_1\hat{\varphi} _{t}
 \overline{(i\xi \hat{\omega } - l \hat{\varphi})})
 +\operatorname{Re}(\rho_1\hat{\varphi} _{t}\overline{(i\xi \hat{\omega } 
 - l \hat{\varphi})}_t) \\
& +\operatorname{Re}\Big(ik\xi ( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  )\overline{(i\xi \hat{\omega } 
- l \hat{\varphi})}\Big)  +k_0l|i\xi \hat{\omega } - l \hat{\varphi}|^2 
-\operatorname{Re}\Big(l\gamma \hat{\theta}_1\overline{(i\xi \hat{\omega } 
- l \hat{\varphi})}\Big)= 0.
\end{align*}
Inequality \eqref{e4} implies 
\begin{equation}\label{e34}
\begin{aligned}
&\frac{d Re}{dt}\Big(-\rho_1\hat{\varphi} _{t}\overline{(i\xi \hat{\omega } 
 - l \hat{\varphi})}\Big)-\frac{\rho_1}{m_1}\operatorname{Re}
 (\hat{\varphi}_t\overline{\hat{\theta}}_{1t}) 
 -\frac{\rho_1k_1}{m_1}\operatorname{Re}(\xi^2\hat{\varphi}_t
 \overline{\hat{\theta}}_{1}) \\
& +\operatorname{Re}(ik\xi ( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )
 \overline{(i\xi \hat{\omega } - l \hat{\varphi})})
 +k_0l|i\xi \hat{\omega } - l \hat{\varphi}|^2 \\
&  -\operatorname{Re}\Big(l\gamma \hat{\theta}_1\overline{(i\xi \hat{\omega } 
- l \hat{\varphi})}\Big)= 0.
\end{aligned}
\end{equation}
On the other hand, multiplying \eqref{e1} by $-\frac{\overline{\hat{\theta}}_1}{m_1}$ 
and taking real part, it follows that
\begin{equation}\label{e35}
\begin{aligned}
&\frac{d Re}{dt}(-\frac{\rho_1}{m_1}\hat{\varphi} _{t}\overline{\hat{\theta}}_1)
+\frac{\rho_1}{m_1}\operatorname{Re}(\hat{\varphi} _{t}\overline{\hat{\theta}}_{1t})
+\operatorname{Re}\Big( \frac{ik}{m_1}\xi \overline{\hat{\theta}}_1 ( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  )\Big) \\
& +\operatorname{Re}\Big(\frac{k_0l}{m_1}\overline{\hat{\theta}}_1( i\xi \hat{\omega } 
- l \hat{\varphi} ) \Big)
-\frac{l\gamma}{m_1} |\hat{\theta}_1|^2 = 0.
\end{aligned}
\end{equation}
Adding \eqref{e34} and \eqref{e35},
\begin{align*}
\frac{d}{dt}T_1(\xi,t)+k_0l|i\xi \hat{\omega } - l \hat{\varphi}|^2 
&\leq \frac{\rho_1k_1}{m_1}|\xi|^2|\hat{\varphi}_t||\hat{\theta}_{1}|
-\operatorname{Re}( ik\xi ( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  )\overline{(i\xi \hat{\omega } - l \hat{\varphi})}) \\
&\quad +l\gamma |\hat{\theta}_1||i\xi \hat{\omega } - l \hat{\varphi}|  
+\frac{k}{m_1}|\xi| |\hat{\theta}_1||i\xi \hat{\varphi} 
 - \hat{\psi} -l \hat{\omega }| \\
&\quad +\frac{k_0l}{m_1}|\hat{\theta}_1|| i\xi \hat{\omega } 
 - l \hat{\varphi}|+\frac{l\gamma}{m_1} |\hat{\theta}_1|^2,
\end{align*}
applying Young's inequality, \eqref{e10} follows.
\end{proof}

\begin{lemma}\label{lem3''}
The functional
\begin{equation*}
T_2(\xi,t)=\operatorname{Re}(i\rho_1\xi\hat{\omega}_t \overline{(i\xi \hat{\omega } 
- l \hat{\varphi})} + i\frac{\rho_1}{m_1}\xi\hat{\omega}_t\overline{\hat{\theta}_1}),
\end{equation*}
satisfies
\begin{equation}\label{e36}
\begin{aligned}
&\frac{d}{dt}T_2(\xi,t) +\frac{k_0}{2}|\xi|^2|i\xi \hat{\omega }
 - l \hat{\varphi}|^2 \\
&\leq \frac{\rho_1k_1}{m_1}|\xi|^3|\hat{\omega}_t| |\hat{\theta}_1|
+\operatorname{Re}(ikl\xi(i\xi\hat{\varphi}-\hat{\psi}
 -l\hat{\omega})\overline{(i\xi \hat{\omega } - l \hat{\varphi})}) \\
&\quad +\frac{kl}{m_1}|\xi||\hat{\theta}_1||i\xi\hat{\varphi}
 -\hat{\psi}-l\hat{\omega}|+C_3 |\xi|^2|\hat{\theta}_1|^2,
\end{aligned}
\end{equation}
where $C_3$ is a positive constant.
\end{lemma}

\begin{proof}
Multiplying \eqref{e3} by $i\xi\overline{(i\xi \hat{\omega } - l \hat{\varphi})}$ 
and taking real part, we obtain
\begin{equation}\label{e37}
\begin{aligned}
&\frac{d}{dt} \operatorname{Re}(i\rho_1\xi\hat{\omega}_{t}
 \overline{(i\xi \hat{\omega } - l \hat{\varphi})})
 -\operatorname{Re}(i\rho_1\xi\hat{\omega}_{t}\overline{(i\xi \hat{\omega } 
 - l \hat{\varphi})}_t)+k_0|\xi|^2 | i\xi \hat{\omega } - l \hat{\varphi}|^2 \\
&- \operatorname{Re}(ikl\xi( i\xi \hat{\varphi} - \hat{\psi} 
-l \hat{\omega }  )\overline{(i\xi \hat{\omega } - l \hat{\varphi})})\\
&\quad- \operatorname{Re}\Big( \gamma \xi^2 \hat{\theta}_1
\overline{(i\xi \hat{\omega }
 - l \hat{\varphi})}\Big)=0,
\end{aligned}
\end{equation}
using \eqref{e4}, it follows that
\begin{equation}\label{e38}
\begin{aligned}
&\frac{d}{dt} \operatorname{Re}(i\rho_1\xi\hat{\omega}_{t}
\overline{(i\xi \hat{\omega } - l \hat{\varphi})})+\frac{\rho_1}{m_1}
\operatorname{Re}(i\xi\hat{\omega}_t\overline{\hat{\theta}}_{1t}) 
+\frac{\rho_1k_1}{m_1}\operatorname{Re}(i\xi^3\hat{\omega}_t
\overline{\hat{\theta}}_{1}) \\
&+k_0|\xi|^2 | i\xi \hat{\omega } - l \hat{\varphi}|^2
 - \operatorname{Re}(ikl\xi( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )
 \overline{(i\xi \hat{\omega } - l \hat{\varphi})}) \\
&\quad - \operatorname{Re}( \gamma \xi^2 \hat{\theta}_1\overline{(i\xi \hat{\omega } 
 - l \hat{\varphi})})=0.
\end{aligned}
\end{equation}

On the other hand, multiplying \eqref{e3} by
 $\frac{i\xi}{m_1}\overline{\hat{\theta}}_1$ and taking real part,
\begin{equation}\label{e39}
\begin{aligned}
&\frac{d}{dt} \operatorname{Re}(\frac{i\rho_1\xi}{m_1}\hat{\omega}_{t}
 \overline{\hat{\theta}}_1)-\frac{\rho_1}{m_1}\operatorname{Re}
 (i\xi\hat{\omega}_{t}\overline{\hat{\theta}}_{1t})
 +\operatorname{Re}( \frac{k_0\xi^2}{m_1} ( i\xi \hat{\omega } 
 - l \hat{\varphi})\overline{\hat{\theta}}_1)  \\
&- \operatorname{Re}(\frac{ikl\xi}{m_1}( i\xi \hat{\varphi}
  - \hat{\psi} -l \hat{\omega }  )\overline{\hat{\theta}}_1)
 - \frac{\gamma }{m_1}|\xi|^2|\hat{\theta}_1|^2=0.
\end{aligned}
\end{equation}
Adding \eqref{e38} and \eqref{e39}, applying Young's inequality, \eqref{e36} follows.
\end{proof}

\begin{lemma}\label{lem3}
Consider the functional
\begin{equation*}
J_2(\xi,t):=lT_1(\xi,t)+T_2(\xi,t).
\end{equation*}
Then, there exists $\delta > 0$ such that
\begin{align*}
\frac{d}{dt}J_2(\xi,t)+k_0\delta| i\xi \hat{\omega } - l \hat{\varphi}|^2
& \leq \frac{\rho_1lk_1}{m_1} |\xi|^2|\hat{\varphi}_t||\hat{\theta}_{1}|
 +\frac{\rho_1k_1}{m_1} |\xi|^3|\hat{\omega}_t||\hat{\theta}_{1}| \\
&\quad  + \frac{2kl}{m_1}|\xi|| i\xi \hat{\varphi} - \hat{\psi} 
 -l \hat{\omega }||\hat{\theta}_1|+C_4(1+\xi^2)|\hat{\theta}_1|^2,
\end{align*}
where $C_4$ is a positive constant.
\end{lemma}

\begin{proof}
Lemmas \eqref{lem3'} and \eqref{lem3''} imply that
\begin{align*}
&\frac{d}{dt}J_2(\xi,t) +\frac{k_0}{2}(l^2+\xi^2)|i\xi \hat{\omega }
 - l \hat{\varphi}|^2 \\
&\leq \frac{\rho_1lk_1}{m_1}|\xi|^2|\hat{\varphi}_t|
 |\hat{\theta}_1|+\frac{2kl}{m_1}|\xi||\hat{\theta}_1||i\xi\hat{\varphi}
 -\hat{\psi}-l\hat{\omega}|
 +\frac{\rho_1k_1}{m_1}|\xi|^3|\hat{\omega}_t||\hat{\theta}_1|
 +C_4(1+ \xi^2)|\hat{\theta}_1|^2.
\end{align*}
Note that there exists $\delta >0$ such that $2\delta \leq \frac{l^2+\xi^2}{1+\xi^2}$. 
Thus,
\begin{equation}\label{e43}
\begin{aligned}
&\frac{d}{dt}J_2(\xi,t) +k_0\delta(1+\xi^2)|i\xi \hat{\omega } 
- l \hat{\varphi}|^2\\
& \leq \frac{\rho_1lk_1}{m_1}|\xi|^2|\hat{\varphi}_t|
 |\hat{\theta}_1|+\frac{2kl}{m_1}|\xi||\hat{\theta}_1||i\xi\hat{\varphi}
 -\hat{\psi}-l\hat{\omega}|\\
&\quad  +\frac{\rho_1k_1}{m_1}|\xi|^3|\hat{\omega}_t||\hat{\theta}_1|
 +C_4(1+ \xi^2)|\hat{\theta}_1|^2.
\end{aligned}
\end{equation}
\end{proof}

\begin{lemma}\label{lem4}
Consider the functional
\begin{equation*}
J_3(\xi,t)=\operatorname{Re}(-\rho_2 \hat{\psi}_t\overline{( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  )}-i\frac{\rho_1b}{k}\xi\hat{\psi}
\overline{\hat{\varphi}_t}).
\end{equation*}
If $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$, then
\begin{equation}\label{e12}
\begin{aligned}
&\frac{d}{dt}J_3(\xi,t) +\frac{k}{2}|i\xi \hat{\varphi} -\hat{\psi} 
-l \hat{\omega }|^2 \\
&\leq \rho_2 |\hat{\psi}_t|^2   +\rho_2 l \operatorname{Re}( \hat{\psi}_t
 \overline{\hat{\omega}_t}) 
-bl\operatorname{Re}( i\xi\hat{\psi}\overline{( i\xi \hat{\omega } - l \hat{\varphi} )})
+ \frac{bl\gamma}{k} |\xi| |\hat{\psi}||\hat{\theta}_1|
+C_5|\xi|^2|\hat{\theta}_2|^2.
\end{aligned}
\end{equation}
Moreover, if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$, then
\begin{equation}\label{e12'}
\begin{aligned}
&\frac{d}{dt}J_3(\xi,t) +\frac{k}{2}|i\xi \hat{\varphi} -\hat{\psi} 
-l \hat{\omega }|^2 \\
&\leq \rho_2 |\hat{\psi}_t|^2 
+\rho_2 l \operatorname{Re}( \hat{\psi}_t\overline{\hat{\omega}_t}) 
+(\rho_2-\frac{b\rho_1}{k}) \operatorname{Re}( i\xi\hat{\psi}_t
 \overline{\hat{\varphi}_t}) \\
&\quad -\frac{k_0bl}{k}\operatorname{Re}( i\xi\hat{\psi}\overline{( i\xi \hat{\omega }
- l \hat{\varphi} )})  + \frac{bl\gamma}{k} |\xi| |\hat{\psi}||\hat{\theta}_1|
+C_5|\xi|^2|\hat{\theta}_2|^2,
\end{aligned}
\end{equation}
where $C_5$ is a positive constant.
\end{lemma}

\begin{proof}
Multiplying \eqref{e2} by 
$-\overline{ ( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )}$ and taking 
real part,
\begin{equation}\label{e11}
\begin{aligned}
&\frac{d}{dt}\operatorname{Re}\Big(-\rho_2\hat{\psi}_{t}
 \overline{ ( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )}\Big) 
 - \operatorname{Re}(i\rho_2\xi\hat{\psi}_{t}\overline{\hat{\varphi}_t}) 
 -\rho_2 |\hat{\psi}_t|^2 \\
&-\operatorname{Re}\Big(\rho_2 l\hat{\psi}_t \overline{\hat{\omega }_t}\Big)
-\operatorname{Re}(b\xi ^{2}\hat{\psi }\overline{ ( i\xi \hat{\varphi} 
-\hat{\psi} -l \hat{\omega }  )}) \\
&+ k| i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  |^2 
 -\operatorname{Re}\Big( i\gamma \xi  \hat {\theta}_2\overline{ 
( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )}\Big) =0.
\end{aligned}
\end{equation}
First, suppose that
\begin{equation}\label{swp}
\frac{\rho_1}{\rho_2} =\frac{k}{b}\quad\text{and}\quad k=k_0\,.
\end{equation}
By \eqref{e1} and \eqref{swp},
\begin{align*}
&\operatorname{Re}\Big(b\xi ^{2}\hat{\psi }\overline{ ( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  )}\Big) \\
&= b\operatorname{Re}\Big(i\xi\hat{\psi }
\overline{i\xi ( i\xi \hat{\varphi} -\hat{\psi} -l \hat{\omega }  )}\Big) \\
&=\frac{\rho_1 b}{k}\operatorname{Re}(i\xi\hat{\psi}\overline{\hat{\varphi} _{tt}}) 
  -bl\operatorname{Re}(i\xi\hat{\psi} \overline{( i\xi \hat{\omega } 
 - l \hat{\varphi} )}) 
 +\frac{bl\gamma}{k}\operatorname{Re}(i\xi\hat{\psi}\overline{\hat{\theta}_1}) \\
&=\rho_2\frac{d}{dt} \operatorname{Re}(i\xi\hat{\psi}\overline{\hat{\varphi} _{t}}) 
 -\rho_2 \operatorname{Re}(i\xi\hat{\psi}_t\overline{\hat{\varphi} _{t}}) 
 -bl\operatorname{Re}\Big(i\xi\hat{\psi} \overline{( i\xi \hat{\omega } 
 - l \hat{\varphi} )}\Big)
  +\frac{bl\gamma}{k}\operatorname{Re}(i\xi\hat{\psi}\overline{\hat{\theta}_1}).
\end{align*}
Substituting this in \eqref{e11}, we have
\begin{align*}
&\frac{d}{dt}J_3(\xi,t)+k| i\xi \hat{\varphi} 
 - \hat{\psi} -l \hat{\omega }  |^2 \\
&\leq \rho_2 |\hat{\psi}_t|^2 
 + \operatorname{Re}(\rho_2 l\hat{\psi}_t \overline{\hat{\omega }_t})
 -bl\operatorname{Re}(i\xi\hat{\psi} \overline{( i\xi \hat{\omega } 
 - l \hat{\varphi} )}) 
 +\frac{bl\gamma}{k} |\xi| |\hat{\psi}||\hat{\theta}_1|\\
&\quad +\gamma |\xi| |\hat {\theta}_2|| i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|.
\end{align*}
Applying Young's inequality, \eqref{e12} follows. Now, suppose that
\begin{equation}\label{dwp}
\frac{\rho_1}{\rho_2} \neq \frac{k}{b} \quad \text{or} \quad k\neq k_0.
\end{equation}
Proceeding as above, \eqref{e1} implies 
\begin{align*}
\operatorname{Re}\Big(b\xi ^{2}\hat{\psi }\overline{ ( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  )}\Big)
&=\frac{\rho_1 b}{k}\frac{d}{dt}\operatorname{Re}(i\xi\hat{\psi}
 \overline{\hat{\varphi} _{t}}) - \frac{\rho_1 b}{k}\operatorname{Re}
 (i\xi\hat{\psi}_t\overline{\hat{\varphi} _{t}}) \\
&\quad -\frac{k_0}{k}bl\operatorname{Re}(i\xi\hat{\psi} 
 \overline{( i\xi \hat{\omega } - l \hat{\varphi} )})
+\frac{bl\gamma}{k}\operatorname{Re}(i\xi\hat{\psi}\overline{\hat{\theta}_1}).
\end{align*}
Substituting in \eqref{e11} and applying Young's inequality, we obtain \eqref{e12'}.
\end{proof}

\begin{lemma}\label{lem5}
Let $0<\varepsilon_1 < \frac{\rho_2l^2}{2\rho_1}$ and consider the functional
\begin{equation*}
J_4(\xi,t)=\operatorname{Re}(\frac{\rho_2^2l^2}{\rho_1}\hat{\psi}_t
\overline{\hat{\psi}}-\rho_2l\hat{\omega}_t\overline{\hat{\psi}}).
\end{equation*}
Then
\begin{equation}\label{e13'}
\begin{aligned}
&\frac{d}{dt}J_4(\xi,t)+b(\frac{\rho_2l^2}{\rho_1}
 -\frac{\varepsilon_1}{2}) \xi^2|\hat{\psi}|^2 \\
&\leq  \frac{\rho_2^2l^2}{\rho_1}|\hat{\psi}_t|^2 
 - \rho_2l\operatorname{Re}(\overline{\hat{\psi}}_t\hat{\omega}_t) 
 + \frac{\rho_2k_0l}{\rho_1}\operatorname{Re}( i\xi \hat{\psi}
 \overline{( i\xi \hat{\omega } - l \hat{\varphi} )}) \\
&\quad + C(\varepsilon_1)(|\hat{\theta}_1|^2+|\hat{\theta}_2|^2),
\end{aligned}
\end{equation}
where $ C(\varepsilon_1)$ is a positive constant.
\end{lemma}

\begin{proof}
Multiplying \eqref{e2} by $\overline{ \hat{\psi}}$ and taking real part,
\begin{equation}\label{ee12}
\frac{d}{dt}\operatorname{Re}(\rho_2\hat{\psi}_{t}
\overline{ \hat{\psi}})-\rho_2|\hat{\psi}_t|^2
+b\xi ^{2}|\hat{\psi }|^2
- \operatorname{Re}( k\overline{ \hat{\psi}}( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  ) )
+\operatorname{Re}( i\gamma \xi\overline{ \hat{\psi}}  \hat {\theta}_2) =0.
\end{equation}
Then \eqref{e3} implies 
\begin{align*}
&\frac{d}{dt}\operatorname{Re}(\rho_2\hat{\psi}_{t}\overline{ \hat{\psi}}) 
- \frac{d}{dt}\operatorname{Re}(\frac{\rho_1}{l}\overline{\hat{\psi}}\hat{\omega}_{t})
+b\xi ^{2}|\hat{\psi }|^2 \\
&= \rho_2|\hat{\psi}_t|^2-\frac{\rho_1}{l}\operatorname{Re}(\overline{\hat{\psi}_t}
 \hat{\omega}_{t}) 
 - \frac{k_0}{l}\operatorname{Re}(i\xi\overline{\hat{\psi}} ( i\xi \hat{\omega }
 - l \hat{\varphi} ))
+\frac{\gamma}{l}\operatorname{Re}(i\xi\overline{\hat{\psi}} \hat{\theta}_1) 
 - \operatorname{Re}( i\gamma \xi\overline{ \hat{\psi}}  \hat {\theta}_2).
\end{align*}
Multiplying by $\frac{\rho_2 l^2}{\rho_1}$, we have
\begin{align*}
\frac{d}{dt}J_4(\xi,t)+\frac{b\rho_2 l^2}{\rho_1}\xi ^{2}|\hat{\psi }|^2 
&= \frac{\rho_2^2 l^2}{\rho_1}|\hat{\psi}_t|^2-\rho_2l
 \operatorname{Re}(\overline{\hat{\psi}_t}\hat{\omega}_{t}) 
 - \frac{\rho_2k_0l}{\rho_1}\operatorname{Re}(i\xi\overline{\hat{\psi}} 
 ( i\xi \hat{\omega } - l \hat{\varphi} )) \\
&\quad +\frac{\gamma\rho_2 l}{\rho_1}\operatorname{Re}
 (i\xi\overline{\hat{\psi}} \hat{\theta}_1) 
 - \frac{\gamma\rho_2 l^2}{\rho_1}\operatorname{Re}
 ( i\xi\overline{ \hat{\psi}}  \hat {\theta}_2).
\end{align*}
Applying Young's inequality and using the wave speeds of propagation, we obtain
\eqref{e13'}.
\end{proof}

\begin{lemma}\label{lem6}
Let $0<\varepsilon_1 < \frac{\rho_2l^2}{2\rho_1}$ and consider 
$K(\xi,t)=J_3(\xi,t)+J_4(\xi,t)$. If $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and 
$k=k_0$, then
\begin{align*}
&\frac{d}{dt}K(\xi,t)+(\frac{\rho_2l^2}{\rho_1}-\varepsilon_1) b\xi^2|\hat{\psi}|^2 
+ \frac{k}{2}|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 \\
&\leq \rho_2s_1|\hat{\psi}_t|^2 
 + C(\varepsilon_1)|\hat{\theta}_1|^2 
+  C(\varepsilon_1)(1+\xi^2)|\hat{\theta}_2|^2.
\end{align*}
Moreover, if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$  or $k\neq k_0$, then
\begin{align*}
&\frac{d}{dt}K(\xi,t)+(\frac{\rho_2l^2}{\rho_1}-\varepsilon_1) b\xi^2|\hat{\psi}|^2 
 + \frac{k}{2}|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 \\
&\leq \rho_2s_1|\hat{\psi}_t|^2 + (\frac{\rho_2}{\rho_1}
 -\frac{b}{k})k_0l \operatorname{Re}(i\xi \hat{\psi}(\overline{i\xi\hat{\omega}
 -l\hat{\varphi}})) + (\rho_2-\frac{b\rho_1}{k}) \operatorname{Re}(i\xi \hat{\psi}_t
 \overline{\hat{\varphi}}_t)  \\
&\quad + C(\varepsilon_1)|\hat{\theta}_1|^2 
 +  C(\varepsilon_1)(1+\xi^2)|\hat{\theta}_2|^2,
\end{align*}
where $s_1= \frac{\rho_2 l^2}{\rho_1}+1$.
\end{lemma}

The above lemma follows from Lemmas \ref{lem4} and \ref{lem5}, 
applying Young's inequality.
For the next lemma, it is necessary to observe that
\begin{gather*}
(i\xi\hat{\varphi} - \hat{\psi} - l \hat{\omega})_t - i\xi\hat{\varphi}_t 
+ \hat{\psi}_t +l \hat{\omega}_t = 0 \label{e16} \\
(i\xi\hat{\omega} - l \hat{\varphi})_t - i\xi\hat{\omega}_t +l\hat{\varphi}_t = 0
 \label{e17}
\end{gather*}

\begin{lemma}\label{lem7}
Consider the functional
\begin{equation*}
H(\xi,t)=\rho_1 \operatorname{Re}\Big( (i\xi \hat{\varphi}- \hat{\psi} 
-l \hat{\omega })\overline{\hat{\omega}_t}\Big)
+\rho_1 \operatorname{Re}((i\xi \hat{\omega } - l \hat{\varphi})
\overline{\hat{\varphi}_t}).
\end{equation*}
If $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$, then
\begin{equation}\label{e15}
\begin{aligned}
&\frac{d}{dt}H(\xi,t)+\rho_1 l |\hat{\varphi}_t|^2
 +\frac{\rho_1 l}{2}|\hat{\omega}_t|^2 \\
&\leq \frac{\rho_2 k}{2 b l}|\hat{\psi}_t|^2 +\frac{3kl}{2}|i\xi \hat{\varphi}
 -\hat{\psi} -l \hat{\omega }|^2 
+\frac{3k_0l}{2}|i\xi \hat{\omega } 
- l \hat{\varphi}|^2 +C_6(1+\xi^2)|\hat{\theta}_1|^2.
\end{aligned}
\end{equation}
Moreover, if $\frac{\rho_1}{\rho_2} \neq\frac{k}{b}$ or $k\neq k_0$, then
\begin{equation}\label{e15'}
\begin{aligned}
&\frac{d}{dt}H(\xi,t)+\rho_1 l |\hat{\varphi}_t|^2
 +\frac{\rho_1 l}{2}|\hat{\omega}_t|^2 \\
&\leq \frac{\rho_1}{2l}|\hat{\psi}_t|^2 
 +C_1(k,k_0)|i\xi \hat{\varphi} -\hat{\psi} -l \hat{\omega }|^2 \\
&\quad +C_2(k,k_0)(1+\xi^2)|i\xi \hat{\omega } - l \hat{\varphi}|^2
 +C_6(1+\xi^2)|\hat{\theta}_1|^2,
\end{aligned}
\end{equation}
where $C_1(k,k_0)$ and $C_6$ are positive constants.
\end{lemma}

\begin{proof}
Multiplying \eqref{e1} by $\overline{(i\xi \hat{\omega } - l \hat{\varphi})}$, 
\eqref{e17} by $\rho_1\overline{\hat{\varphi}_t}$, adding these equalities 
and taking the real part, we obtain
\begin{equation}\label{e20}
\begin{aligned}
&\frac{d}{dt}\operatorname{Re}(\rho_1(i\xi\hat{\omega}
 - l \hat{\varphi})\overline{\hat{\varphi}_t}) 
 -\rho_1\operatorname{Re}( i\xi\hat{\omega}_t \overline{\hat{\varphi}_t})
 +\rho_1l|\hat{\varphi}_t|^2 \\
&-k\operatorname{Re}(i\xi ( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )
 \overline{(i\xi \hat{\omega } - l \hat{\varphi})}) \\
&-k_0l| i\xi \hat{\omega } - l \hat{\varphi}|^2 
+l\gamma \operatorname{Re}(\hat{\theta}_1\overline{(i\xi \hat{\omega } 
- l \hat{\varphi})}) = 0.
\end{aligned}
\end{equation}
Multiplying \eqref{e3} by 
$\overline{( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega })}$, \eqref{e16} by 
$\rho_1\overline{\hat{\omega}_t}$, adding, and  taking the real part,
\begin{equation}\label{e23}
\begin{aligned}
&\frac{d}{dt}\operatorname{Re}\Big(\rho_1(i\xi\hat{\varphi} - \hat{\psi} 
 - l \hat{\omega})\overline{\hat{\omega}_t}\Big) 
 -\rho_1\operatorname{Re}( i\xi\hat{\varphi}_t\overline{\hat{\omega}_t})
 +\rho_1 \operatorname{Re}( \hat{\psi}_t\overline{\hat{\omega}_t})
  +\rho_1l |\hat{\omega}_t|^2 \\
&- kl| i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 
 -k_0\operatorname{Re}\Big(i\xi ( i\xi \hat{\omega } 
 - l \hat{\varphi} )\overline{( i\xi \hat{\varphi} - \hat{\psi}
  -l \hat{\omega })}\Big) \\
&\quad + \gamma \operatorname{Re}\Big(i \xi \hat{\theta}_1\overline{( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega })}\Big)=0.
\end{aligned}
\end{equation}
Adding \eqref{e20} and \eqref{e23},
\begin{align*}
&\frac{d}{dt}H(\xi,t) +\rho_1l |\hat{\varphi}_t|^2 
 +\rho_1l |\hat{\omega}_t|^2  \\
&\leq kl| i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 
 + k_0l| i\xi \hat{\omega } - l \hat{\varphi}|^2 
 +\gamma |\xi||\hat{\theta}_1|| i\xi \hat{\varphi} - \hat{\psi} 
 -l \hat{\omega }| \\
&\quad + \gamma l |\hat{\theta}_1|| i\xi \hat{\omega }  - l \hat{\varphi}| 
+|k-k_0||\xi||i\xi\hat{\omega}-l\hat{\varphi}||i\xi\hat{\varphi}
 -\hat{\psi}-l\hat{\omega}|+\rho_1|\hat{\psi}_t||\hat{\omega}_t|.
\end{align*}
Applying Young's inequality and using the wave propagation properties,
\eqref{e15} and \eqref{e15'} follow.
\end{proof}

To prove our main result, we need to establish a functional equivalent 
to the energy \eqref{energytypeII} in a polynomial sense. In particular, 
this kind of functional gives some dissipative terms of the vector solution. 
First, let us define
\begin{equation}
\mathcal{L}_1(\xi,t) 
=\begin{cases}
J_1+\varepsilon_2 \xi^2 K+\xi^2J_2+\varepsilon_3\xi^2H, \\
\quad \text{if $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$}, \\[4pt]
\frac{\xi^2}{(1+\xi^2+\xi^4)}\big( \lambda_1  \varepsilon_3 J_1 
+ \frac{\xi^2}{(1+\xi^2+\xi^4)}(\varepsilon_3\lambda_2  K 
 + J_2+\varepsilon_3H ) \big),\\
\quad \text{if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$}.
\end{cases}
\end{equation}
where $\lambda_1, \lambda_2, \varepsilon_2, \varepsilon_3$ are positive 
constants to be determined later.

\begin{proposition}\label{prop8}
There exist constants $M, M' > 0$ such that, if $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ 
and $k=k_0$, then
\begin{equation}\label{e24}
\begin{aligned}
&\frac{d}{dt}\mathcal{L}_1(\xi,t) +M\xi^2 \big\{ b \xi^2|\hat{\psi}|^2 
 + k|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 
 +\rho_2|\hat{\psi}_t|^2 \ \\
& +k_0|i\xi \hat{\omega } - l \hat{\varphi}|^2 
 + \rho_1|\hat{\varphi}_t|^2+\rho_1|\hat{\omega}_t|^2\big\}\\
& \leq C(\varepsilon_1,\varepsilon_2,\varepsilon_3)(1+\xi^2+\xi^4+\xi^6)\xi^2
|\hat{\theta}_1|^2 
 +C(\varepsilon_1,\varepsilon_2)(1+\xi^2+\xi^4)|\hat{\theta}_2|^2.
\end{aligned}
\end{equation}
Moreover, if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$, we obtain
\begin{equation}\label{e24'}
\begin{aligned}
&\frac{d}{dt}\mathcal{L}_1(\xi,t) +M'\frac{\xi^4}{(1+\xi^2+\xi^4)^2} 
\big\{b \xi^2|\hat{\psi}|^2 + k|i\xi \hat{\varphi} - \hat{\psi} 
-l \hat{\omega }|^2 +\rho_2|\hat{\psi}_t|^2  \\
&+k_0|i\xi \hat{\omega } - l \hat{\varphi}|^2 + \rho_1|\hat{\varphi}_t|^2
 +\rho_1|\hat{\omega}_t|^2\big\} \\
&\leq C(\varepsilon_1,\varepsilon_3,\lambda_1,\lambda_2)
(1+\xi^2)\xi^2(|\hat{\theta}_1|^2 +|\hat{\theta}_2|^2 )
\end{aligned}
\end{equation}
\end{proposition}

\begin{proof}
First, we suppose that $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$. 
By using  Lemmas \ref{lem2} and \ref{lem6}, we have
\begin{align*}
&\frac{d}{dt}\{ J_1(\xi,t)+\varepsilon_2  \xi^2 K(\xi,t)\} 
+b\varepsilon_2(\frac{\rho_2l^2}{\rho_1}-\varepsilon_1) \xi^4|\hat{\psi}|^2
 + k\frac{\varepsilon_2}{2}\xi^2|i\xi \hat{\varphi} 
 - \hat{\psi} -l \hat{\omega }|^2 \\
&+\rho_2(\frac{m_2}{2}-s_1\varepsilon_2)\xi^2|\hat{\psi}_t|^2 \\
&\leq b|\xi|^3|\hat{\psi}||\hat{\theta}_2|
 + k|\xi||\hat{\theta}_2| | i\xi \hat{\varphi} 
 + \hat{\psi} +l \hat{\omega } |
 + C(\varepsilon_1, \varepsilon_2)\xi^2|\hat{\theta}_1|^2
 + C(\varepsilon_1,\varepsilon_2)(1+\xi^2)\xi^2|\hat{\theta}_2|^2.
\end{align*}
On the other hand, By Lemmas \ref{lem3} and\ref{lem7}, it follows that
\begin{align*}
&\frac{d}{dt}\{ \xi^2J_2(\xi,t)+\varepsilon_3\xi^2H(\xi,t) \} 
 + k_0\delta\xi^2|i\xi \hat{\omega } - l \hat{\varphi}|^2 
 + \rho_1\varepsilon_3 l \xi^2|\hat{\varphi}_t|^2
 +\frac{\rho_1 \varepsilon_3 l}{2}\xi^2|\hat{\omega}_t|^2 \\
&\leq \frac{\rho_1lk_1}{m_1}|\xi|^4|\hat{\varphi}_t||\hat{\theta}_1|
 +\frac{\rho_1k_1}{m_1}|\xi|^5|\hat{\theta}_1||\hat{\omega}_t| 
 +\frac{2kl}{m_1}|\xi|^3|\hat{\theta}_1||i\xi \hat{\varphi} -\hat{\psi} 
 -l \hat{\omega }|\\
&\quad +C_4(1+\xi^2) \xi^2 |\hat{\theta}_1|^2
 +\frac{3k_0\varepsilon_3 l}{2}\xi^2|i\xi \hat{\omega } 
 - l \hat{\varphi}|^2 +\frac{3kl\varepsilon_3}{2}|\xi|^2|i\xi \hat{\varphi}
  -\hat{\psi} -l \hat{\omega }|^2 \\
&\quad +\frac{\rho_2 \varepsilon_3 k}{2 b l}\xi^2|\hat{\psi}_t|^2
 +C(\varepsilon_3)(1+\xi^2)\xi^2|\hat{\theta}_1|^2.
\end{align*}
Adding and using Young's inequality, we obtain
\begin{align*}
&\frac{d}{dt}\mathcal{L}_1(\xi,t) +b\varepsilon_2(\frac{\rho_2l^2}{\rho_1}
 -2\varepsilon_1) \xi^4|\hat{\psi}|^2 + k(\frac{\varepsilon_2}{4}
 -2l\varepsilon_3)\xi^2|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 \\
&+\rho_2(\frac{m}{2}-s_1\varepsilon_2-\frac{k}{2bl}\varepsilon_3)\xi^2|\hat{\psi}_t|^2
 +k_0(\delta-\frac{3l}{2}\varepsilon_3)\xi^2|i\xi \hat{\omega } 
 - l \hat{\varphi}|^2 \\
&\quad  +\frac{\rho_1l}{2}\varepsilon_3 |\xi|^2|\hat{\varphi}_t|^2 
+\frac{\rho_1l }{4}\varepsilon_3|\xi|^2|\hat{\omega}_t|^2  \\
& \leq C(\varepsilon_1,\varepsilon_2,\varepsilon_3)(1+\xi^2+\xi^4
 +\xi^6)\xi^2|\hat{\theta}_1|^2 +C(\varepsilon_1,\varepsilon_2)(1+\xi^2+\xi^4)
 |\hat{\theta}_2|^2.
\end{align*}
We choose our constants as follows:
\[
\varepsilon_1 < \frac{\rho_2 l^2}{2 \rho_1}, \quad
\varepsilon_2 < \frac{m_2}{2s_1}, \quad 
\varepsilon_3 < \min\big\{ \frac{2\delta}{3l}, \frac{\varepsilon_2}{8l},
\frac{2bl}{k}(\frac{m_2}{2}-s_1\varepsilon_2)\big\}.
\]
Consequently, we deduce that there exists $M>0$, such that
\begin{align*}
&\frac{d}{dt}\mathcal{L}_1(\xi,t)+M\xi^2 \big\{ b \xi^2|\hat{\psi}|^2 
 + k|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 
 +\rho_2|\hat{\psi}_t|^2 \\
&+k_0|i\xi \hat{\omega } - l \hat{\varphi}|^2 
 + \rho_1|\hat{\varphi}_t|^2+\rho_1|\hat{\omega}_t|^2\big\} \\
&\leq C(\varepsilon_1,\varepsilon_2,\varepsilon_3)(1+\xi^2+\xi^4
 +\xi^6)\xi^2|\hat{\theta}_1|^2 +C(\varepsilon_1,\varepsilon_2)(1+\xi^2+\xi^4)
 |\hat{\theta}_2|^2.
\end{align*}
At last, we assume that $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$. 
Consider the functional
\[
P_1 =  \frac{\xi^2}{1+\xi^2+\xi^4} \lambda_1 \varepsilon_3 J_1(\xi,t)
+\frac{\xi^4}{(1+\xi^2+\xi^4)^2}\lambda_2 \varepsilon_3 K(\xi,t).
\]
By Lemmas \ref{lem2} and \ref{lem6} and by using Young's inequality, it follows that
\begin{align*}
&\frac{d}{dt} P_1(\xi,t) +\big(\frac{\rho_2l^2}{\rho_1}-3\varepsilon_1\big) 
 \frac{\lambda_2\varepsilon_3b\xi^6}{(1+\xi^2+\xi^4)^2} |\hat{\psi}|^2 
 + \frac{\varepsilon_3\lambda_2k\xi^4}{4(1+\xi^2+\xi^4)^2}|i\xi \hat{\varphi} 
 - \hat{\psi} -l \hat{\omega }|^2  \\
&+\big(\frac{\lambda_1m_2}{2}-C(\varepsilon_4,\lambda_2)\big)
 \frac{\rho_2\varepsilon_3\xi^4}{(1+\xi^2+\xi^4)}|\hat{\psi}_t|^2 \\
&\leq \frac{\lambda_2\varepsilon_3\varepsilon_4 \xi^4|\hat{\varphi}_t|^2}{(1+\xi ^2
 +\xi^4)^2}  +  C(\varepsilon_1,\lambda_2)\varepsilon_3
 \frac{(1+\xi^2)\xi^4|i\xi\hat{\omega}-l\hat{\varphi}|^2}{(1+\xi^2+\xi^4)^2} \\
&\quad  +C(\varepsilon_1,\varepsilon_3,\lambda_1, \lambda_2)(1+ \xi^2)
 \xi^2(|\hat{\theta}_1|^2+ |\hat{\theta}_2|).
\end{align*}
In the above estimate we used the following inequalities:
\begin{equation}\label{ert}
\frac{1}{(1+\xi^2+\xi^4)}\leq 1, \quad \frac{1+\xi^4}{(1+\xi^2+\xi^4)}\leq 1, 
\quad \frac{\xi^2}{(1+\xi^2+\xi^4)}\leq 1.
\end{equation}
On the other hand, we consider the functional
\[
P_2 =  \frac{\xi^4}{(1+\xi^2+\xi^4)^2}( J_2(\xi,t)+\varepsilon_3 H(\xi,t)).
\]
By \eqref{e43} in Lemma \ref{lem3}, Lemma \ref{lem7}, Young's inequality and
\eqref{ert}, we obtain
\begin{align*}
&\frac{d}{dt}P_2 + k_0(\delta- C(k,k_0)\varepsilon_3)
 \frac{(1+\xi^2)\xi^4}{(1+\xi^2+\xi^4)^2}|i\xi \hat{\omega }  - l \hat{\varphi}|^2\\
&+\frac{\rho_1\varepsilon_3 l \xi^4}{2(1+\xi^2+\xi^4)^2}|\hat{\varphi}_t|^2 
 +\frac{\rho_1\varepsilon_3 l\xi^4}{4(1+\xi^2+\xi^4)^2}|\hat{\omega}_t|^2 \\
& \leq \frac{\rho_1 \varepsilon_3 \xi^4}{2l(1+\xi^2+\xi^4)}|\hat{\psi}_t|^2 
+C(k,k_0)\varepsilon_3\frac{\xi^4}{(1+\xi^2+\xi^4)^2}|i\xi \hat{\varphi}
 -\hat{\psi} -l \hat{\omega }|^2 \\
&\quad +C(\varepsilon_3)(1+\xi^2)\xi^2|\hat{\theta}_1|^2.
\end{align*}
Thus, adding the above estimates of $P_1$ and $P_2$, we obtain
\begin{align*}
&\frac{d}{dt}\mathcal{L}_1(\xi,t) +(\frac{\rho_2l^2}{\rho_1}-3\varepsilon_1)
 \frac{\lambda_2\varepsilon_3b\xi^6}{(1+\xi^2+\xi^4)^2}|\hat{\psi}|^2 \\
&+ (\frac{\lambda_2}{4}-C(k,k_0))\frac{\varepsilon_3k\xi^4}{(1+\xi^2+\xi^4)^2}
 |i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2  \\
&+\Big(\frac{\lambda_1m_2}{2}-C(\varepsilon_4,\lambda_2)
 -\frac{\rho_1}{2l\rho_2}\Big)\frac{\rho_2\varepsilon_3\xi^4}{(1+\xi^2+\xi^4)}
 |\hat{\psi}_t|^2  +\frac{\rho_1 \varepsilon_3l\xi^4}{4(1+\xi^2+\xi^4)^2}
 |\hat{\omega}_t|^2 \\
&+(\delta-C(k,k_0,\varepsilon_1,\lambda_2)\varepsilon_3)
 \frac{k_0(1+\xi^2)\xi^4}{(1+\xi^2+\xi^4)^2}|i\xi \hat{\omega }
  - l \hat{\varphi}|^2\\
&\quad  + (\frac{l}{2}-\frac{\varepsilon_4\lambda_2}{\rho_1})
 \frac{\rho_1\varepsilon_3\xi^4}{(1+\xi^2+\xi^4)^2}|\hat{\varphi}_t|^2\\
&\leq C(\varepsilon_1,\varepsilon_3,\lambda_1,\lambda_2)(1+\xi^2)
 \xi^2(|\hat{\theta}_1|^2 +|\hat{\theta}_2|^2)
\end{align*}
We choose our constants as follows:
\begin{gather*}
\varepsilon_1 < \frac{\rho_2 l^2}{3\rho_1}, \quad
\lambda_2 > 4C(k,k_0), \quad
\varepsilon_4 < \frac{\rho_1 l}{2\lambda_2}, \\
\lambda_1 > \frac{\rho_1+2l\rho_2C(\varepsilon_4,\lambda_2) }{m_2l \rho_2},\quad
\varepsilon_3 < \frac{\delta}{C(k,k_0,\varepsilon_2,\lambda_2)}.
\end{gather*}
Consequently, by using \eqref{ert}, we deduce that there exists $M'>0$, such that
\begin{align*}
&\frac{d}{dt}\mathcal{L}_1(\xi,t) +M'\frac{\xi^4}{(1+\xi^2+\xi^4)^2} 
\big\{ b \xi^2|\hat{\psi}|^2 + k|i\xi \hat{\varphi} 
 - \hat{\psi} -l \hat{\omega }|^2 +\rho_2|\hat{\psi}_t|^2 \\
&+k_0|i\xi \hat{\omega } - l \hat{\varphi}|^2 
 + \rho_1|\hat{\varphi}_t|^2+\rho_1|\hat{\omega}_t|^2\big\} \\
&\leq C(\varepsilon_1,\varepsilon_3,\lambda_1,\lambda_2)
(1+\xi^2)\xi^2(|\hat{\theta}_1|^2+|\hat{\theta}_2|^2 )
\end{align*}
\end{proof}

With the Proposition \ref{prop8} in hand and making some appropriate combinations,
we build a Lyapunov functional $\mathcal{L}$, which  plays a crucial role in the proof 
of our main result.

\begin{theorem}\label{teo1}
For any $t \geq 0$ and $\xi \in \mathbb{R}$, we obtain the following decay rates
\begin{equation}\label{eq27'}
\hat{E}(\xi,t) \leq \begin{cases}
C \hat{E}(\xi,0) e^{-\beta s_1(\xi)t}, & \text{if $\frac{\rho_1}{\rho_2} =\frac{k}{b}$
 and $k=k_0$}, \\[4pt]
C' \hat{E}(\xi,0) e^{-\beta' s_2(\xi)t}, & \text{if $\frac{\rho_1}{\rho_2} 
\neq\frac{k}{b}$ or $k\neq k_0$},
\end{cases}
\end{equation}
where $C, \beta, C',\beta'$ are positive constants and
\begin{equation*}
s_1(\xi)=\frac{\xi^4}{(1+\xi^2+\xi^4+\xi^6+\xi^8)}, \quad 
s_2(\xi)=\frac{\xi^4}{(1+\xi^2)(1+\xi^2+\xi^4)^2}.
\end{equation*}
\end{theorem}

\begin{proof}
Consider the Lyapunov functional
\begin{equation}\label{f1}
\mathcal{L}(\xi,t) = \begin{cases}
\xi ^2\mathcal{L}_1(\xi,t)+N(1+\xi^2+\xi^4+\xi^6+\xi^8)\hat{E}(\xi,t), 
& \text{if $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$}, \\
\mathcal{L}_1(\xi,t)+N'(1+\xi^2)\hat{E}(\xi,t), 
& \text{if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$}.
\end{cases}
\end{equation}
where $N$ and $N'$ are positive constants to be fixed later.
First, we suppose that $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$.  
By Lemma \ref{lem1} and Proposition \ref{prop8},
\begin{align*}
\frac{d}{dt}\mathcal{L}(\xi,t) 
&\leq - M\xi^4 \big\{ b \xi^2|\hat{\psi}|^2 + k|i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }|^2 +\rho_2|\hat{\psi}_t|^2+k_0|i\xi \hat{\omega }
 - l \hat{\varphi}|^2 \\
&\quad  +\rho_1|\hat{\varphi}_t|^2+\rho_1|\hat{\omega}_t|^2\big\} \\
&\quad -(2\gamma \eta N-C(\varepsilon_1,\varepsilon_2,\varepsilon_3))
(1+\xi^2+\xi^4+\xi^6+\xi^8)\xi^2(|\hat{\theta}_1|^2 +|\hat{\theta}_2|^2 ),
\end{align*}
where $\eta = \min\{\frac{k_1}{m_1},\frac{k_2}{m_2}\}$. On the other hand, 
by definition of $\mathcal{L}_1$, there exists $M_1 >0$, such that
\[
|\xi^2\mathcal{L}_1(\xi,t)| \leq M_1 (1+\xi^2+\xi^4+\xi^6) \hat{E}(\xi,t).
\]
It follows that
\begin{equation}\label{e29}
\begin{aligned}
&(N-M_1)(1+\xi^2+\xi^4+\xi^6+\xi^8)\hat{E}(\xi,t) \\
&\leq \mathcal{L}(\xi,t) \\
&\leq (N+M_1)(1+\xi^2+\xi^4+\xi^6+\xi^8) \hat{E}(\xi,t).
\end{aligned}
\end{equation}
Choosing $N > \max (M_1, \frac{C(\varepsilon_1,\varepsilon_2,\varepsilon_3)}
{2\gamma \eta})$ and by using
\[
(1+\xi^2+\xi^4+\xi^6+\xi^8) \geq \xi^2,
\]
there exists $M_2>0$, such that
\begin{equation}\label{e28}
\frac{d}{dt}\mathcal{L}(\xi,t) \leq - M_2\xi^4 \hat{E}(\xi,t).
\end{equation}
Note that \eqref{e29} implies 
\begin{equation}\label{e30}
\frac{d}{dt}\mathcal{L}(\xi,t) 
\leq - \beta\frac{\xi^4}{(1+\xi^2+\xi^4+\xi^6+\xi^8)} \mathcal{L}(\xi,t),
\end{equation}
where $\beta=\frac{M_2}{N+M_1}$. By using Gronwall inequality, it follows that
\begin{equation}\label{e31}
\mathcal{L}(\xi,t) \leq \mathcal{L}(\xi,0) e^{-\beta s_1(\xi)t}.
\end{equation}
Now,  \eqref{e29} yields 
\[
\hat{E}(\xi,t)  \leq C \hat{E}(\xi,0) e^{-\beta s_1(\xi) t},
\]
where $C= \frac{N+M_1}{N-M_1} > 0$.


 At last, we assume that $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$.
 By Lemma \ref{lem1} and Proposition  \ref{prop8},
\begin{align*}
\frac{d}{dt}\mathcal{L}(\xi,t) 
&\leq - M'\frac{\xi^4}{(1+\xi^2+\xi ^4)^2} \big\{ b \xi^2|\hat{\psi}|^2 
+ k|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2
 +\rho_2|\hat{\psi}_t|^2  \\
&\quad +k_0|i\xi \hat{\omega } - l \hat{\varphi}|^2 + \rho_1|\hat{\varphi}_t|^2
+\rho_1|\hat{\omega}_t|^2\big\} \\
&\quad -(2\gamma \eta N'-C(\varepsilon_1,\varepsilon_3,\lambda_1,\lambda_2))
(1+\xi^2)\xi^2(|\hat{\theta}_1|^2 +|\hat{\theta}_2|^2 )
\end{align*}
On the other hand, by definition of $\mathcal{L}_2$ and by using Young's inequality,
there exists $M'_1 >0$ such that
\begin{equation}\label{e29'}
(N'-M'_1)(1+\xi^2)\hat{E}(\xi,t) \leq \mathcal{L}(\xi,t)
\leq (N'+M'_1)(1+\xi^2) \hat{E}(\xi,t).
\end{equation}
Choosing $N' > \max (M'_1, \frac{C(\varepsilon_1,
\varepsilon_3,\lambda_1,\lambda_2)}{2\gamma \eta})$ and by using
\[\
(1+\xi^2) \geq \frac{\xi^2}{(1+\xi^2+\xi^4)^2},
\]
there exists $M'_2>0$, such that
\begin{equation}\label{e28'}
\frac{d}{dt}\mathcal{L}(\xi,t) \leq - M'_2\frac{\xi^4}{(1+\xi^2+\xi^4)^2} \hat{E}(\xi,t).
\end{equation}
From \eqref{e29'}, we obtain
\begin{equation}\label{e30'}
\frac{d}{dt}\mathcal{L}(\xi,t) \leq - \beta'\frac{\xi^4}{(1+\xi^2)(1+\xi^2+\xi^4)^2}
\mathcal{L}(\xi,t),
\end{equation}
where $\beta'=\frac{M'_2}{N'+M'_1}$. By using Gronwall inequality, we 
conclude that
\begin{equation}\label{e31'}
\mathcal{L}(\xi,t) \leq \mathcal{L}(\xi,0) e^{-\beta' s_2(\xi)t}.
\end{equation}
The last inequality together with \eqref{e29'} leads to the second 
inequality of the theorem, which completes the proof.
\end{proof}

\subsection{Thermoelastic Bresse system of type III}

In this subsection, we establish decay rates for the Fourier image of the 
solutions of Thermoelastic Bresse system of Type III. Taking Fourier 
Transform in \eqref{ee2}, we obtain the  ODE system
\begin{gather}
\rho_1\hat{\varphi} _{tt}-ik\xi ( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  ) -k_0l( i\xi \hat{\omega } - l \hat{\varphi} ) 
+l\gamma \hat{\theta}_{1t} = 0 \quad \text{in $\mathbb{R} \times (0,\infty)$}, \label{eq1}\\
\rho_2\hat{\psi}_{tt}+b\xi ^{2}\hat{\psi }- k( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  )+i\gamma \xi  \hat {\theta}_{2t}=0
 \quad \text{in $\mathbb{R} \times (0,\infty)$},\label{eq2}\\
\rho_1\hat{\omega}_{tt}-ik_0\xi ( i\xi \hat{\omega } - l \hat{\varphi} )
- kl( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )
+ i\gamma \xi \hat{\theta}_{1t}=0 \quad \text{in $\mathbb{R} \times (0,\infty)$},\label{eq3}\\
\hat{\theta}_{1tt} +k_1\xi ^{2}\hat{\theta}_1+\alpha_1\xi ^{2}\hat{\theta}_{1t}
+m_1 ( i\xi \hat{\omega } - l \hat{\varphi} )_t=0 
\quad \text{in $\mathbb{R} \times (0,\infty)$},\label{eq4} \\
\hat{\theta}_{2tt} +k_2\xi ^{2}\hat{\theta}_2+\alpha_2\xi ^{2}\hat{\theta}_{2t}
+im_2 \xi \hat{\psi}_{t} =0 \quad \text{in $\mathbb{R} \times (0,\infty)$}.\label{eq5}
\end{gather}
The energy functional associated with the above system is 
\begin{equation}\label{energytypeIII}
\begin{aligned}
\hat{\mathbb{E}}( \xi,t  )
&=\rho_1|\hat{\varphi}_{t} |^{2}+\rho_2|\hat{\psi}_{t}|^{2}
+\rho_1 |\hat{\omega}_{t}|^{2}+\frac{\gamma}{m_1}|\hat{\theta}_{1t} |^{2}
+\frac{k_1\gamma}{m_1}\xi^2|\hat{\theta}_1 |^{2}
+\frac{\gamma}{m_2}|\hat{\theta}_{2t} |^{2} \\
&\quad +\frac{k_2\gamma}{m_2}\xi^2|\hat{\theta}_2 |^{2}
+b| \xi|^{2}|\hat{\psi}|^{2}
+k|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^{2} 
+ k_0|i\xi \hat{\omega } - l \hat{\varphi}|^{2}
\end{aligned}
\end{equation}

\begin{lemma}\label{lemq1}
Let $\hat{\mathbb{E}}$ the energy functional associated with
 system \eqref{eq1}-\eqref{eq5}. Then
\begin{gather}\label{eq7}
\frac{d}{dt}\hat{\mathbb{E}}(\xi,t)=-2\gamma\xi^2(\frac{\alpha_1}{m_1}|\hat{\theta}_{1t}|^2
+\frac{\alpha_2}{m_2}|\hat{\theta}_{2t}|^2).
\end{gather}
\end{lemma}

\begin{proof}
Multiplying \eqref{eq1} by $\overline{\hat{\varphi}}_t$, \eqref{eq2} by 
$\overline{\hat{\psi}}_t$, \eqref{eq3} by $\overline{\hat{\omega}}_t$, 
\eqref{eq4} by $\frac{\gamma}{m_1}\overline{\hat{\theta}_{1t}}$, \eqref{eq5}
 by $\frac{\gamma}{m_2}\overline{\hat{\theta}_{2t}}$,
adding these equalities and taking the real part, \eqref{eq7} follows.
\end{proof}

To establish the main result of this subsection and based in the approach 
done in the previous subsection, we establish the following  lemmas:

\begin{lemma}\label{lem9}
The functional
\begin{equation*}
\mathbb{J}_1(\xi,t)=\operatorname{Re}( i\rho_2\xi 
\hat{\psi}_t\overline{\hat{\theta}_{2t}})+\operatorname{Re}( ik_2\rho_2\xi^3 
\hat{\psi}\overline{\hat{\theta}_{2}}),
\end{equation*}
satisfies
\begin{equation}\label{eq9}
\begin{aligned}
&\frac{d}{dt}\mathbb{J}_1(\xi,t) + \frac{m_2\rho_2}{2}\xi^2|\hat{\psi}_t|^2 \\
&\leq k_2\rho_2|\xi|^3|\hat{\psi}||\hat{\theta}_{2t}| 
 +b|\xi|^3|\hat{\psi}||\hat{\theta}_{2t}| 
 + k|\xi||\hat{\theta}_{2t}| | i\xi \hat{\varphi} 
 - \hat{\psi} -l \hat{\omega }|\\
&\quad +  C_1(1+\xi^2)\xi^2|\hat{\theta}_{2t}|^2,
\end{aligned}
\end{equation}
where $C_1$ is a positive constant.
\end{lemma}

\begin{proof}
Multiplying \eqref{eq5} by $-i\rho_2\xi \overline{\hat{\psi}_t}$ and
 taking real part, we obtain
\begin{align*}
&\frac{d}{dt}\operatorname{Re}(-i\rho_2\xi \overline{\hat{\psi}_t}
\hat{\theta}_{2t})+\operatorname{Re}(i\rho_2\xi \overline{\hat{\psi}_{tt}}
\hat{\theta}_{2t}) -\frac{d}{dt}\operatorname{Re}(ik_2\rho_2\xi^3 
\overline{\hat{\psi}}\hat{\theta}_2) \\
&\quad +\operatorname{Re}(ik_2\rho_2\xi^3 \overline{\hat{\psi}}\hat{\theta}_{2t}) 
-\operatorname{Re}(i\alpha_2\rho_2\xi^3 \overline{\hat{\psi}_t}
 \hat{\theta}_{2t})+m_2\rho_2 \xi^2 |\hat{\psi}_{t}|^2 =0.
\end{align*}
By \eqref{eq2}, it follows that
\begin{align*}
\frac{d}{dt}\mathbb{J}_1(\xi,t)+m_2\rho_2 \xi^2 |\hat{\psi}_{t}|^2
&\leq k_2\rho_2|\xi|^3 |\hat{\psi}||\hat{\theta}_{2t}|
+\alpha_2\rho_2|\xi|^3 |\hat{\psi}_t||\hat{\theta}_{2t}|+b|\xi|^3 |\hat{\psi}| 
|\hat{\theta}_{2t}| \\
&\quad + k|\xi||\hat{\theta}_{2t}|| i\xi \hat{\varphi} 
 - \hat{\psi} -l \hat{\omega } |
+\gamma \xi^2 |\hat {\theta}_{2t}|^2.
\end{align*}
Applying Young's inequality, \eqref{eq9} follows.
\end{proof}

\begin{lemma}\label{lem10'}
The functional
\begin{equation*}
\mathbb{T}_1(\xi,t)=\operatorname{Re}\Big(-\rho_1\hat{\varphi}_t 
\overline{(i\xi \hat{\omega } - l \hat{\varphi})} 
- \frac{\rho_1}{m_1}\hat{\varphi}_t\overline{\hat{\theta}_{1t}}\Big),
\end{equation*}
satisfies
\begin{equation}\label{eq10}
\begin{aligned}
&\frac{d}{dt}\mathbb{T}_1(\xi,t) +\frac{k_0l}{2}|i\xi \hat{\omega } 
 - l \hat{\varphi}|^2  \\
&\leq \frac{\alpha_1\rho_1}{m_1}|\xi|^2|\hat{\varphi}_t||\hat{\theta}_{1t}|
 -\operatorname{Re}(ik\xi(i\xi\hat{\varphi}-\hat{\psi}-l\hat{\omega})
 \overline{(i\xi \hat{\omega } - l \hat{\varphi})}) \\
&\quad -\frac{k}{m_1}\operatorname{Re}(i\xi\overline{\hat{\theta}_{1t}}
 (i\xi\hat{\varphi}-\hat{\psi}-l\hat{\omega})) 
 +\frac{\rho_1k_1}{m_1}\operatorname{Re}(\xi^2\hat{\varphi}_t
 \overline{\hat{\theta}}_{1}) +C_2 |\hat{\theta}_{1t}|^2,
\end{aligned}
\end{equation}
where $C_2$ is a positive constant.
\end{lemma}

\begin{proof}
Multiplying \eqref{eq1} by $-\overline{(i\xi \hat{\omega } - l \hat{\varphi})}$ 
and taking real part, we have
\begin{align*}
&\frac{d}{dt} \operatorname{Re}(-\rho_1\hat{\varphi} _{t}
\overline{(i\xi \hat{\omega } - l \hat{\varphi})})
 +\operatorname{Re}(\rho_1\hat{\varphi} _{t}\overline{(i\xi \hat{\omega } 
 - l \hat{\varphi})}_t)\\
&+\operatorname{Re}(ik\xi ( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )
\overline{(i\xi \hat{\omega } - l \hat{\varphi})}) 
+k_0l|i\xi \hat{\omega } - l \hat{\varphi}|^2 -\operatorname{Re}(l\gamma \hat{\theta}_{1t}\overline{(i\xi \hat{\omega } - l \hat{\varphi})})= 0,
\end{align*}
by using \eqref{eq4}, we have
\begin{align*}
&\frac{d Re}{dt}(-\rho_1\hat{\varphi} _{t}\overline{(i\xi \hat{\omega } 
 - l \hat{\varphi})})-\frac{\rho_1}{m_1}\frac{d}{dt}\operatorname{Re}
 (\hat{\varphi}_t\overline{\hat{\theta}}_{1t})
 +\frac{\rho_1}{m_1}\operatorname{Re}(\hat{\varphi}_{tt}\overline{\hat{\theta}}_{1t})\\
&-\frac{\rho_1k_1}{m_1}\operatorname{Re}(\xi^2\hat{\varphi}_t
 \overline{\hat{\theta}}_{1}) 
-\frac{\alpha_1\rho_1}{m_1}\operatorname{Re}(\xi^2\hat{\varphi}_t
 \overline{\hat{\theta}}_{1t}) +\operatorname{Re}( ik\xi ( i\xi \hat{\varphi} 
 - \hat{\psi} -l \hat{\omega }  )\overline{(i\xi \hat{\omega } - l \hat{\varphi})})\\
&\quad +k_0l|i\xi \hat{\omega } - l \hat{\varphi}|^2
 -\operatorname{Re}(l\gamma \hat{\theta}_{1t}\overline{(i\xi \hat{\omega } 
 - l \hat{\varphi})})= 0.
\end{align*}
Note that \eqref{eq1} implies 
\begin{align*}
&\frac{d}{dt} \mathbb{T}_1(\xi,t) + \frac{k}{m_1}\operatorname{Re}
\Big( i\xi\overline{\hat{\theta}}_{1t}( i\xi \hat{\varphi} - \hat{\psi}
 -l \hat{\omega }  )\Big) 
 +\operatorname{Re}\Big( ik\xi ( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )
 \overline{(i\xi \hat{\omega } - l \hat{\varphi})}\Big) \\
&+k_0l|i\xi \hat{\omega } - l \hat{\varphi}|^2
-\frac{\rho_1k_1}{m_1}\operatorname{Re}(\xi^2\hat{\varphi}_t
 \overline{\hat{\theta}}_{1})-\frac{\alpha_1\rho_1}{m_1}
 \operatorname{Re}(\xi^2\hat{\varphi}_t\overline{\hat{\theta}}_{1t}) \\
&\leq  \frac{l\gamma}{m_1} |\hat{\theta}_{1t} |^2 
+(l\gamma+\frac{k_0 l}{m_1})|\hat{\theta}_{1t}|| i\xi \hat{\omega } 
- l \hat{\varphi}|.
\end{align*}
Applying Young's inequality, we obtain \eqref{eq10}.
\end{proof}

\begin{lemma}\label{lem10''}
The functional
\begin{equation*}
\mathbb{T}_2(\xi,t)=\operatorname{Re}\Big(i\rho_1\xi\hat{\omega}_t \overline{(i\xi \hat{\omega }
 - l \hat{\varphi})} + i\frac{\rho_1}{m_1}\xi\hat{\omega}_t
\overline{\hat{\theta}_{1t}}\Big),
\end{equation*}
satisfies
\begin{equation}\label{eq36}
\begin{aligned}
&\frac{d}{dt}\mathbb{T}_2(\xi,t) +\frac{k_0}{2}|\xi|^2|i\xi \hat{\omega }
 - l \hat{\varphi}|^2 \\
&\leq \frac{\alpha_1\rho_1}{m_1}|\xi|^3|\hat{\omega}_t||\hat{\theta}_{1t}| 
 +\operatorname{Re}\Big( ikl\xi(i\xi\hat{\varphi}-\hat{\psi}
 -l\hat{\omega})\overline{(i\xi \hat{\omega } - l \hat{\varphi})}\Big)\\
&\quad +\frac{kl}{m_1}\operatorname{Re}( i\xi\overline{\hat{\theta}}_{1t}
 ( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  ))
-\frac{k_1\rho_1}{m_1}\operatorname{Re}(i\xi^3\hat{\omega}_t
\overline{\hat{\theta}}_{1})+C_3 |\xi|^2|\hat{\theta}_{1t}|^2,
\end{aligned}
\end{equation}
where $C_3$ is a positive constant.
\end{lemma}

\begin{proof}
Multiplying \eqref{eq3} by $i\xi\overline{(i\xi \hat{\omega } - l \hat{\varphi})}$ 
and taking real part,
\begin{align*}
&\frac{d}{dt} \operatorname{Re}\Big(i\rho_1\xi\hat{\omega}_{t}
\overline{(i\xi \hat{\omega } - l \hat{\varphi})}\Big)
-\operatorname{Re}\Big(i\rho_1\xi\hat{\omega}_{t}\overline{(i\xi \hat{\omega } 
- l \hat{\varphi})}_t\Big)+k_0|\xi|^2 | i\xi \hat{\omega } - l \hat{\varphi}|^2 \\
&- \operatorname{Re}\Big(ikl\xi( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega } 
 )\overline{(i\xi \hat{\omega } - l \hat{\varphi})}\Big)
- \operatorname{Re}\Big(\gamma \xi^2 \hat{\theta}_{1t}\overline{(i\xi \hat{\omega } 
- l \hat{\varphi})}\Big)=0 ,
\end{align*}
by using \eqref{eq4}, we have
\begin{align*}
&\frac{d}{dt} \operatorname{Re}\Big(i\rho_1\xi\hat{\omega}_{t}
\overline{(i\xi \hat{\omega } - l \hat{\varphi})}\Big)
+\frac{\rho_1}{m_1}\frac{d}{dt}\operatorname{Re}(i\xi\hat{\omega} _{t}
\overline{\hat{\theta}}_{1t})-\frac{\rho_1}{m_1}
\operatorname{Re}(i\xi\hat{\omega} _{tt}\overline{\hat{\theta}}_{1t}) \\
&+\frac{\rho_1k_1}{m_1}\operatorname{Re}(i\xi ^3\hat{\omega} _{t}
 \overline{\hat{\theta}}_{1})
+\frac{\alpha_1\rho_1}{m_1}\operatorname{Re}(i\xi^3\hat{\omega} _{t}
 \overline{\hat{\theta}}_{1t})+k_0|\xi|^2 | i\xi \hat{\omega } - l \hat{\varphi}|^2 \\
&- \operatorname{Re}\Big(ikl\xi( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )
\overline{(i\xi \hat{\omega } - l \hat{\varphi})}\Big)
 - \operatorname{Re}\Big( \gamma \xi^2 \hat{\theta}_{1t}\overline{(i\xi \hat{\omega } 
- l \hat{\varphi})}\Big)=0.
\end{align*}
Note that, \eqref{eq3} implies that
\begin{align*}
&\frac{d}{dt}\mathbb{T}_2(\xi,t) -\frac{kl}{m_1}\operatorname{Re}
( i\xi\overline{\hat{\theta}}_{1t}( i\xi \hat{\varphi}
 - \hat{\psi} -l \hat{\omega }  ))  
-\operatorname{Re}\Big( ikl\xi ( i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }  )
\overline{(i\xi \hat{\omega } - l \hat{\varphi})}\Big)\\
&+k_0\xi^2|i\xi \hat{\omega } - l \hat{\varphi}|^2
+\frac{k_1\rho_1}{m_1}\operatorname{Re}(i\xi^3\hat{\omega}_t
\overline{\hat{\theta}}_{1})+\frac{\alpha_1\rho_1}{m_1}
\operatorname{Re}(i\xi^3\hat{\omega}_t\overline{\hat{\theta}}_{1t})\\
&\leq  \frac{\gamma\xi^2}{m_1} |\hat{\theta}_{1t} |^2 
+(\gamma+\frac{k_0 }{m_1})\xi^2|\hat{\theta}_{1t}|| i\xi \hat{\omega } 
- l \hat{\varphi}|.
\end{align*}
Applying Young's inequality, we obtain \eqref{eq36}.
\end{proof}

\begin{lemma}\label{lem10}
Consider the functional
\begin{equation*}
\mathbb{J}_2(\xi,t):=l\mathbb{T}_1(\xi,t)+\mathbb{T}_2(\xi,t)+\frac{\rho_1 k_1}{m_1}
\operatorname{Re}\Big( \xi^2\overline{\hat{\theta}}_{1}(i\xi\hat{\omega}
-l\hat{\varphi})\Big).
\end{equation*}
Then, there exists $\delta > 0$ such that
\begin{equation}\label{eq41}
\begin{aligned}
&\frac{d}{dt}\mathbb{J}_2(\xi,t)+k_0\delta| i\xi \hat{\omega }
 - l \hat{\varphi}|^2  \\
&\leq \frac{\alpha_1\rho_1l}{m_1} |\xi|^2|\hat{\varphi}_t||\hat{\theta}_{1t}| 
+\frac{\alpha_1\rho_1}{m_1} |\xi|^3|\hat{\omega}_t||\hat{\theta}_{1t}| 
+C_4(1+\xi^2)|\hat{\theta}_{1t}|^2
\end{aligned}
\end{equation}
where $C_3$ is a positive constant.
\end{lemma}

\begin{proof}
By Lemma \eqref{lem10'} and Lemma \eqref{lem10''},
\begin{align*}
&\frac{d}{dt} \mathbb{J}_2(\xi,t)+\frac{k_0}{2}(l^2+\xi^2)|i\xi \hat{\omega } 
- l \hat{\varphi}|^2 \\
&\leq \frac{\rho_1l\alpha_1}{m_1}|\xi|^2|\hat{\varphi}_t||\hat{\theta}_{1t}|
+\frac{\rho_1k_1}{m_1}\xi^2|\hat{\theta}_{1t}||i\xi\hat{\omega}-l\hat{\varphi}| 
 +\frac{\alpha_1\rho_1}{m_1}|\xi|^3|\hat{\omega}_t||\hat{\theta}_{1t}|
+C_4(1+ \xi^2)|\hat{\theta}_{1t}|^2
\end{align*}
Note that there exists $\delta >0$ such that $4\delta \leq \frac{l^2+\xi^2}{1+\xi^2}$.
 Thus,
\begin{equation}\label{eq43}
\begin{aligned}
&\frac{d}{dt} \mathbb{J}_2(\xi,t)+2k_0\delta(1+\xi^2)|i\xi \hat{\omega } 
 - l \hat{\varphi}|^2  \\
&\leq \frac{\rho_1l\alpha_1}{m_1}|\xi|^2|\hat{\varphi}_t||\hat{\theta}_{1t}| 
 +\frac{\rho_1k_1}{m_1}(1+\xi^2)|\hat{\theta}_{1t}||i\xi\hat{\omega}
 -l\hat{\varphi}| \\
&\quad +\frac{\alpha_1\rho_1}{m_1}|\xi|^3|\hat{\omega}_t| |\hat{\theta}_{1t}| 
 +C_4(1+ \xi^2)|\hat{\theta}_{1t}|^2.
\end{aligned}
\end{equation}
Applying Young's inequality, \eqref{eq41} follows.
\end{proof}

\begin{lemma}\label{lem11}
Consider the functional
\begin{equation*}
\mathbb{J}_3(\xi,t)=\operatorname{Re}\Big(-\rho_2 \hat{\psi}_t\overline{( i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }  )}-i\frac{\rho_1b}{k}\xi\hat{\psi}
\overline{\hat{\varphi}_t}\Big).
\end{equation*}
If $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$, then
\begin{equation}\label{eq12}
\begin{aligned}
&\frac{d}{dt}\mathbb{J}_3(\xi,t) +\frac{k}{2}|i\xi \hat{\varphi} -\hat{\psi} 
 -l \hat{\omega }|^2 \\
&\leq \rho_2 |\hat{\psi}_t|^2 +\rho_2 l \operatorname{Re}
( \hat{\psi}_t\overline{\hat{\omega}_t}) 
-bl\operatorname{Re}\Big( i\xi\hat{\psi}\overline{( i\xi \hat{\omega } 
- l \hat{\varphi} )}\Big)  \\
&\quad + \frac{bl\gamma}{k} |\xi| |\hat{\psi}||\hat{\theta}_{1t}|
+C_5|\xi|^2|\hat{\theta}_{2t}|^2
\end{aligned}
\end{equation}
Moreover, if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$, then
\begin{equation}\label{eq12'}
\begin{aligned}
&\frac{d}{dt}\mathbb{J}_3(\xi,t) +\frac{k}{2}|i\xi \hat{\varphi} -\hat{\psi}
 -l \hat{\omega }|^2  \\
&\leq \rho_2 |\hat{\psi}_t|^2 +\rho_2 l \operatorname{Re}
 ( \hat{\psi}_t\overline{\hat{\omega}_t}) \\
&\quad +(\rho_2-\frac{b\rho_1}{k}) \operatorname{Re}
( i\xi\hat{\psi}_t\overline{\hat{\varphi}_t})
-\frac{k_0bl}{k}\operatorname{Re}\Big( i\xi\hat{\psi}
\overline{( i\xi \hat{\omega } - l \hat{\varphi} )}\Big) \\
&\quad + \frac{bl\gamma}{k} |\xi| |\hat{\psi}||\hat{\theta}_{1t}|
+C_5|\xi|^2|\hat{\theta}_{2t}|^2,
\end{aligned}
\end{equation}
where $C_5$ is a positive constant.
\end{lemma}

\begin{proof}
Proceeding as proof of Lemma \ref{lem4}, we obtain \eqref{eq12} and \eqref{eq12'}.
\end{proof}

\begin{lemma}\label{lem12}
Let $0<\varepsilon_1 < \frac{\rho_2l^2}{2\rho_1}$ and consider the functional
\begin{equation*}
\mathbb{J}_4(\xi,t)=\operatorname{Re}(\frac{\rho_2^2l^2}{\rho_1}
\hat{\psi}_t\overline{\hat{\psi}}-\rho_2l\hat{\omega}_t\overline{\hat{\psi}}).
\end{equation*}
If $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$, then
\begin{equation}\label{eq13}
\begin{aligned}
&\frac{d\mathbb{J}_4(\xi,t)}{dt}+b(\frac{\rho_2l^2}{\rho_1}
 -\frac{\varepsilon_1}{2}) \xi^2|\hat{\psi}|^2  \\
&\leq \frac{\rho_2^2l^2}{\rho_1}|\hat{\psi}_t|^2 
 - \rho_2l\operatorname{Re}(\overline{\hat{\psi}}_t\hat{\omega}_t) 
+ bl\operatorname{Re}( i\xi \hat{\psi}\overline{( i\xi \hat{\omega } 
- l \hat{\varphi} )}) + C(\varepsilon_1)(|\hat{\theta}_{1t}|^2
+|\hat{\theta}_{2t}|^2)
\end{aligned}
\end{equation}
Moreover, If $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$, then
\begin{equation}\label{eq13'}
\begin{aligned}
&\frac{d}{dt}\mathbb{J}_4(\xi,t)+b(\frac{\rho_2l^2}{\rho_1}-\frac{\varepsilon_1}{2}) 
\xi^2|\hat{\psi}|^2  \\
&\leq  \frac{\rho_2^2l^2}{\rho_1}|\hat{\psi}_t|^2 
 - \rho_2l\operatorname{Re}(\overline{\hat{\psi}}_t\hat{\omega}_t) \\
&\quad + \frac{\rho_2k_0l}{\rho_1}
\operatorname{Re}\Big( i\xi \hat{\psi}\overline{( i\xi \hat{\omega }
 - l \hat{\varphi} )}\Big)
+ C(\varepsilon_1)(|\hat{\theta}_{1t}|^2+|\hat{\theta}_{2t}|^2),
\end{aligned}
\end{equation}
where $ C(\varepsilon_1)$ is a positive constant.
\end{lemma}

\begin{proof}
Proceeding as proof of Lemma \ref{lem5}, we obtain  \eqref{eq13} and \eqref{eq13'}.
\end{proof}

\begin{lemma}\label{lem13}
Let $0<\varepsilon_1 < \frac{\rho_2l^2}{2\rho_1}$ and consider 
$\mathbb{K}(\xi,t)=\mathbb{J}_3(\xi,t)+\mathbb{J}_4(\xi,t)$, If $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ 
and $k=k_0$, then
\begin{align*}
&\frac{d}{dt}\mathbb{K}(\xi,t)+b(\frac{\rho_2l^2}{\rho_1}-\varepsilon_1) \xi^2|\hat{\psi}|^2 
+ \frac{k}{2}|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 \\
&\leq \rho_2s_1|\hat{\psi}_t|^2 
 + C(\varepsilon_1)|\hat{\theta}_{1t}
|^2 +  C(\varepsilon_1)(1+\xi^2)|\hat{\theta}_{2t}|^2
\end{align*}
Moreover, if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$  or $k\neq k_0$, then
\begin{align*}
&\frac{d}{dt}\mathbb{K}(\xi,t)+b(\frac{\rho_2l^2}{\rho_1}-\varepsilon_1) \xi^2|\hat{\psi}|^2 
+ \frac{k}{2}|i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 \\
&\leq  \rho_2s_1|\hat{\psi}_t|^2 
+ k_0(\frac{\rho_2}{\rho_1}-\frac{b}{k})l \operatorname{Re}(i\xi \hat{\psi}
(\overline{i\xi\hat{\omega}-l\hat{\varphi}})) 
+ (\rho_2-\frac{b\rho_1}{k}) \operatorname{Re}(i\xi \hat{\psi}_t
\overline{\hat{\varphi}}_t)  \\
&\quad +C(\varepsilon_1)|\hat{\theta}_{1t}|^2 
+  C(\varepsilon_1)(1+\xi^2)|\hat{\theta}_{2t}|^2,
\end{align*}
where $s_1= \frac{\rho_2 l^2}{\rho_1}+1$.
\end{lemma}

The above lemma follows from Lemmas \ref{lem11} and \ref{lem12}, applying 
Young's inequality.

\begin{lemma}\label{lem14}
Consider the functional
\begin{equation*}
\mathbb{H}(\xi,t)=\rho_1 \operatorname{Re}\Big( (i\xi \hat{\varphi}
- \hat{\psi} -l \hat{\omega })\overline{\hat{\omega}_t}\Big)
+\rho_1 \operatorname{Re}((i\xi \hat{\omega } - l \hat{\varphi})
\overline{\hat{\varphi}_t}).
\end{equation*}
If $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$, then
\begin{equation}\label{eq15}
\begin{aligned}
&\frac{d}{dt}\mathbb{H}(\xi,t)+\rho_1 l |\hat{\varphi}_t|^2
+\frac{\rho_1 l}{2}|\hat{\omega}_t|^2 \\
&\leq \frac{\rho_2 k}{2 b l}|\hat{\psi}_t|^2 +\frac{3kl}{2}|i\xi \hat{\varphi} 
-\hat{\psi} -l \hat{\omega }|^2 
+\frac{3k_0l}{2}|i\xi \hat{\omega } - l \hat{\varphi}|^2 
+C_6(1+\xi^2)|\hat{\theta}_{1t}|^2
\end{aligned}
\end{equation}
Moreover, if $\frac{\rho_1}{\rho_2} \neq\frac{k}{b}$ or $k\neq k_0$, then
\begin{equation}\label{eq15'}
\begin{aligned}
&\frac{d}{dt}\mathbb{H}(\xi,t)+\rho_1 l |\hat{\varphi}_t|^2
 +\frac{\rho_1 l}{2}|\hat{\omega}_t|^2 \\
&\leq \frac{\rho_1}{2l}|\hat{\psi}_t|^2 +C_1(k,k_0)|i\xi \hat{\varphi} 
-\hat{\psi} -l \hat{\omega }|^2 
+C_2(k,k_0)(1+\xi^2)|i\xi \hat{\omega } - l \hat{\varphi}|^2 \\
&\quad +C_6(1+\xi^2)|\hat{\theta}_{1t}|^2,
\end{aligned}
\end{equation}
where $C_1(k,k_0), C_6$ are positive constants.
\end{lemma}

\begin{proof}
Proceeding as proof of Lemma \ref{lem7}, we obtain \eqref{eq15} and \eqref{eq15'}.
\end{proof}

\begin{lemma}\label{lem15}
The functional
\begin{align*}
\mathbb{S}(\xi,t)&= \frac{\gamma}{m_1}\operatorname{Re}(\xi^2\hat{\theta}_{1t}
\overline{\hat{\theta}}_{1})
+\frac{\gamma}{m_2}\operatorname{Re}(\xi^2\hat{\theta}_{2t}
 \overline{\hat{\theta}}_{2}) + \frac{\gamma}{2}\xi^4(\frac{\alpha_1}{m_1}
 |\hat{\theta}_{1t}|^2+ \frac{\alpha_2}{m_2}| \hat{\theta}_{2t}|^2  ) \\
&\quad + \gamma \operatorname{Re}( i\xi^3\hat{\psi}\overline{\hat{\theta}}_2
 + \xi^2\overline{\hat{\theta}}_1(i \xi \hat{\omega} - l\hat{\varphi}))
\end{align*}
satisfies
\begin{equation}\label{eqq1}
\begin{aligned}
&\frac{d}{dt}\mathbb{S}(\xi,t)+\frac{k_1\gamma}{m_1}\xi^4|\hat{\theta}_1|^2
+\frac{k_2\gamma}{m_2}\xi^4|\hat{\theta}_2|^2 \\
&\leq \gamma\xi^2 |\hat{\theta}_{1t}||i\xi\hat{\omega}-l\hat{\varphi}|
+\gamma|\xi|^3|\hat{\psi}||\hat{\theta}_{2t}| 
+\frac{\gamma}{m_1} \xi^2|\hat{\theta}_{1t}|^2
+\frac{\gamma}{m_2}\xi^2 |\hat{\theta}_{2t}|^2
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Multiplying \eqref{eq4} by $\frac{\gamma}{m_1}\xi^2 \overline{\hat{\theta}}_1$ 
and taking real part, we obtain
\begin{equation}\label{eqq2}
\begin{aligned}
&\frac{d}{dt}\big\{ \frac{\gamma}{m_1}\operatorname{Re}(\xi^2\hat{\theta}_{1t}
 \overline{\hat{\theta}}_{1}) + \frac{\alpha_1\gamma}{2m_1}\xi^4|
 \hat{\theta}_{1t}|^2    + \gamma \operatorname{Re}( \xi^2\overline{\hat{\theta}}_1
 (i \xi \hat{\omega} - l\hat{\varphi}))\big\}\\
&+k_1\frac{\gamma}{m_1}\xi^4|\hat{\theta}_1|^2  \\
&\leq  \frac{\gamma}{m_1}\xi^2|\hat{\theta}_{1t}|^2 
 +\gamma\xi^2|\hat{\theta}_{1t}||i \xi \hat{\omega} - l\hat{\varphi}|.
\end{aligned}
\end{equation}
Moreover, multiplying \eqref{eq5} by 
$\frac{\gamma}{m_2}\xi^2 \overline{\hat{\theta}}_2$ and taking real part,
\begin{equation}\label{eqq3}
\begin{aligned}
&\frac{d}{dt}\big\{ \frac{\gamma}{m_2}\operatorname{Re}(\xi^2
 \hat{\theta}_{2t}\overline{\hat{\theta}}_{2})  
+ \frac{\alpha_2\gamma}{2m_2}\xi^4| \hat{\theta}_{2t}|^2
 + \gamma \operatorname{Re}( i\xi^3\hat{\psi}\overline{\hat{\theta}}_2)\big\} 
 +k_2\frac{\gamma}{m_2}\xi^4|\hat{\theta}_2|^2 \\
&\leq  \frac{\gamma}{m_2}\xi^2|\hat{\theta}_{2t}|^2
+\gamma |\xi|^3|\hat{\psi}||\hat{\theta}_{2t}|.
\end{aligned}
\end{equation}
Adding \eqref{eqq2} and \eqref{eqq3}, we obtain \eqref{eqq1}
\end{proof}

Now, Consider the functional
\begin{equation}
\mathcal{L}_2(\xi,t) =
\begin{cases}
\mathbb{J}_1(\xi,t)+\varepsilon_2 \xi^2 \mathbb{K}(\xi,t) +\xi^2\mathbb{J}_2(\xi,t) 
+\varepsilon_3\xi^2\mathbb{H}(\xi,t)+\mathbb{S}(\xi,t), \\
\quad \text{if $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$}, \\[4pt]
\frac{\xi^2}{(1+\xi^2+\xi^4)}\big( \lambda_1  \varepsilon_3 \mathbb{J}_1 
 + \frac{1}{(1+\xi^2+\xi^4)}(\varepsilon_3\lambda_2  \xi^2\mathbb{K} 
+ \xi^2\mathbb{J}_2+\varepsilon_3\xi^2\mathbb{H} +\mathbb{S} ) \big), \\
\quad \text{if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$}.
\end{cases}
\end{equation}
where $\lambda_1, \lambda_2, \varepsilon_2, \varepsilon_3$ are positive 
constants to be determined later.

\begin{proposition}\label{prop16}
There exist constants $M, M' > 0$ such that  if $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ 
and $k=k_0$, then
\begin{equation}\label{eq24}
\begin{aligned}
&\frac{d}{dt}\mathcal{L}_2(\xi,t) +M\xi^2 \big\{ b \xi^2|\hat{\psi}|^2 + k|i\xi \hat{\varphi}
 - \hat{\psi} -l \hat{\omega }|^2 +\rho_2|\hat{\psi}_t|^2+k_0|i\xi \hat{\omega } 
- l \hat{\varphi}|^2  \\
& + \rho_1|\hat{\varphi}_t|^2+\rho_1|\hat{\omega}_t|^2
 +\frac{k_1\gamma}{m_1}\xi^2|\hat{\theta}_{1}|^2 
 +\frac{k_2\gamma}{m_2}\xi^2|\hat{\theta}_{2}|^2 \big\}  \\
&\leq C(\varepsilon_1,\varepsilon_2,\varepsilon_3)(1+\xi^2+\xi^4
+\xi^6)\xi^2|\hat{\theta}_{1t}|^2 +C(\varepsilon_1,\varepsilon_2)(1+\xi^2+\xi^4)
|\hat{\theta}_{2t}|^2
\end{aligned}
\end{equation}
Moreover, if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$, we obtain
\begin{equation}\label{eq24'}
\begin{aligned}
&\frac{d}{dt}\mathcal{L}_2(\xi,t) +M'\frac{\xi^4}{(1+\xi^2+\xi^4)^2} 
\big\{ b \xi^2|\hat{\psi}|^2 + k|i\xi \hat{\varphi} - \hat{\psi} 
 -l \hat{\omega }|^2 +\rho_2|\hat{\psi}_t|^2\  \\
&+k_0|i\xi \hat{\omega } - l \hat{\varphi}|^2 + \rho_1|\hat{\varphi}_t|^2
 +\rho_1|\hat{\omega}_t|^2+\frac{k_1\gamma}{m_1}\xi^2|\hat{\theta}_{1}|^2 
 +\frac{k_2\gamma}{m_2}\xi^2|\hat{\theta}_{2}|^2 \}   \\
&\leq C(\varepsilon_1,\varepsilon_3,\lambda_1,\lambda_2)
(1+\xi^2)\xi^2(|\hat{\theta}_1|^2 +|\hat{\theta}_2|^2 ).
\end{aligned}
\end{equation}
\end{proposition}

\begin{proof}
We can prove \eqref{eq24} and \eqref{eq24'} following the ideas used on 
the proof of Proposition \ref{prop8}, thus we omit some details.
First, we suppose that $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$. 
By Lemmas \ref{lem9}, \ref{lem13}, \ref{lem10} and \ref{lem14}, 
it follows that
\begin{align*}
&\frac{d}{dt}\{\mathcal{L}_2(\xi,t)-\mathbb{S}(\xi,t)\} +(\frac{\rho_2l^2}{\rho_1}-2\varepsilon_1)
\varepsilon_2b \xi^4|\hat{\psi}|^2 
+ (\frac{\varepsilon_2}{4}-\frac{3l}{2}\varepsilon_3)k\xi^2|i\xi \hat{\varphi} 
- \hat{\psi} -l \hat{\omega }|^2 \\
&+(\frac{m}{2}-s_1\varepsilon_2-\frac{k}{2bl}\varepsilon_3)\rho_2\xi^2
 |\hat{\psi}_t|^2 +(\delta-\frac{3l}{2}\varepsilon_3)k_0\xi^2|i\xi \hat{\omega } 
- l \hat{\varphi}|^2 \\
&+ \frac{\rho_1l}{2}\varepsilon_3 |\xi|^2|\hat{\varphi}_t|^2
+\frac{\rho_1l }{4}\varepsilon_3|\xi|^2|\hat{\omega}_t|^2  \\
&\leq C(\varepsilon_1,\varepsilon_2,\varepsilon_3)(1+\xi^2+\xi^4
+\xi^6)\xi^2|\hat{\theta}_{1t}|^2 +C(\varepsilon_1,\varepsilon_2)
(1+\xi^2+\xi^4)|\hat{\theta}_{2t}|^2.
\end{align*}
Adding $\mathbb{S}(\xi,t)$ in the above inequality, applying Lemma \ref{lem15} and 
Young's inequality, we obtain
\begin{align*}
&\frac{d}{dt}\mathcal{L}_2(\xi,t) +(\frac{\rho_2l^2}{\rho_1}-3\varepsilon_1)\varepsilon_2b 
\xi^4|\hat{\psi}|^2 + (\frac{\varepsilon_2}{4}-\frac{3l}{2}\varepsilon_3)k\xi^2|i\xi 
\hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 \\
&+(\frac{m}{2}-s_1\varepsilon_2-\frac{k}{2bl}\varepsilon_3)\rho_2\xi^2
 |\hat{\psi}_t|^2 +(\delta-2l\varepsilon_3)k_0\xi^2|i\xi \hat{\omega } 
 - l \hat{\varphi}|^2 \\
&+ \frac{\rho_1l}{2}\varepsilon_3 |\xi|^2|\hat{\varphi}_t|^2
 +\frac{\rho_1l }{4}\varepsilon_3|\xi|^2|\hat{\omega}_t|^2
 +\frac{k_1\gamma}{m_1}\xi^4|\hat{\theta}_{1}|^2
 +\frac{k_2\gamma}{m_2}\xi^4|\hat{\theta}_{2}|^2 \\
&\leq C(\varepsilon_1,\varepsilon_2,\varepsilon_3)(1+\xi^2+\xi^4+\xi^6)\xi^2
 |\hat{\theta}_{1t}|^2 +C(\varepsilon_1,\varepsilon_2)(1+\xi^2+\xi^4)
 |\hat{\theta}_{2t}|^2.
\end{align*}
We choose our constants as follows:
\[
\varepsilon_1 < \frac{\rho_2 l^2}{3 \rho_1}, \quad
\varepsilon_2 < \frac{m_2}{2s_1}, \quad 
\varepsilon_3 < \min\{ \frac{\delta}{2l}, 
\frac{\varepsilon_2}{6l},\frac{2bl}{k}(\frac{m_2}{2}-s_1\varepsilon_2)\}
\]
Consequently, we deduce that there exists $M>0$, such that \eqref{eq24} holds.

Second, we assume that  $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ and $k\neq k_0$. 
By Lemmas \ref{lem9}, \ref{lem13}, the estimate \eqref{eq43} in Lemma \ref{lem10},
 Lemma  \ref{lem14}, adding these inequalities and by using Young's inequality,
we obtain
\begin{align*}
&\frac{d}{dt}\big\{ \mathcal{L}_2(\xi,t)- \frac{\xi^2\mathbb{S}(\xi,t)}{(1+\xi^2+\xi^4)^2}\big\}
+(\frac{\rho_2l^2}{\rho_1}-3\varepsilon_1)
 \frac{\lambda_2\varepsilon_3b\xi^6}{(1+\xi^2+\xi^4)^2}|\hat{\psi}|^2  \\
&+\frac{\rho_1 \varepsilon_3l\xi^4}{4(1+\xi^2+\xi^4)^2}|\hat{\omega}_t|^2
+(\frac{\lambda_1m_2}{2}-C(\varepsilon_4,\lambda_2)-\frac{\rho_1}{\rho_2})
 \frac{\rho_2\varepsilon_3\xi^4}{(1+\xi^2+\xi^4)}|\hat{\psi}_t|^2 \\
&+ (\frac{\lambda_2}{4}-C(k,k_0))\frac{\varepsilon_3k\xi^4}{(1+\xi^2+\xi^4)^2}
 |i\xi \hat{\varphi} - \hat{\psi} -l \hat{\omega }|^2 \\
&+(\delta-C(k,k_0,\varepsilon_1,\lambda_2)\varepsilon_3)
 \frac{k_0(1+\xi^2)\xi^4}{(1+\xi^2+\xi^4)^2}|i\xi \hat{\omega } 
 - l \hat{\varphi}|^2
 + (\frac{l}{2}-\frac{\varepsilon_4\lambda_2}{\rho_1})
 \frac{\rho_1\varepsilon_3\xi^4}{(1+\xi^2+\xi^4)^2}|\hat{\varphi}_t|^2 \\
&\leq C(\varepsilon_1,\varepsilon_3,\lambda_1,\lambda_2)(1+\xi^2)\xi^2
 (|\hat{\theta}_{1t}|^2 +|\hat{\theta}_{2t}|^2).
\end{align*}
In the last estimate, we used also the inequalities \eqref{ert}. 
Adding $\frac{\xi^2}{(1+\xi^2+\xi^4)^2}\mathbb{S}(\xi,t)$ in the above inequality, 
applying Lemma \ref{lem15} and Young's inequality, it follows that
\begin{align*}
&\frac{d}{dt}\mathcal{L}_2(\xi,t)+(\frac{\rho_2l^2}{\rho_1}
 -4\varepsilon_1)\frac{\lambda_2\varepsilon_3b\xi^6}{(1+\xi^2
 +\xi^4)^2}|\hat{\psi}|^2 \\
&+ (\frac{\lambda_2}{4}-C(k,k_0))
 \frac{\varepsilon_3k\xi^4}{(1+\xi^2+\xi^4)^2}|i\xi \hat{\varphi} 
 - \hat{\psi} -l \hat{\omega }|^2  \\
&+(\frac{\lambda_1m_2}{2}-C(\varepsilon_4,\lambda_2)
 -\frac{\rho_1}{\rho_2})\frac{\rho_2\varepsilon_3\xi^4}{(1+\xi^2+\xi^4)}
 |\hat{\psi}_t|^2
 +\frac{\rho_1l \varepsilon_3\xi^4}{4(1+\xi^2+\xi^4)^2}|\hat{\omega}_t|^2 \\
&+(\delta-C(k,k_0,\varepsilon_1,\lambda_2)\varepsilon_3)
 \frac{k_0(1+\xi^2)\xi^4}{(1+\xi^2+\xi^4)^2}|i\xi \hat{\omega } 
 - l \hat{\varphi}|^2 +
(\frac{l}{2}-\frac{\varepsilon_4\lambda_2}{\rho_1})
 \frac{\rho_1\varepsilon_2\xi^4}{(1+\xi^2+\xi^4)^2}|\hat{\varphi}_t|^2 \\
&+\frac{k_1\gamma}{m_1}\frac{\xi^6}{(1+\xi^2+\xi^4)^2}|\hat{\theta}_{1}|^2
 +\frac{k_2\gamma}{m_2}\frac{\xi^6}{(1+\xi^2+\xi^4)^2}|\hat{\theta}_{2}|^2 \\
&\leq  C(\varepsilon_1,\varepsilon_3,\lambda_1,\lambda_2)
 (1+\xi^2)\xi^2(|\hat{\theta}_{1t}|^2 +|\hat{\theta}_{2t}|^2).
\end{align*}
We choose our constants as follows:
\begin{gather*}
\varepsilon_1 < \frac{\rho_2 l^2}{4\rho_1}, \quad
\lambda_2 > 4C(k,k_0), \quad
\varepsilon_4 < \frac{\rho_1 l}{2\lambda_2}, \\
\lambda_1 > \frac{2(\rho_1+C(\varepsilon_4,\lambda_2)\rho_2) }{m_2 \rho_2},\quad 
\varepsilon_3 < \frac{\delta}{C(k,k_0,\varepsilon_2,\lambda_2)}.
\end{gather*}
Consequently, by using \eqref{ert}, we deduce that there exists $M'>0$, 
such that \eqref{eq24'} holds.
\end{proof}

\begin{theorem}\label{teo1'}
For any $t \geq 0$ and $\xi \in \mathbb{R}$, we obtain the following decay rates,
\begin{equation}\label{eq27}
\hat{\mathbb{E}}(\xi,t) 
\leq \begin{cases}
C \hat{\mathbb{E}}(\xi,0) e^{-\beta s_1(\xi)t}, 
 &\text{if $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and $k=k_0$}, \\[4pt]
C' \hat{\mathbb{E}}(\xi,0) e^{-\beta' s_2(\xi)t}, 
 &\text{if $\frac{\rho_1}{\rho_2} \neq\frac{k}{b}$ or $k\neq k_0$},
\end{cases}
\end{equation}
where $C, \beta, C',\beta'$ are positive constants and
\begin{equation*}
s_1(\xi)=\frac{\xi^4}{(1+\xi^2+\xi^4+\xi^6+\xi^4)}, \quad 
s_2(\xi)=\frac{\xi^4}{(1+\xi^2)(1+\xi^2+\xi^4)^2}.
\end{equation*}
\end{theorem}

\begin{proof}
We prove \eqref{eq27}, by Proposition \ref{prop16} and using the same
approach done in the proof of Theorem \ref{teo1}, Thus, we omit the details.
\end{proof}


\section{Main result}

In this section, we establish decay estimates of the solutions to 
 systems \eqref{ee1} and \eqref{ee2}. For Bresse system \eqref{ee1},
 thermoelasticity of Type I, we consider the vector solution
\begin{equation}\label{vectorsolution1}
\begin{aligned}
V_1&:=\Big( \rho_1^{1/2}\varphi_{t},\rho_2^{1/2}\psi_{t},
\rho_1^{1/2}\omega_{t},(\frac{\gamma}{m_1})^{1/2}\theta_1, 
(\frac{\gamma}{m_2})^{1/2}\theta_2,b^{1/2}\psi_x,   \\
&\quad k^{1/2}(\varphi_x - \psi_x -l \omega_x), k_0^{1/2}( \omega_x - l\varphi)\Big)
\end{aligned}
\end{equation}
and for  Bresse system \eqref{ee2}, thermoelasticity of type III,
\begin{equation}\label{vectorsolution2}
\begin{aligned}
V_2&:=\Big( \rho_1^{1/2}\varphi_{t},\rho_2^{1/2}\psi_{t}, \rho_1^{1/2}\omega_{t},
(\frac{\gamma}{m_1})^{1/2}\theta_{1t}, (\frac{k_1\gamma}{m_1})^{1/2}\theta_{1x},
 (\frac{\gamma}{m_2})^{1/2}\theta_{2t}, \\
&\quad  (\frac{k_2\gamma}{m_2})^{1/2}\theta_{2x},b^{1/2}\psi_x,k^{1/2}
(\varphi_x - \psi_x -l \omega_x), k_0^{1/2}( \omega_x - l\varphi)\Big)
\end{aligned}
\end{equation}
Note that
\begin{equation}\label{eq45}
\hat{E}(\xi,t)=|\hat{V}_1(\xi,t)|^2,\quad 
\hat{\mathbb{E}}(\xi,t)=|\hat{V}_2(\xi,t)|^2,
\end{equation}
where $\hat{E}$ and $\hat{\mathbb{E}}$ are defined in \eqref{energytypeII} 
and \eqref{energytypeIII}, respectively.
We are now in a  position to prove our main result.

\begin{proof}[Proof of Theorem \ref{teo2}]
Applying the Plancherel identity and \eqref{eq45}, we have
\begin{gather*}
\|\partial^k_xV_1(t)\|_{L^2(\mathbb{R})}^2 
= \|(i\xi)^k\hat{V_1}(t)\|_{L^2(\mathbb{R})}^2 
= \int_{\mathbb{R}}|\xi|^{2k}\hat{E}(\xi,t)^2 d\xi, \\
\|\partial^k_xV_2(t)\|_{L^2(\mathbb{R})}^2 
= \|(i\xi)^k\hat{V_2}(t)\|_{L^2(\mathbb{R})}^2
 = \int_{\mathbb{R}}|\xi|^{2k}\hat{\mathbb{E}}(\xi,t)^2 d\xi.
\end{gather*}
By Theorems \ref{teo1} and \ref{teo1'}, it follows that
\begin{align*}
\|\partial^k_xV_j(t)\|_2^2 
&\leq C\int_{\mathbb{R}}|\xi|^{2k}e^{-\beta s(\xi)t}\hat{V_j}(0,\xi)^2 d\xi \\
&\leq C\int_{|\xi|\leq 1}|\xi|^{2k}e^{-\beta s_i(\xi)t}\hat{V_j}^2(0,\xi) d\xi
  + C\int_{|\xi|\geq 1}|\xi|^{2k}e^{-\beta  s_i(\xi)t}\hat{V_j}^2(0,\xi) d\xi\\
&=I_1+I_2,  \quad (i,j=1,2).
\end{align*}
It is not difficult to see that if $\frac{\rho_1}{\rho_2} =\frac{k}{b}$ and 
$k=k_0$, then the function $s_1(\xi)$  satisfies
\begin{equation}\label{e46}
\begin{gathered}
s_1(\xi) \geq \frac{1}{5}\xi^4 \quad\text{if }|\xi|\leq 1, \\
s_1(\xi) \geq \frac{1}{5}\xi^{-4} \quad\text{if } |\xi|\geq 1\,.
\end{gathered}
\end{equation}
Thus, we estimate $I_1$ as follows,
\begin{equation}
\begin{aligned}
I_1 &\leq C \|\hat{V_j^0}\|_{L^\infty}^2 \int_{|\xi|\leq 1}|\xi|^{2k}
 e^{-\frac{\beta}{5}\xi^4t} d\xi \\
& \leq C_1\|\hat{V_j^0}\|_{L^\infty}^2( 1+t)^{-\frac{1}{4}(1+2k)} \\
& \leq C_1( 1+t)^{-\frac{1}{4}(1+2k)} \|V_j^0\|_{L^1}^2, \quad j=1,2.
\end{aligned}
\end{equation}
On the other hand,  using the second inequality in \eqref{e46}, we obtain
\begin{align*}
I_2 
&\leq C\int_{|\xi|\geq 1}|\xi|^{2k}e^{-\frac{\beta}{5}\xi^{-4}t}
 \hat{V_j^0}^2(\xi)d\xi \\
&\leq C \sup_{|\xi| \geq 1}\{ |\xi |^{-2l}e^{-\frac{\beta}{5} \xi ^{-4}t} \} 
 \int_{\mathbb{R}}|\xi|^{2(k+l)}\hat{V_j^0}^2(\xi)d\xi   \\
&\leq C_2 (1+t)^{-\frac{l}{2}} \|\partial_x^{k+l}V_j^0\|_2^2,\quad j=1,2.
\end{align*}
Combining the estimates of $I_1$ and $I_2$, we obtain \eqref{e32}.
On the other hand, if $\frac{\rho_1}{\rho_2} \neq \frac{k}{b}$ or $k\neq k_0$, 
the function $s_2(\xi)$ satisfies
\begin{equation}\label{e46'}
\begin{gathered}
s_2(\xi) \geq \frac{1}{18}\xi^4\quad \text{if }|\xi|\leq 1, \\
s_2(\xi) \geq \frac{1}{18}\xi^{-6}\quad\text{if } |\xi|\geq 1
\end{gathered}
\end{equation}
Thus, we estimate $I_1$ as follows,
\begin{align*}
I_1 
&\leq C \|\hat{V_j^0}_0\|_{L^\infty}^2 
\int_{|\xi|\leq 1}|\xi|^{2k}e^{-\frac{\beta}{18}\xi^4t} d\xi \\
&\leq C_1\|\hat{V_j^0}\|_{L^\infty}^2( 1+t)^{-\frac{1}{4}(1+2k)} \\
&\leq C_1( 1+t)^{-\frac{1}{4}(1+2k)}\|V_j^0\|_{L^1}^2
\end{align*}
Moreover, by using the second inequality in \eqref{e46'}, it follows that
\begin{align*}
I_2 
&\leq C\int_{|\xi|\geq 1}|\xi|^{2k}e^{-\frac{\beta}{18}\xi^{-6}t}
 \hat{V_j^0}^2(\xi)d\xi  \\
&\leq C \sup_{|\xi| \geq 1}\{ |\xi |^{-2l}e^{-\frac{\beta}{18} \xi ^{-6}t} \} 
 \int_{\mathbb{R}}|\xi|^{2(k+l)}\hat{V_j^0}^2(\xi)d\xi   \\
&\leq C_2 (1+t)^{-\frac{l}{3}} \|\partial_x^{k+l}V_j^0\|_2^2
\end{align*}
Combining the estimates of $I_1$ and $I_2$, we obtain \eqref{e32'}.
\end{proof}

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\end{thebibliography}


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