\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 173, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/173\hfil Compactness of the difference \dots ] {Compactness of the difference between the porous thermoelastic semigroup and its decoupled semigroup} \author[E. M. Ait Benhassi, J. E. Benyaich, H. Bouslous, L. Maniar \hfil EJDE-2015/173\hfilneg] {El Mustapha Ait Benhassi, Jamal Eddine Benyaich, \\ Hammadi Bouslous, Lahcen Maniar } \address{ Universit\'e Cadi Ayyad, Facult\'e des Sciences Semlalia, LMDP, UMMISCO (IRD- UPMC), Marrakech 40000, B.P. 2390, Maroc. Fax. 00212524437409} \email{m.benhassi@uca.ma} \email{jbenyaich@gmail.com} \email{bouslous@uca.ma} \email{maniar@uca.ma} \thanks{Submitted October 14, 2014. Published June 22, 2015.} \subjclass[2010]{34G10, 47D06} \keywords{Porous thermoelastic system; semigroup; compactness; \hfill\break\indent norm continuity; fractional powers; essential spectrum} \begin{abstract} Under suitable assumptions, we prove the compactness of the difference between the porous thermoelastic semigroup and its decoupled one. This will be achieved by proving the norm continuity of this difference and the compactness of the difference between the resolvents of their generators. Applications to porous thermoelastic systems are given. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} \label{intro} An increasing interest to determine the decay behavior of solutions of several porous elastic and thermoelastic problems has been discovered recently. The theory of porous elastic material was established first by Cowin and Nunziato \cite{cowin1, cowin0, cowin2}. In a recent paper the authors of \cite {quintanilla} proved a slow decay of solution of porous elastic system with boundary Dirichlet conditions in one dimensional case. After, Casas and Quintanilla \cite{casas}, proved the exponential decay of a porous thermoelastic system. This problem has recently been the focus of interest of Glowinsky and Lada \cite{glowinski1, glowinski2, glowinski3}. In this work, we consider the abstract porous thermoelastic model \begin{gather} \label{damped1} \ddot w_1(t) + A_1 w_1(t) + C_1 w_2(t) +C_2\theta (t)= 0, \quad t \geq 0, \\ \label{damped2} \ddot w_2(t) + A_2 w_2(t) -C_1^* w_1(t) -C_3 \theta (t)+DD^* \dot w_2(t) = 0,\quad t \geq 0, \\ \label{damped3} \dot \theta(t) + A_3\theta (t) - C_2^* \dot w_1(t)+C_3^* \dot w_2(t) = 0,\quad t \geq 0, \\ \label{output} w_1(0) = w_1^0, \quad \dot w_1(0)=w_1^1, \quad w_2(0) = w_2^0, \quad \dot w_2(0)=w_2^1, \quad \theta (0) = \theta^0, \end{gather} with its decoupled system \begin{gather} \label{cdamped1} \begin{aligned} &\ddot w_1(t) + A_1 w_1(t) + C_1 w_2(t)+ C_2 A^{-1}_3C_2^* \dot w_1(t) -C_2 A^{-1}_3C_3^* \dot w_2(t)\\ & = 0, \quad t \geq 0, \end{aligned} \\ \label{cdamped2} \begin{aligned} &\ddot w_2(t) + A_2 w_2(t) -C_1^* w_1(t)-C_3A^{-1}_3C_2^* \dot w_1(t) +( C_3A^{-1}_3C_3^* +DD^*) \dot w_2(t)\\ & = 0,\quad t \geq 0, \end{aligned} \\ \label{cdamped3} \dot \theta(t) =- A_3\theta (t) + C_2^* \dot w_1(t)-C_3^* \dot w_2(t), \quad t \geq 0, \\ \label{coutput} w_1(0) = w_1^0, \quad \dot w_1(0)=w_1^1, \quad w_2(0) = w_2^0, \quad \dot w_2(0)=w_2^1, \quad \theta (0) = \theta ^0. \end{gather} The corresponding porous elastic system is given by the first and second equations in the decoupled system \eqref{cdamped1}-\eqref{coutput}, \begin{gather} \label{tdamped1} \begin{aligned} &\ddot w_1(t) + A_1 w_1(t) + C_1 w_2(t)+ C_2 A^{-1}_3C_2^* \dot w_1(t) -C_2 A^{-1}_3C_3^* \dot w_2(t)\\ &= 0, \quad t \geq 0, \end{aligned}\\ \label{tdamped2} \begin{aligned} &\ddot w_2(t) + A_2 w_2(t) -C_1^* w_1(t)-C_3A^{-1}_3C_2^* \dot w_1(t) +(C_3A^{-1}_3C_3^* +DD^*) \dot w_2(t) \\ &= 0, \quad t \geq 0, \end{aligned} \\ \label{toutput} w_1(0) = w_1^0, \quad \dot w_1(0)=w_1^1, \quad w_2(0) = w_2^0, \quad \dot w_2(0)=w_2^1. \end{gather} In this article, we first show the existence of solution of problems determined by systems \eqref{damped1}-\eqref{output}, \eqref{cdamped1}-\eqref{coutput} and \eqref{tdamped1}-\eqref{toutput} using the Lumer-Phillips theorem from the theory of semigroups \cite[Corollary 3.20]{nagel}]. Second we address the problem of compactness of difference between the porous-thermoelasticity $C_0$-semigroup $(\mathcal{T} (t))_{t\geq0}$ generated by the system \eqref{damped1}-\eqref{output} and the $C_0$-semigroup $(\mathcal{T}_d (t)) _{t\geq0}$ generated by its decoupled system \eqref{cdamped1}-\eqref{coutput}. As in \cite{Ait Ben Hassi}, we prove the norm continuity of $ t \longmapsto \mathcal{T} (t) - \mathcal{T}_d(t) $ for $t > 0$, and we show the compactness of the difference $ R(\lambda,\mathcal{A}) - R(\lambda,\mathcal{A}_d)$ for every $ \lambda$ in $\rho (\mathcal{A}) \cap \rho (\mathcal{A}_d)$, where $\mathcal{A}$ and $\mathcal{A}_d$ are the generators of $(\mathcal{T} (t))_{t\geq0}$ and $(\mathcal{T}_d (t)) _{t\geq0}$, respectively. These two results together with \cite[Theorem 2.3]{huang} lead to the compactness of the difference $\mathcal{T}(t) - \mathcal{T}_d(t)$. This yields that the essential spectrums $\sigma_e(\mathcal{T} (t))$, and $ \sigma_e(\mathcal{T}_d (t))$ coincide. In the case where the operators $A_3^{-1}$ and $A_1^{-1/2}C_1 A_2^{-1} $ are compact, following a similar argument as in \cite {henry}, we prove that $\sigma_e(\mathcal{S} (t)) = \sigma_e(\mathcal{T}_d (t))$, where $(\mathcal{S} (t)) _{t\geq0}$ is the $C_0$-semigroup generated by the system \eqref{tdamped1}-\eqref{toutput}. Consequently one can derive stability results on the first semigroup from the ones of the third semigroup. Finally two applications to a porous thermoelastic system are given. In the first application where $A_i^{-1}$, $i=1,2$ are compact but $A_3^{-1}$ is not compact, we show that only the two essential spectrums $\sigma_e(\mathcal{T} (t))$, and $ \sigma_e(\mathcal{T}_d (t))$ coincide. The second application is similar to the one given by Glowinsky and Lada in \cite {glowinski3}, where the exponential stability of porous thermoelastic system is derived from the corresponding decoupled system. In this application, following a different approach and using the compactness of $A_i^{-1}$, $i=1,2,3$, we obtain the same stability result first for the simpler porous elastic system, then the property is derived for the original porous thermoelastic system. \section{Main results} In what follows, $A_i :\mathcal{D}(A_i) \subset H_i \to H_i$, $i=1,2,3$, be self-adjoint positive operators with bounded inverses, and $H_i$ be Hilbert spaces equipped with the norm $\|\cdot\|_{H_i}$, $ i=1,2,3$. The operator $A_i$ can be extended (or restricted) to each $H_{i,\alpha}$, such that it becomes a bounded operator \begin{equation} \label{A0ext} A_i : H_{i,\alpha} \to H_{i,\alpha - 1}, \quad \forall \alpha\in\mathbb{R}, \end{equation} where for $\alpha\geq 0$, $H_{i,\alpha}=\mathcal{D}(A_i^{\alpha})$, with the norm $\|z \|_{i,\alpha}=\|A_i^\alpha z\|_{H_i}$ and for $\alpha \leq 0$, $H_{i,\alpha} =H_{i,-\alpha}^*$, the dual of $H_{i,-\alpha}$ with respect to the pivot space $H_i$. The operator $D\in \mathcal{L}(H_2)$ and $D^*$ its adjoint. The coupled operators $C_i$, i=1,2,3, satisfy \begin{itemize} \item[(C1)] $D(C_1) \subset H_{2} \to H_1$, with adjoint $C_1^*$ such that $ D(A_2^{1/2}) \hookrightarrow D(C_1)$ and $D(A_1^{1/2}) \hookrightarrow D(C_1^*)$. \item[(C2)] $D(C_2) \subset H_{3}\to H_1$ with adjoint $C_2^*$ such that $D(A_3^{1/2}) \hookrightarrow D(C_2)$ and $D(A_1^{1/2}) \hookrightarrow D(C_2^*)$. \item[(C3)] $D(C_3) \subset H_{3} \to H_2$ with adjoint $C_3^*$ such that \begin{equation} \label{coupcond} D(A_3^{1/2}) \hookrightarrow D(C_3)\quad\text{and}\quad D(A_2^{1/2}) \hookrightarrow D(C_3^*). \end{equation} \end{itemize} Set $$ \mathcal{H}:= H_{1,1/2} \times H_{2,1/2} \times H_1 \times H_2\times H_{3}, $$ in this Hilbert space we introduce the new inner product \begin{align*} \Big\langle \left( \begin{smallmatrix}w_1\\w_2\\v_1\\v_2 \\\theta \end{smallmatrix}\right), \left(\begin{smallmatrix}\widetilde{w}_1 \\ \widetilde{w_2} \\\widetilde{v}_1 \\\widetilde{v}_2\\\widetilde{\theta} \end{smallmatrix}\right)\Big\rangle &=\langle w_1,\widetilde{w}_1\rangle _{H_{1,1/2}} +\langle w_2,\widetilde{w_2} \rangle _{H_{2,1/2}} +\langle v_1,\widetilde{v}_1\rangle _{H_1} +\langle v_2,\widetilde{v}_2\rangle _{H_2}\\ &\quad +\langle \theta,\widetilde{\theta}\rangle _{H_3} +\Re (\langle C_1^* w_1,\widetilde{w}_2\rangle _{H_{2}} - \langle w_2,C_1^* \widetilde{w}_1\rangle _{H_{2}}). \end{align*} The associated norm of this inner product coincides with the canonical norm of $\mathcal{H}$. We can rewrite \eqref{damped1}-\eqref{output} and \eqref{cdamped1}-\eqref{coutput} as the first order evolution equations in $\mathcal{H}$, \begin{gather*} \frac{d\eta}{dt}=\mathcal{A} \eta,\quad \eta \in \mathcal{H} ,\\ \eta (0)=(w_1^0,w_2^0,w_1^1,w_2^1,\theta ^0), \end{gather*} and \begin{gather*} \frac{d \overline{\eta}}{dt}=\mathcal{A}_d \overline{\eta}, \quad \overline{\eta} \in \mathcal{H},\\ \overline{\eta} (0)=(w_1^0,w_2^0,w_1^1,w_2^1,\theta ^0), \end{gather*} respectively, where $\mathcal{A}$ is the unbounded linear operator defined by \begin{equation} \label{opn} \mathcal{A} : \mathcal{D}(\mathcal{A}) \subset \mathcal{H} \to \mathcal{H}, \quad \mathcal{A} = \left(\begin{smallmatrix} 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & I & 0 \\ - A_1 & -C_1 & 0 & 0 & -C_2 \\ C_1^* & -A_2 & 0 & -DD^* & C_3 \\ 0 & 0 & C_2^* &-C_3^* &-A_3\\ \end{smallmatrix}\right), \end{equation} with \begin{equation} \label{dom opn} \mathcal{D}(\mathcal{A}) =\mathcal{D}(A_1)\times \mathcal{D}( A_2)\times \mathcal{D}( A_1^{1/2 }) \times \mathcal{D}( A_2^{1/2})\times \mathcal{D}( A_3 ), \end{equation} and the operator $\mathcal{A}_d$ associated to the decoupled system \begin{equation} \label{opnd} \begin{aligned} \mathcal{A}_d: \mathcal{D}(\mathcal{A}_d) &=\mathcal{D}(\mathcal{A}) \subset \mathcal{H} \to \mathcal{H}, \, \mathcal{A}_d \\ &= \left(\begin{smallmatrix} 0 & 0 & I & 0 & 0 \\ 0 & 0 & 0 & I & 0 \\ - A_1 & -C_1 & -C_2 A^{-1}_3C_2^* & C_2 A^{-1}_3C_3^* & 0 \\ C_1^* & -A_2 & C_3 A^{-1}_3C_2^*& -C_3A^{-1}_3C_3^* -DD^* & 0 \\ 0 & 0 & C_2^* &-C_3^* &-A_3\\ \end{smallmatrix}\right). \end{aligned} \end{equation} We rewrite the coupled second order system \eqref{tdamped1}-\eqref{toutput} on the Hilbert space $$ \mathcal{H}_c:= H_{1,1/2} \times H_{2,1/2} \times H_1 \times H_2, $$ as the first order evolution equation \begin{gather*} \frac{d\widetilde{\eta}}{dt}=\mathcal{M} \widetilde{\eta},\quad \widetilde{\eta} \in \mathcal{H}_c ,\\ \widetilde{\eta} ^0=(w_1^0,w_2^0,w_1^1,w_2^1), \end{gather*} and $\mathcal{M} : \mathcal{D}(\mathcal{M}) \subset \mathcal{H}_c \to \mathcal{H}_c$, is the unbounded linear operator defined by \begin{equation} \label{opn1} \mathcal{M} = \left(\begin{smallmatrix} 0 & 0 & I & 0 \\ 0 & 0 & 0 & I \\ - A_1 & -C_1 & -C_2 A^{-1}_3C_2^* & C_2 A^{-1}_3C_3^* \\ C_1^* & -A_2 & C_3 A^{-1}_3C_2^*& -C_3A^{-1}_3C_3^* -DD^* \\ \end{smallmatrix}\right), \end{equation} with \begin{equation} \label{dom opn1} \mathcal{D}(\mathcal{M}) =\mathcal{D}(A_1)\times \mathcal{D}( A_2)\times \mathcal{D}( A_1^{1/2 }) \times \mathcal{D}( A_2^{1/2}). \end{equation} Now we formulate the main results of this paper. \begin{theorem}\label{generation0} The operators $\mathcal{A}$, $\mathcal{A}_d$ and $\mathcal{M}$ generate strongly continuous contraction semigroups $(\mathcal{T}(t))_{t\geq0}$, $(\mathcal{T}_d(t))_{t\geq0}$ on $\mathcal{H}$ and $(\mathcal{S}(t))_{t\geq0}$ on $\mathcal{H}_c$. \end{theorem} \begin{theorem} \label{obscoup3} Assume that \begin{equation} \label{assumption1} A_1^{-1/2}C_2A_3^{-1},\quad \,\,A_1^{-1/2}C_1A_2^{-1}, \quad A_2^{-1/2}C_3 A_3^{-1}, \end{equation} are compact operators from $H_3$ to $H_1$, from $H_2$ to $H_1$ and from $H_3$ to $H_2$ respectively. Then $\mathcal{T}(t)- \mathcal{T}_d(t)$ is compact for every $t\geq 0$. \end{theorem} As a consequence of Theorem \ref{obscoup3}, we have the following particular results. \begin{corollary} \label{obscoup} Assume that the operators $A_i^{-1}$, $i=1,2$, are compact. Then $\mathcal{T}(t)-\mathcal{T}_d(t)$ is compact for every $t\geq 0$. \end{corollary} \begin{corollary}\label{obscoup2} Assume that the operators $A_3^{-1}$ and $A_1^{-1/2}C_1 A_2^{-1} $ are compact. Then $\sigma_e(\mathcal{T}(t))=\sigma_e(\mathcal{S}(t))$ for $t \geq 0$. \end{corollary} \section{Well-posedness results} In this section we use Lumer-Phillips theorem (see \cite[Corollary 3.20]{nagel}) for the proof of Theorem \ref{generation0}. \subsection{Porous thermoelastic system} To show that the operator $(\mathcal{A}, \mathcal{D}(\mathcal{A}))$ defined by \eqref{opn}-\eqref{dom opn} generates a contraction semigroup on the Hilbert $\mathcal{H}$, we need the following technical lemma. \begin{lemma}\label{lemfond1} The operator $\mathcal{A}$ is invertible in $\mathcal{H}$ and $\mathcal{A}^{-1}$ is bounded on $\mathcal{H}$. \end{lemma} \begin{proof} Given a vector $\left(\begin{smallmatrix}f_1\\ f_2\\ f_3\\ f_4\\ f_5 \end{smallmatrix}\right) \in\mathcal{H}$, we need $\left(\begin{smallmatrix}w_1\\w_2\\v_1\\v_2\\w_3 \end{smallmatrix}\right) \in\mathcal{D}(\mathcal{A})$, such that \begin{align*} \mathcal{A} \left(\begin{smallmatrix}w_1\\w_2\\v_1\\v_2\\w_3 \end{smallmatrix}\right) =\left(\begin{smallmatrix}f_1\\ f_2\\ f_3\\ f_4\\ f_5 \end{smallmatrix}\right). \end{align*} We have \[ \mathcal{A} \left(\begin{smallmatrix}w_1\\w_2\\v_1\\v_2\\w_3 \end{smallmatrix}\right) =\left(\begin{smallmatrix}f_1\\ f_2\\ f_3\\ f_4\\ f_5 \end{smallmatrix}\right) \Leftrightarrow \begin{cases} v_1=f_1, \\ v_2=f_2, \\ A_1w_1 + C_1 w_2 +C_2 w_3=-f_3, \\ - C_1^* w_1 +A_2w_2 +DD^* v_2- C_3 w_3=-f_4,\\ -C_2^* v_1 +C_3^* v_2+ A_3w_3= -f_5. \end{cases} \] Hence \begin{align*} & \mathcal{A} \left(\begin{smallmatrix}w_1\\w_2\\v_1\\v_2\\w_3 \end{smallmatrix}\right) =\left(\begin{smallmatrix}f_1\\ f_2\\ f_3\\ f_4\\ f_5 \end{smallmatrix}\right) \Leftrightarrow \begin{cases} v_1=f_1, \\ v_2=f_2, \\ A_1w_1 + C_1 w_2 +C_2 w_3= -f_3, \\ -C_1^* w_1 + A_2w_2 - C_3 w_3=-f_4-DD^* f_2, \\ w_3= -A^{-1}_3(f_5-C_2^* f_1 +C_3^*f_2), \\ \end{cases}\\ & \Leftrightarrow \begin{cases} v_1=f_1, \\ v_2=f_2, \\ A_1w_1 + C_1 w_2 =C_2A^{-1}_3(f_5-C_2^* f_1 +C_3^*f_2) -f_3=K_1, \\ -C_1^* w_1 +A_2w_2 = -C_3A^{-1}_3(f_5-C_2^* f_1 +C_3^*f_2) -f_4-DD^* f_2=K_2,\\ w_3= -A^{-1}_3(f_5-C_2^* f_1 +C_3^*f_2), \end{cases}\\ &\Leftrightarrow \begin{cases} v_1=f_1, \\ v_2=f_2, \\ w_1=- A_1^{-1} C_1 w_2+A_1^{-1}K_1, \\ (C_1^*A_1^{- 1} C_1+A_2) w_2=K_2+C_1^*A_1^{-1}K_1 ,\\ w_3=- A^{-1}_3(f_5-C_2^* f_1 +C_3^*f_2). \end{cases} \end{align*} We have $$ v_1=f_1 \in H_{1,1/2}, \quad v_2=f_2\in H_{2,1/2},\quad w_3=- A^{-1}_3(f_5-C_2^* f_1 +C_3^*f_2) \in \mathcal{D}(A_3 ). $$ Suppose that we have found $w_2$ with the appropriate regularity. Then, $$ w_1=- A_1^{-1} C_1 w_2+A_1^{-1}K_1 \in \mathcal{D}(A_1 ). $$ We now solve the equation \begin{equation} \label{eq w_2} (C_1^*A_1^{- 1} C_1+A_2) w_2=K_2+C_1^*A_1^{-1}K_1. \end{equation} To find $w_2$ we introduce a bilinear form $\Lambda$ on $\mathcal{D}(A_2^{1/2})$, defined by $$ \Lambda(\eta,\zeta)= \langle A_1^{-1/2}C_1\eta,A_1^{-1/2}C_1\zeta\rangle + \langle A_2^{\frac{1}{2}}\eta,A_2^{\frac{1}{2}}\zeta\rangle . $$ Since $\Lambda$ is a bilinear continuous and coercive form on $\mathcal{D}(A_2^{1/2})$, the Lax-Milgram Lemma leads to the existence and uniqueness of $w_2\in \mathcal{D}(A_2^{1/2})$ solution to the equation \eqref{eq w_2}. Moreover $K_2+C_1^*A_1^{-1}K_1- C_1^*A_1^{- 1} C_1w_2 \in H_2$ and $ [(A_2)_{-1}]^{-1}H_2 =\mathcal{D}(A_2) $, (where $(A_2)_{-1}$ is an extension of $A_2$), then $w_2\in \mathcal{D}(A_2)$, (see \cite[Proposision 5]{Maniar}). Set $B_1=(C_1^*A_1^{- 1} C_1+A_2)^{-1}$, then we have \begin{align*} &\mathcal{A} \left(\begin{smallmatrix}w_1\\w_2\\v_1\\v_2\\w_3 \end{smallmatrix}\right) =\left(\begin{smallmatrix}f_1\\ f_2\\ f_3\\ f_4\\ f_5 \end{smallmatrix}\right) \Leftrightarrow \begin{cases} v_1=f_1, \\ v_2=f_2, \\ w_1=-A_1^{- 1} C_1 w_2+A_1^{-1}K_1, \\ w_2=B_1K_2 +B_1C_1^*A_1^{-1}K_1 , \\ w_3= -A^{-1}_3(f_5-C_2^* f_1 +C_3^*f_2), \end{cases} \end{align*} \begin{align*} &\Leftrightarrow \begin{cases} v_1=f_1, \\ v_2=f_2, \\ w_1=(-A_1^{- 1} C_1B_1C_3A^{-1}_3C_2^* +A_1^{- 1} C_1B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_2^* \\ \qquad -A_1^{-1}C_2A_3^{-1}C_2^*)f_1 +(A_1^{- 1} C_1 B_1C_3A^{-1}_3C_3^*+A_1^{- 1} C_1 B_1DD^*\\ \qquad -A_1^{- 1} C_1B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_3^* +A_1^{-1}C_2A_3^{-1}C_3^*)f_2\\ \qquad +(A_1^{- 1} C_1B_1C_1^*A_1^{-1}-A_1^{-1})f_3+A_1^{- 1}C_1B_1 f_4\\ \qquad +(A_1^{- 1} C_1 B_1C_3A^{-1}_3 -A_1^{- 1} C_1 B_1C_1^*A_1^{-1}C_2A^{-1}_3+A_1^{-1}C_2A_3^{-1})f_5, \\ w_2=(B_1C_3A^{-1}_3C_2^* -B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_2^* )f_1 \\ \qquad +(- B_1C_3A^{-1}_3C_3^*- B_1DD^*+B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_3^*)f_2\\ \qquad -B_1C_1^*A_1^{-1}f_3-B_1f_4+(- B_1C_3A^{-1}_3 + B_1C_1^*A_1^{-1}C_2A^{-1}_3)f_5 , \\ w_3= -A^{-1}_3(f_5-C_2^* f_1 +C_3^*f_2). \end{cases} \end{align*} Thus, \begin{equation} \label{opn 1} \mathcal{A}^{-1} = \left(\begin{smallmatrix} a_{11} & a_{12} & a_{13} & A_1^{- 1} C_1B_1 & a_{15} \\ a_{21} & a_{22} & -B_1C_1^*A_1^{-1} & -B_1 &a_{25} \\ I&0&0&0&0 \\ 0 & I & 0 & 0 & 0 \\ A^{-1}_3 C_2^*& -A^{-1}_3C_3^* & 0&0 &-A^{-1}_3\\ \end{smallmatrix} \right), \end{equation} where \begin{gather*} a_{11}=-A_1^{- 1} C_1B_1C_3A^{-1}_3C_2^* +A_1^{- 1} C_1B_1C_1^*A_1^{-1}C_2 A_3^{-1} C_2^* -A_1^{-1}C_2A_3^{-1}C_2^*, \\ \begin{aligned} a_{12}&=A_1^{- 1} C_1 B_1C_3A^{-1}_3C_3^*+A_1^{- 1} C_1 B_1DD^* -A_1^{- 1} C_1B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_3^*\\ &\quad +A_1^{-1}C_2A_3^{-1}C_3^*, \end{aligned} \\ a_{13}= A_1^{- 1} C_1B_1C_1^*A_1^{-1}-A_1^{-1}, \\ a_{15}= A_1^{- 1} C_1 B_1C_3A^{-1}_3 -A_1^{- 1} C_1 B_1C_1^*A_1^{-1}C_2A^{-1}_3 +A_1^{-1}C_2A_3^{-1}, \\ a_{21}=B_1C_3A^{-1}_3C_2^* -B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_2^* ,\\ a_{22}=- B_1C_3A^{-1}_3C_3^*-B_1DD^*+B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_3^*,\\ a_{25}=- B_1C_3A^{-1}_3 + B_1C_1^*A_1^{-1}C_2A^{-1}_3. \end{gather*} The boundedness of the operator $\mathcal{A}^{-1}$ follows by the assumptions \eqref{coupcond}. \end{proof} Now, to prove that the operator $\mathcal{A}$ generates a strongly continuous contraction semigroup $(\mathcal{T}(t))_{t\geq0}$ on $\mathcal{H}$, we have only to show that $(\mathcal{A},\mathcal{D}(\mathcal{A}))$ is a dissipative operator on $\mathcal{H}$ and $\lambda I-\mathcal{A}$ is surjective for some $\lambda >0$. For every $\left(\begin{smallmatrix}w_1\\w_2\\v_1\\v_2\\w_3 \end{smallmatrix}\right)\in \mathcal{D}(\mathcal{A})$, by the Cauchy-Schwartz inequality, we have \begin{align*} &\Re\bigg(\bigg\langle \mathcal{A} \left(\begin{smallmatrix}w_1\\w_2\\v_1\\v_2\\w_3\end{smallmatrix}\right), \left(\begin{smallmatrix}w_1\\w_2\\v_1\\v_2\\w_3 \end{smallmatrix}\right) \bigg\rangle \bigg) = \Re\bigg(\bigg\langle \left(\begin{smallmatrix} v_1 \\ v_2\\ - A_1w_1 - C_1 w_2 -C_2 w_3 \\ C_1^* w_1 -A_2w_2 -DD^* v_2+ C_3 w_3\\ C_2^* v_1 -C_3^* v_2- A_3w_3 \end{smallmatrix}\right), \left(\begin{smallmatrix}w_1\\w_2\\v_1\\v_2\\w_3 \end{smallmatrix}\right) \bigg\rangle \bigg)\\ &=\Re \Big(\langle v_1,w_1\rangle_{H_{1,1/2}} +\langle v_2,w_2\rangle_{H_{2,1/2}}-\langle A_1 w_1,v_1\rangle_{H_{1}} -\langle C_1w_2,v_1\rangle_{H_{1}}\\ &\quad -\langle C_2w_3,v_1\rangle_{H_{1}} +\langle C_1^* w_1,v_2\rangle_{H_{2}} -\langle A_2 w_2,v_2\rangle_{H_{2}}-\langle DD^* v_2, v_2\rangle_{H_{2}} \\ &\quad +\langle C_3 w_3,v_2\rangle_{H_2} +\langle C_2^* v_1,w_3\rangle_{H_{3}} - \langle C_3^*v_2,w_3\rangle_{H_3} -\langle A_3 w_3,w_3\rangle_{H_{3}}\\ &\quad +\langle C_1^* v_1,w_2\rangle_{H_{2}}- 0$, follows as in Theorem \ref{generation0}. %\end{proof} \subsection{Porous elastic system} As above, we can compute the operator \begin{equation} \label{opn10} \mathcal{M}^{-1} = \left(\begin{matrix} b_{11} & b_{12} & b_{13} & A_1^{- 1} C_1B_1 \\ b_{21} & b_{22} & -B_1C_1^*A_1^{-1} & -B_1 \\ I&0&0&0 \\ 0 & I & 0 & 0 \\ \end{matrix}\right), \end{equation} where $B_1=(C_1^*A_1^{- 1} C_1+A_2)^{-1}$, and \begin{gather*} b_{11}=-A_1^{- 1} C_1B_1 C_3A^{-1}_3C_2^* +A_1^{- 1} C_1B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_2^* -A_1^{-1}C_2A_3^{-1}C_2^*, \\ \begin{aligned} b_{12}&=A_1^{- 1} C_1 B_1C_3A^{-1}_3C_3^*+A_1^{- 1} C_1 B_1DD^*-A_1^{- 1} C_1B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_3^*\\ &\quad +A_1^{-1}C_2A_3^{-1}C_3^*, \end{aligned} \\ b_{13}= A_1^{- 1} C_1B_1C_1^*A_1^{-1}-A_1^{-1}, \\ b_{21}=B_1C_3A^{-1}_3C_2^* -B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_2^* ,\\ b_{22}= -B_1C_3A^{-1}_3C_3^*- B_1DD^*+B_1C_1^*A_1^{-1}C_2 A_3^{-1}C_3^*, \end{gather*} and show that the operator $\mathcal{M} $ generates a strongly continuous contraction semigroup $(\mathcal{S}(t))_{t\geq0}$ on $\mathcal{H}_c$. \section{Compactness result} In this section we prove the compactness of the difference $\mathcal{T}(t) - \mathcal{T}_d(t)$, we use \cite[Theorem 2.3]{huang}, where it is sufficient to prove the norm continuity of the difference between the two semigroups, and the compactness of the difference between the resolvents of their generators. To show the first assertion, we need the following technical lemma, see \cite[Theorem 1.4.3]{lunardi} . \begin{lemma}\label{tec lem} The map $t\mapsto A_3^\alpha e^{-A_3t}$ is norm continuous on $(0,\infty ) $ for all $ \alpha \geq 0$. \end{lemma} Now we can show the following norm continuity result. \begin{theorem} The map $t\mapsto \mathcal{T}(t)-\mathcal{T}_d(t)$ is norm continuous on $(0,\infty)$. \end{theorem} \begin{proof} Let $t> 0$ and $x_0=\left(\begin{smallmatrix} w_1^0 \\ w_2^0 \\ w_1^1\\ w_2^1\\ w_3^0\\ \end{smallmatrix}\right) \in \mathcal{D}(\mathcal{A})$ such that $\| x_0 \| \leq 1$. Let us write \[ \mathcal{T}(t)x_0-\mathcal{T}_d(t)x_0 =\left( \begin{smallmatrix} w_1(t)-\overline{w}_1(t) \\ w_2(t)-\overline{w}_2(t) \\ v_1(t)-\overline{v}_1(t)\\ v_2(t)-\overline{v}_2(t) \\ w_3(t)-\overline{w}_3(t) \\ \end{smallmatrix}\right) =\int _0^t \mathcal{T}(t-s) \left( \begin{smallmatrix} 0 \\ 0 \\ f(s) \\ g(s)\\ 0 \\ \end{smallmatrix} \right)\, ds, \] where \begin{gather*} f(s)=C_2A_3^{-1}C_2^*\overline{v}_1(s)- C_2A_3^{-1 }C_3^* \overline{v}_2(s)-C_2 \overline{w}_3(s),\\ g(s) =-C_3A_3^{-1}C_2^*\overline{v}_1(s)+ C_3 A_3^{-1 }C_3^* \overline{v}_2(s)+C_3 \overline{w}_3(s). \end{gather*} Let $0 0$, $$ \left \{\mathcal{T}_d(t)(w_1^0,w_2^0,w_1^1,w_2^1,w_3^0) -(\mathcal{S}(t)(w_1^0,w_2^0,w_1^1,w_2^1);0): \|(w_1^0,w_2^0,w_1^1,w_2^1,w_3^0)\| \leq 1 \right\} $$ is a compact set in $\mathcal{H}$, i.e. that \begin{align*} \Big\{& e^{-A_3 t} w_3^0 +\int _0^t e^{-A_3(t-\sigma) } C_2^* \overline{v}_1 (\sigma) d \sigma - \int _0^t e^{-A_3(s-\sigma)}C_3^* \overline{v}_2(\sigma) d \sigma : \\ &\| (w_1^0,w_2^0,w_1^1,w_2^1,w_3^0)\| \leq 1 \Big\} \end{align*} is a compact set in $H_3$, where $(\overline{w}_1(\sigma),\overline{w}_2(\sigma),\overline{v}_1(\sigma), \overline{v}_2(\sigma))= \mathcal{S}(\sigma)(w_1^0,w_2^0,w_1^1,w_2^1)$. Since \begin{align*} &(w_1^0,w_2^0,w_1^1,w_2^1,w_3^0)\\ &\to A_3^{1/2}e^{-A_3 t} w_3^0 +\int _0^t A_3 e^{-A_3(t-\sigma) } (C_2 A_3^{-1/2})^* \overline{v}_1 (\sigma) d \sigma \\ &\quad - \int _0^t A_3e^{-A_3(s-\sigma)}(C_3 A_3^{-1/2})^* \overline{v}_2(\sigma) d\sigma \end{align*} is bounded with values in $H_3$ (we have used the Lemma \ref{tec lem} and Lebesgue's theorem) and $A_3^{-1/2}$ is compact, the result follows. \end{proof} \begin{remark} \rm (1) If we have the conditions $C_3A_3^{-\gamma}$, $C_2A_3^{-\gamma}$ and $C_1A_2^{-\gamma}$ are compact for some $\gamma <1$ then the assumptions \eqref{assumption1} are satisfied and we have the compactness of the difference between $\,\mathcal{T}(t)- \mathcal{T}_d(t)$ for every $t\geq 0$, which is similar to Henry's condition in \cite{henry}. (2) If we suppose that $A_1^{-1}$ and $A_2^{-1}$ are compacts we have $\sigma_e(\mathcal{T}(t))=\sigma_e(\mathcal{T}_d(t))$ for $t \geq 0$ but to have $\sigma_e(\mathcal{T}_d(t))=\sigma_e(\mathcal{S}(t))$ for $t \geq 0$ we need the condition $A_3^{-1}$ is compact. \end{remark} \section{Applications} We give two illustrating examples of Theorem \ref{obscoup3} and Corollary \ref{obscoup2}. \subsection*{Application 1} We give one application of Theorem \ref{obscoup3}. Let $\Omega $ the bounded open Jelly Roll set proposed in \cite{simon}, $$ \Omega=\{(x,y)\in \mathbb{R}^2:\, \frac{1}{2}< r<1 \}\,\backslash \Gamma, $$ where $\Gamma$ is the curve in $\mathbb{R}^2$ given in polar coordinates by $$ r(\phi)=\frac{\frac{3\pi}{2}+\arctan (\phi)}{2\pi},\quad -\infty <\phi < \infty. $$ We consider the initial and boundary problem \begin{equation} \label{app1} \begin{gathered} \ddot u(t,x) - \Delta _e u(t,x) - b \nabla \phi(t,x) + c \nabla \theta (t,x)= 0 \quad \text{in } (0,+\infty) \times \Omega, \\ \ddot\phi(t,x) - (a\Delta - \alpha I) \phi(t,x) +b\operatorname{div}\, u(t,x) -d \theta (t,x)+ r \dot \phi(t,x) = 0 \\ \text{in } (0,+\infty) \times \Omega, \\ \dot \theta(t,x) - (\Delta - kI)\theta (t,x) + c\operatorname{div} \dot u(t,x)+d \dot \phi(t,x) = 0 \quad \text{in }(0,+\infty) \times \Omega, \\ u=0, \quad \phi=0, \quad \frac{\partial \theta}{\partial n}=0\quad \text{on } (0,+\infty) \times \partial \Omega, \\ u(0)=u^0, \quad \dot u(0)=u^1, \quad \phi(0) = \phi^0, \quad \dot \phi(0) = \phi^1, \quad \theta (0) = \theta^0, \quad \text{in } \Omega, \end{gathered} \end{equation} where $n$ denotes the outer uniter normal vector to $\partial \Omega$, $ \Delta _e:= \mu \Delta +(\mu+\lambda)\nabla \operatorname{div}$, and $\mu$, $\lambda$, $a$, $b$, $c$, $d$, $r$, $\alpha$, $k$ are positive constants. To fit this system into the abstract setting of \eqref{damped1}-\eqref{output}, we take \begin{gather*} H_1 = L^2(\Omega)^2,\quad H_2 =H_3=L^2(\Omega),\quad H_{1,\frac{1}{2}} = (H_0^1(\Omega))^2 ,\quad H_{2,\frac{1}{2}} = H_0^1(\Omega), \\ \mathcal{H}= \mathcal{H}_c \times L^2(\Omega), \quad \mathcal{H}_c=(H_0^1(\Omega))^2 \times H_0^1(\Omega)\times L^2(\Omega)^2 \times L^2(\Omega), \\ A_1 = - \Delta _e, \quad \mathcal{D}(A_1) =\mathcal{D}(-\Delta _D)= (H^2(\Omega) \cap H^1_0(\Omega))^2,\\ A_2 = - (a \Delta -\alpha I), \quad \mathcal{D}(A_2) = \mathcal{D}(-\Delta _D)=H^2(\Omega) \cap H^1_0(\Omega),\\ A_3= -( \Delta -kI), \quad \mathcal{D}(A_3) = \mathcal{D}(- \Delta _N). \end{gather*} We recall that $u, \phi,\theta$ are the displacement vector, the volume fraction and the temperature. The Dirichlet and Neumann Laplacian operators $\Delta_D$ and $\Delta_N$ are the unique positive self adjoint operators associated to the closed quadratic form on $H_0^1(\Omega)$ and $H^1(\Omega)$ respectively \[ \langle \Delta f,g\rangle =\int _\Omega \nabla f \nabla g dx. \] The operator $DD^*=rI_{H_2}$, and the coupled operators \begin{gather*} C_1 =- b \nabla, \quad C_2=c \nabla,\quad C_1^* = b \operatorname{div}, \quad C_2^*=-c \operatorname{div},\quad C_3=d\,I_{H_3}, \\ \mathcal{D}(C_1)=\mathcal{D}(C_2)= H^1(\Omega), \quad \mathcal{D}(C_2^*)=\mathcal{D}(C_1^*)=\{u\in H^1(\Omega)^2 : u\cdot\overrightarrow{n}=0 \,\,\text{in } \partial \Omega \}. \end{gather*} Note that the conditions \eqref{coupcond} are verified and we have $A_1^{-1}$ and $A_2^{-1}$ are compact from $H_1$ and $H_2$ respectively, then the assumptions \eqref{assumption1} are satisfied, consequently the Theorem \ref{obscoup3} is satisfied. To show $\sigma_e(\mathcal{T}_d(t))=\sigma_e(\mathcal{S}(t))$ for $t \geq 0$, we need the compactness of $A_3^{-1}$, but from \cite{simon}, $A_3^{-1}$ is not compact. \subsection*{Application 2} We give an application of Corollary \ref{obscoup2}. Let $\Omega \subset \mathbb{R}^2 $ a bounded open domain with boundary $\partial \Omega$ having regularity of class $C^2$, and satisfies the following condition: \begin{itemize} \item[(A1)] If $\varphi \in (H_0^1(\Omega))^2$ such that \begin{equation} \label{cond A1} \begin{gathered} -\Delta \varphi=\gamma^2 \varphi\quad\text{in } \Omega,\\ \operatorname{div}\varphi=0\quad\text{in } \Omega,\\ \varphi=0\quad \text{in } \partial \Omega. \end{gathered} \end{equation} for some $\gamma \in \mathbb{R}$, then $\varphi=0$. \end{itemize} We consider the initial and boundary problem \begin{equation} \label{app 2} \begin{gathered} \ddot u(t,x) - \Delta _e \,u(t,x) - b\nabla \phi(t,x) + c\nabla \theta (t,x)= 0 \quad \text{in } (0,+\infty) \times \Omega,\\ \ddot\phi(t,x) - (a\, \Delta - \alpha I) \phi (t,x) +b\operatorname{div} u(t,x) + r \dot \phi(t,x) = 0 \quad\text{in } (0,+\infty) \times \Omega,\\ \dot \theta(t,x) - \Delta \theta (t,x) + c\operatorname{div} \dot u(t,x) = 0\quad \text{in } (0,+\infty) \times \Omega,\\ u=0, \quad \phi=0, \quad \theta=0 \quad\text{on } (0,+\infty) \times \partial \Omega, \\ u(0)=u^0, \quad \dot u(0)=u^1, \quad \phi(0) = \phi^0, \quad \dot \phi(0) = \phi^1, \theta(0) = \theta^0 \quad \text{in } \Omega, \end{gathered} \end{equation} where $ \Delta _e:= \mu \Delta +(\mu+\lambda)\nabla \operatorname{div}$ is Lam\'{e} operator, $\mu$, $\lambda$, $a$, $b$, $c$, $r$, $\alpha$ are positive constants, and the condition $(\lambda + \mu)\alpha > b^2$ is satisfied. To fit this system into the abstract setting of \eqref{damped1}-\eqref{output}, we take \begin{gather*} H_1 = L^2(\Omega)^2, \quad H_2 =H_3=L^2(\Omega),\quad H_{1,\frac{1}{2}} = (H_0^1(\Omega))^2 , \quad H_{2,\frac{1}{2}} = H^1_0(\Omega),\\ \mathcal{H}= \mathcal{H}_c \times L^2(\Omega),\quad \text{where } \mathcal{H}_c=(H_0^1(\Omega))^2 \times H^1_0(\Omega)\times L^2(\Omega)^2 \times L^2(\Omega), \\ A_1 = - \Delta _e, \quad \quad\mathcal{D}(A_1) =\mathcal{D}(-\Delta _D)= (H^2(\Omega) \cap H^1_0(\Omega))^2,\\ A_2 = - (a \Delta -\alpha I), \quad \mathcal{D}(A_2) =H^2(\Omega) \cap H^1_0(\Omega),\\ A_3= - \Delta ,\quad \mathcal{D}(A_3) = \mathcal{D}( A_2 ). \end{gather*} The operator $DD^*=rI_{H_2}$, and the coupled operators \begin{gather*} C_1 =- b \nabla, \quad C_1^* = b \operatorname{div},\quad C_2=c \nabla,\quad C_2^*=-c \operatorname{div},\quad C_3=0, \\ \mathcal{D}(C_1)=\mathcal{D}(C_2)= H^1(\Omega), \quad \mathcal{D}(C_2^*)=\mathcal{D}(C_1^*)=\{u\in H^1(\Omega)^2 : u\cdot\overrightarrow{n}=0 \quad\text{on } \partial \Omega \}. \end{gather*} The decoupled system corresponding to system \eqref{app 2} is given by \begin{equation} \label{dapp 2} \begin{gathered} \ddot u(t,x) - \Delta _e \,u(t,x) - b\nabla \phi(t,x) + c^2P \dot u(t,x) = 0\quad \text{in } (0,+\infty) \times \Omega,\\ \ddot\phi(t,x) - (a \Delta - \alpha I) \phi (t,x) +b\operatorname{div} u(t,x) + r \dot \phi(t,x) = 0 \quad \text{in } (0,+\infty) \times \Omega, \\ \dot \theta(t,x) - \Delta \theta (t,x) + c\operatorname{div} \dot u(t,x) = 0 \quad\text{in } (0,+\infty) \times \Omega,\\ u=0, \quad \phi=0, \quad \theta=0 \quad\text{on } (0,+\infty) \times \partial \Omega, \\ u(0)=u^0, \quad \dot u(0)=u^1, \quad \phi(0) = \phi^0, \quad \dot \phi(0) = \phi^1, \theta(0) = \theta^0 \quad \text{in } \Omega, \end{gathered} \end{equation} where $P:=\nabla (\Delta)^{-1}div$ the orthogonal projection operator from $L^2(\Omega)^2$ into the subspace $\{\nabla\varphi ; \varphi\in H_0^1(\Omega)\}$. Now we write the porous elastic system given by the first and second equation in decoupled system \eqref{dapp 2} \begin{equation} \label{app 9} \begin{gathered} \ddot u(t,x) - \Delta _e \,u(t,x) - b\nabla \phi(t,x) + c^2 P\dot u(t,x) = 0 \quad \text{in } (0,+\infty) \times \Omega,\\ \ddot \phi(t,x) - (a\Delta - \alpha I) \phi(t,x) +b\operatorname{div} u(t,x) + r \dot \phi(t,x) = 0 \quad \text{in } (0,+\infty) \times \Omega,\\ u=0, \quad \phi=0 \quad \text{on } (0,+\infty) \times \partial \Omega, \\ u(0)=u^0, \quad \dot u(0)=u^1, \quad \phi(0) = \phi^0, \quad \dot \phi(0) = \phi^1 \quad \text{in } \Omega. \end{gathered} \end{equation} Let $(\mathcal{T} (t))_{t\geq0}$ the porous-thermoelastic $C_0$-semigroup generated by the system \eqref{app 2} and $ (\mathcal{S} (t))_{t\geq0}$ the porous elastic $C_0$-semigroup generated by the system \eqref{app 9}. Note that the operators $A_1^{-1}$, $A_2^{-1}$ and $A_3^{-1}$ are compact, consequently the assumptions of Corollary \ref{obscoup2} are satisfied, then $$ \sigma_e(\mathcal{T}(t))=\sigma_e(\mathcal{S}(t))\quad\text{for } t \geq 0. $$ The second aim of this application is to characterize the exponential energy decay of solution of system \eqref{app 9}, and then deduce the one of the coupled systems \eqref{app 2}. Now we show that $(\mathcal{S}(t))_{t\geq0}$ is exponentially stable in $\mathcal{H}_c$, by using a similar argument as in the proof of \cite[theorem 4.4]{glowinski3}. Let $\varepsilon(t):=\mathcal{S}(t)\varepsilon^0$, $t\geq 0$, be the solution of \begin{equation} \label{cau} \frac{d \varepsilon }{dt} = \mathcal{M} \varepsilon, \quad \varepsilon(0)=\varepsilon^0, \end{equation} where $\varepsilon^0:=(u^0,\phi^0,u^1,\phi^1)$. We look for $\varepsilon(t)$ having the form $\overline{\varepsilon}(t)=\Sigma _{l=0}^{\infty}b^l \varepsilon_l(t)$, where $\varepsilon_l(t)\equiv \big{(}u_l(t),\phi_l(t),\dot u_l(t),\dot \phi_l(t)\big{)}$, $\overline{\varepsilon}(0)=\varepsilon ^0 $ and $l \in \{0\}\cup \mathbb{N}$. After the formal substitution into the equation \eqref{cau} we derive equations for $(u_l,\phi_l)$, where $l \in \{0\}\cup \mathbb{N}$. For $(u_0,\phi_0)$ we obtain \begin{gather} \label{app10} \ddot u_0(t,x) - \Delta _e u_0(t,x) + c^2 P\dot u_0(t,x) = 0 \quad \text{in }(0,+\infty) \times \Omega,\\ \label{app11} \ddot \phi_0(t,x) - (a\Delta - \alpha I) \phi_0(t,x) + r \dot \phi_0(t,x) = 0 \quad\text{in } (0,+\infty) \times \Omega,\\ u_0=0, \quad \phi_0=0 \quad\text{on } (0,+\infty) \times \partial \Omega, \nonumber\\ u_0(0)=u^0, \quad \dot u_0(0)=u^1, \quad \phi_0(0) = \phi^0, \quad \dot \phi_0(0) = \phi^1 \quad\text{in } \Omega. \nonumber \end{gather} For $k \in \{0\}\cup \mathbb{N}$, $(u_{k+1},\phi_{k+1})$ will be the solution of problem \begin{gather} \label{app12} \ddot u_{k+1}(t,x) - \Delta _e \,u_{k+1}(t,x) - b\nabla \phi_k(t,x) + c^2 P\dot u_{k+1}= 0 \quad\text{in } (0,+\infty) \times \Omega,\\ \label{app13} \begin{aligned} &\ddot \phi_{k+1}(t,x) - (a\Delta - \alpha I) \phi_{k+1}(t,x) + b\operatorname{div} u_k(t,x) +r \dot \phi_{k+1}(t,x) \\ &= 0 \quad \text{in } (0,+\infty) \times \Omega, \end{aligned} \\ u_{k+1}=0, \quad \phi _{k+1}=0 \quad\text{on } (0,+\infty) \times \partial \Omega, \nonumber \\ u_{k+1}(0)=0,\quad \dot u_{k+1}(0)=0, \quad \phi_{k+1}(0) = 0, \quad \dot \phi_{k+1}(0) = 0 \quad\text{in }\Omega. \nonumber \end{gather} Let $\varepsilon = (u,\phi,v,\psi)\in \mathcal{H}_c$, and define the norms \begin{gather*} \|(u(t),v(t))\|_1^2:= \int _\Omega [\mu | \nabla u(x,t)|^2 + (\lambda+\mu)|\operatorname{div}u(x,t) |^2 + |v(x,t) |^2]dx, \\ \|(\phi(t),\psi(t))\|_2^2:= \int _\Omega [ a| \nabla \phi(x,t)|^2 + \alpha |\phi(x,t) |^2 + |\psi(x,t) |^2]dx,\\ \|\varepsilon\|^2:= \|(u(t),v(t))\|_1^2\,\,+\,\|(\phi(t),\psi(t))\|_2^2. \end{gather*} From \cite{liu2}, there exists $M_1,\gamma_1 >0$, such that \[ \|(\phi_0(t),\dot \phi_0(t))\|_2^2 \leq M_1e^{-\gamma_1 t} \|(\phi^0,\phi^1)\|_2^2, \quad t \geq 0. \] Note that the damped Lam\'{e} system \eqref{app10} has been studied by Zuazua and Lebeau in \cite {lebeauzuazua2} and they proved the exponential decay of solution of \eqref{app10} if the following inequality of observability holds true for some $T,C>0$, i.e, \begin{equation} \label{Iob} \| \varphi^0 \|_{(L^2(\Omega))^2} + \|\varphi ^1 \|_{({ H}^{-1}(\Omega))^2 } \leq C \int_0^T \| \operatorname{div} \varphi \|_{{ H}^{-1}(\Omega)}dt, \end{equation} where $\varphi(t)$ is solution of the Lam\'{e} system \begin{equation} \label{app14} \begin{gathered} \ddot \varphi(t,x) - \Delta _e \,\varphi(t,x) = 0 \quad\text{in } (0,+\infty) \times \Omega,\\ \varphi=0 \quad \text{on } (0,+\infty) \times \partial \Omega, \\ \varphi(0)=\varphi^0, \dot \varphi(0)=\varphi^1 \quad\text{in } \Omega. \end{gathered} \end{equation} Under the condition that \eqref{Iob} is satisfied, we have \[ \|(u_0(t),\dot u_0(t))\|_1^2 \leq M_2e^{-\gamma_2 t} \|(u^0,u^1)\|_1^2,\quad t \geq 0\,, \] for positive constants $M_2,\gamma_2$. Let $\gamma =inf (\gamma_1, \gamma_2)$, we have \begin{equation} \label{app14b} \|(u_0(t),\phi_0(t),\dot u_0(t),\dot \phi_0(t))\| \leq M e^{-\frac{\gamma }{2}t} \|(u^0,\phi^0,u^1,\phi^1)\|,\quad t \geq 0 . \end{equation} Let $(\mathcal{G}(t))_{t\geq0}$ and $(\mathcal{K}(t))_{t\geq0}$ be the contraction $C_0$-semigroups generated by the equations \eqref{app11} and \eqref{app10} respectively, where $(\phi_0(t),\dot \phi_0(t))= \mathcal{G}(t)(\phi^0, \phi^1)$, and $(u_0(t),\dot u_0(t))= \mathcal{K}(t)(u^0, u^1)$. For the solution of system \eqref{app13} we have $$ (\phi_{k+1}(t),\dot \phi_{k+1}(t)) = \int_0 ^t \mathcal{G}(t-s) (0,-\operatorname{div}u_k(s))ds. $$ Then $$ \|(\phi_{k+1}(t),\dot \phi_{k+1}(t))\|_2 \leq \int_0 ^t M_1e^{-\frac{\gamma_1 }{2}(t-s)} \|(0,-\operatorname{div}u_k(s))\|_2ds. $$ Since $\|(0,-\operatorname{div}u_k(s))\|_2 \leq C_1 \|(u_k(s),\dot u_k(s))\|_1$, we have $$ \|(\phi_{k+1}(t),\dot \phi_{k+1}(t))\|_2 \leq \int_0 ^t C_1 M_1e^{-\frac{\gamma_1 }{2}(t-s)} \|(u_k(s),\dot u_k(s))\|_1ds. $$ For the solution of system \eqref{app12} we have $$ (u_{k+1}(t),\dot u_{k+1}(t)) = \int_0 ^t \mathcal{K}(t-s)(0,b \nabla \phi_k(s))ds. $$ Then $$ \|(u_{k+1}(t),\dot u_{k+1}(t))\|_1 \leq \int_0 ^t M_2e^{-\frac{\gamma_2}{2} (t-s)} \|(0,b \nabla \phi_k(s))\|_1ds. $$ Since $\|(0,b \nabla \phi_k(s))\|_1 \leq C_2 b \|(\phi_k(s),\dot \phi_k(s))\|_2 $, we have $$ \|(u_{k+1}(t),\dot u_{k+1}(t))\|_1 \leq \int_0 ^t bC_2 M_2e^{-\frac{\gamma_2}{2}(t-s)} \|(\phi_k(s),\dot \phi_k(s))\|_2ds. $$ Then we have \begin{equation} \label{app15} \begin{aligned} &\|(u_{k+1}(t),\phi_{k+1}(t),\dot u_{k+1}(t),\dot \phi_{k+1}(t))\|\\ &\leq \int_0 ^t M_3 e^{-\frac{\gamma}{2} (t-s)} \|(u_k(s),\phi_{k}(s),\dot u_k(s),\dot \phi_{k}(s))\|ds. \end{aligned} \end{equation} From \eqref{app14b} and \eqref{app15} we deduce that $$ \|(u_{l}(t),\phi_{l}(t),\dot u_{l}(t),\dot \phi_{l}(t))\| \leq M M_3^l \frac{t^l}{l!} e^{-\frac{\gamma}{2}t} \|(u^0,\phi^0, u^1, \phi^1)\|. $$ Let $0< b <\frac{\gamma}{2M_3} $, the sequence $\Sigma _{l=0}^{\infty}b^l \varepsilon_l(t)$ is convergent in $C([0,\tau];\mathcal{H}_c)$ for every $\tau >0$. Let $ \overline{\varepsilon} ^n(t)=\Sigma _{l=0}^{n}b^l \varepsilon_l(t),\,\, n \in \mathbb{N} $ where $\overline{\varepsilon}^n(t)$ is the solution of the problem $$ \frac{d \overline{\varepsilon}^n(t)}{dt}=\mathcal{M} \overline{\varepsilon}^n(t)+\beta_n(t);\quad \overline{\varepsilon}^n(0) =\overline{\varepsilon}^0, $$ where $\beta_n(t):=\big{(}0, b^n \nabla \phi_n(t),0,-b^n \operatorname{div} u_n(t)\big{)}^T$. We have $$ \overline{\varepsilon}^n(t) = \mathcal{S}(t)\overline{\varepsilon}^0+ \int_0^t \mathcal{S}(t-s)\beta_n(s)ds, $$ and \[ \| \overline{\varepsilon}(t)- \mathcal{S}(t) \varepsilon ^0\| = \| \Sigma _{l=n+1}^{\infty}b^l \varepsilon_l(t) + \int_0^t \mathcal{S}(t-s)\beta_n(s)ds \|\to 0, \quad\text{as } n \to \infty, \; \forall n\in \mathbb{N}. \] This means that $ \mathcal{S}(t)\varepsilon ^0=\overline{\varepsilon}(t)$ and $$ \| \mathcal{S}(t)\varepsilon ^0\| \leq \Sigma _{l=0}^{\infty}b^l \| \varepsilon_l(t)\| \leq M \Sigma _{l=0}^{\infty}b^l M_3^l \frac{t^l}{l!} e^{-\frac{\gamma t}{2}} \|\varepsilon^0\| \leq M e^{-\varrho t}\|\varepsilon^0\|, $$ where $\varrho :=\frac{\gamma}{2}-M_3b$. Consequently $(\mathcal{S}(t))_{t\geq0}$ is exponentially stable and $w_e(\mathcal{M})<0$. Since $\sigma_e(\mathcal{T}(t))=\sigma_e(\mathcal{S}(t))$ for $t \geq 0$, then $$ w_e(\mathcal{A})<0. $$ Now we prove that $(\mathcal{T}(t))_{t\geq0}$ is exponentially stable in $\mathcal{H}$, i.e, $\|\mathcal{T}(t)\| \leq Me^{-\delta t}$, $ t \geq 0$, where $M,\delta >0$. From \cite[Theorem 2.9]{glowinski2} the semigroup $(\mathcal{T}(t))_{t\geq0}$ is asymptotically stable in $\mathcal{H}$ i.e, $ \lim _{t\to\infty} \| \mathcal{T}(t)x \|=0$, for every $x\in \mathcal{H}$. Then $s_1(\mathcal{A})\leq0$, where \[ s_1 (\mathcal{A})=\sup\{ \Re \lambda / \lambda\in \sigma (\mathcal{A}) \backslash \sigma _e(\mathcal{A})\}. \] To show that $w_0(\mathcal{A})< 0$, it suffices to prove that $s_1(\mathcal{A})<0$. Suppose that $s_1(\mathcal{A})= 0$, then there exists $ \{\lambda_n\}_1^{\infty} \subset \sigma (\mathcal{A}) \backslash \sigma _e(\mathcal{A})$, such that $\Re \lambda_n\to 0$, as $n \to \,\infty$. $e^{\lambda_n t_0}$ is an eigenvalue of $\mathcal{T}(t_0)$, we have $| e^{\lambda_n t_0}| \leq 1$ and $| e^{\lambda_n t_0}|\to 1$ as $n \to \infty$. Let $y$ be the accumulation point of $\{ e^{\lambda_n t_0}\}_1^{\infty}$ in $\mathbb{C}$. Then $y \in \sigma _e(\mathcal{T}(t_0))$ and $| y|=1$. Thus, \[ r_e(\mathcal{T}(t_0)) \geq 1, \] furthermore \[ r_e(\mathcal{T}(t_0))= e^{w_e(\mathcal{A})t_0}<1. \] This contradiction implies that $ s_1(\mathcal{A})<0$, using $ w_e(\mathcal{A})<0$, we obtain $ w_0(\mathcal{A})<0$. Finally we have proved the uniform stabilization of the energy of solution of system \eqref{app 2}. \begin{thebibliography}{99} \bibitem{Ait Ben Hassi} E. Ait Ben Hassi, H. Bouslous, L. 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