\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 136, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/136\hfil Stability of an $N$-component Timoshenko beam] {Stability of an $N$-component Timoshenko beam with localized Kelvin-Voigt and frictional dissipation} \author[T. K. Maryati, J. E. Mu\~noz Rivera, A. Rambaud, O. Vera \hfil EJDE-2018/136\hfilneg] {Tita K. Maryati, Jaime E. Mu\~noz Rivera, Amelie Rambaud, Octavio Vera} \address{Tita K. Maryati \newline Islamic State University (UIN) Syarif Hidayatullah Jakarta, Indonesia} \email{tita.khalis@uinjkt.ac.id} \address{Jaime E. Mu\~noz Rivera \newline Department of Mathematics, University of B\'io-B\'io, Concepci\'on, Chile} \email{jemunozrivera@gmail.com} \address{Amelie Rambaud \newline Department of Mathematics, University of B\'io-B\'io, Concepci\'on, Chile} \email{amelie.rambaud@yahoo.fr} \address{Octavio Vera \newline Department of Mathematics, University of B\'io-B\'io, Concepci\'on, Chile} \email{octaviovera49@gmail.com} \dedicatory{Communicated by Marco Squassina} \thanks{Submitted February 27, 2018. Published July 1, 2018.} \subjclass[2010]{35B40, 74K10, 35M33, 35Q74} \keywords{Timoshenko's model; beam equation; localized dissipation; \hfill\break\indent viscoelaticity; lack of exponential stability; exponential and polynomial stability} \begin{abstract} We consider the transmission problem of a Timoshenko's beam composed by $N$ components, each of them being either purely elastic, or a Kelvin-Voigt viscoelastic material, or an elastic material inserted with a frictional damping mechanism. Our main result is that the rate of decay depends on the position of each component. More precisely, we prove that the Timoshenko's model is exponentially stable if and only if all the elastic components are connected with one component with frictional damping. Otherwise, there is no exponential stability, but a polynomial decay of the energy as $1/t^2$. We introduce a new criterion to show the lack of exponential stability, Theorem \ref{new}. We also consider the semilinear problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} Here we study a transmission problem of a Timoshenko beam \cite{timo} of length $\ell$ composed by $N$ components, each of them can be of three different types of materials: elastic, viscoelastic, or a material with a frictional damping mechanism as illustrated in Figure \ref{fig1} below, for $N= 5$. \begin{figure}[ht] \begin{center} \setlength{\unitlength}{2.0pt} \begin{picture}(145,37)(-15,-3) \put(-30,-40){\includegraphics[width=14cm]{fig1}} % Ncomp.eps \put(3,4){$0$} \put(34,4){$\ell_1$} \put(47,4){$\ell_2$} \put(68,4){$\ell_3$} \put(94,4){$\ell_4$} \put(115,4){$\ell$} \put(16,22){$I_e$} \put(40,22){$I_v$} \put(58,22){$I_f$} \put(80,22){$I_e$} \put(104,22){$I_f$} \end{picture} \end{center} \caption{An example of five-components beam, where $I_e$ is elastic, $I_f$ is frictional, and $I_v$ is viscoelastic component} \label{fig1} \end{figure} Let us decompose the interval $I = [0, \ell]$ into $N$ subintervals, $ [0,\ell]=\cup_{i=1}^n \overline{I_i}$, such that $I_{i}= ]\ell_{i - 1},\ell_{i}[$ for $i=1,2,\dots, N$ with $\ell_0 = 0$ and $\ell_N = \ell$. Over each interval $I_i$, one type of material is configured. We denote by $I_v$, $I_e$ or $I_f$ the subintervala where the viscoelastic component, elastic component, or the component with frictional mechanism is configured, respectively. In Figure \ref{fig1} the intervals $I_1$ and $I_4$ are of type $I_e$, elastic components, $I_2$ is of viscoelastic type $I_v$, and so on. Let us denote the set $$ \widetilde{I}=\cup_{i=1}^n{I_i}=]0,\ell[\setminus \{\ell_0,\ell_1,\dots,\ell_N\}. $$ The set $\widetilde{I}$ is open and disconnected. The classical linear Timoshenko system given by \begin{gather}\label{eq1} \varrho_1\varphi_{tt}-S_{x} = G_1,\quad \text{in}\quad \widetilde{I}\times \mathbb{R}_{+}, \\ \varrho_2\psi_{tt} -M_{x} +S = G_2,\quad \text{in}\quad \widetilde{I}\times \mathbb{R}_{+},\label{eq2} \end{gather} Here we use the Dirichlet boundary conditions \begin{equation}\label{eq3} \varphi(0,t)=\varphi(\ell,t)=\psi(0,t)=\psi(\ell,t)=0. \end{equation} and the initial conditions \begin{equation}\label{ini} \varphi (x,\,0)=\varphi_{0} (x),\quad \psi (x,\,0)=\psi_{0} (x),\quad \varphi_{t} (x,\,0) = \varphi_1 (x),\quad \psi_{t} (x,\,0) = \psi_1 (x) . \end{equation} Here $S$ and $M$ stand for the shear force and the bending moment respectively, $\varrho_1=\varrho A$ and $\varrho_2=\varrho I^M$, where $\varrho$ is the density of the material, $A$ the cross-sectional area and $I^M$ the second moment of the cross-section area. By $\varphi$ we denote the transversal displacement and by $\psi$ the shear angle displacement. The constitutive equations are given by \begin{equation}\label{def-SM} S(\varphi_{x},\psi) = \kappa(x)\,(\varphi_{x} + \psi)+\kappa_0(x)\,(\varphi_{xt} + \psi_t),\quad M(\psi)=b(x)\psi_{x}+b_0(x)\psi_{xt}, \end{equation} where $\kappa=k'\,G\,A$ and $b=E\,I^M$ are positive functions over $\widetilde{I}$. By $E$, $G$ and $k'$ we are denoting the Young's modulus, the modulus of rigidity and the transverse shear factor, respectively. We denote by $b_0$ and $\kappa_0$, positive functions which characterize the viscosity over $I_v$, vanishing over $I_e\cup I_f$. The localized frictional damping mechanism is described by the source terms \begin{equation}\label{def-FF} G_1(x,t)=-\gamma_1(x)\varphi_t,\quad G_2(x,t)=-\gamma_2(x)\psi_t, \end{equation} where $\gamma_1, \gamma_2$ are positive only on the intervals $I_f$, vanishing over $I_v$ and $I_e$. Therefore the elastic coefficients are discontinuous at the points where different materials are fitted. This characterizes the transmission problem. Hence the functions $\kappa$, $\kappa_0$, $b$, $b_0$, $\gamma_1$, $\gamma_2 :[0,\ell]\to\mathbb{R}$ are such that its restrictions to $I_i$, $i = 1, \dots , N$, are $C^1$ functions, with bounded discontinuities at the nodes $\ell_i$, $i=1,\dots, N-1$; but even so, the stress as well as the bending moment must satisfy the laws of action and reaction at each point, therefore we have that any strong solutions of the problem must verify \begin{equation}\label{eq4} \varphi, \psi, S, M \in H^1(0,\ell). \end{equation} In particular \eqref{eq4} implies the transmission conditions at the interface points $\ell_i$: \begin{equation}\label{eq6} \varphi(\ell_i^{-})=\varphi(\ell_i^{+}),\quad S(\ell_i^{-})=S(\ell_i^{+}),\quad \psi(\ell_i^{-})=\psi(\ell_i^{+}),\quad M(\ell_i^{-})=M(\ell_i^{+}), \end{equation} for $i = 1, \dots N-1$. A typical example of a function $y=\kappa_0(x)$ is given in Figure \ref{fig1}: \begin{center} \setlength{\unitlength}{2.0pt} \begin{picture}(120,35) \put(-30,-40){\includegraphics[width=14cm]{fig2}} % graf2.eps \put(3,7){$0$} \put(34,7){$\ell_1$} \put(47,7){$\ell_2$} \put(68,7){$\ell_3$} \put(94,7){$\ell_4$} \put(115,7){$\ell$} \put(39,17.5){$I_v$} \put(60,22){ $y=\kappa_0(x)$} \end{picture} \end{center} \begin{center} \setlength{\unitlength}{2.0pt} \begin{picture}(120,33) \put(-30,-40){\includegraphics[width=14cm]{fig3}} % Ncomp.eps \put(3,4){$0$} \put(34,4){$\ell_1$} \put(47,4){$\ell_2$} \put(68,4){$\ell_3$} \put(94,4){$\ell_4$} \put(115,4){$\ell$} \put(16,22){$I_e$} \put(40,22){$I_v$} \put(58,22){$I_f$} \put(80,22){$I_e$} \put(104,22){$I_f$} \end{picture} \end{center} A similar graph would hold for function $b_0$. The frictional mechanism is characterized by the functions $y=\gamma_i(x)$, $i = 1, 2$, for the same example is given as follows \begin{center} \setlength{\unitlength}{2.0pt} \begin{picture}(120,35) \put(-30,-40){\includegraphics[width=14cm]{fig4}} % graf3.eps \put(3,7){$0$} \put(34,7){$\ell_1$} \put(47,7){$\ell_2$} \put(68,7){$\ell_3$} \put(94,7){$\ell_4$} \put(115,7){$\ell$} \put(58,17.5){ $I_f$} \put(104,17.5){ $I_f$} \put(5,22){ $y=\gamma_i(x)$} \end{picture} \end{center} The energy of the system \eqref{eq1}--\eqref{ini}, is denoted by \begin{equation} \label{def-E1beam} E(t)=\frac{1}{2}\int_{0}^{\ell}\varrho_1\,|\varphi_{t}|^2 + \varrho_2\,|\psi_{t}|^2 + \kappa\,|\varphi_{x} + \psi|^2 + b\,|\psi_{x}|^2\ dx. \end{equation} It is easy to see that \[ \frac{d}{dt}{E}(t) = -\int_{0}^{l} \kappa_{0}(x)|\varphi_{xt} + \psi_{t}|^2\ dx + b_{0}(x)|\psi_{xt}|^2+\gamma_1(x)|\varphi_{t}|^2 +\gamma_2(x)|\psi_{t}|^2\; dx. \] When $\kappa_0=b_0=\gamma_1=\gamma_2=0$ the system is conservative. Regarding the novelty of our result, previous works on exponential stability consider only the effectiveness of the dissipative mechanism, whether or not it produces exponential stability, thus characterizing the dissipative mechanism as strong or weak respectively. For example to one-dimensional models was shown that the frictional dissipation exponentially stabilizes the model regardless of the position or region where the dissipative mechanism is concentrated, see for example \cite {z3S191, 1088227, 1617324, 1047433, l3MM96, l6AL06} to quote but a few. On the other hand, the dissipation produced by viscous materials, when effective over the whole domain, produces not only exponential stability but also analyticity of the corresponding semigroup. But when it concentrates in only a part of the domain, it loses effectiveness and produces neither exponential stability nor analyticity see \cite{98,Liu}. In this article we consider the two types of dissipative mechanisms, the frictional and the visco elastic dissipation both concentrated within the domain. Our main result is that the resulting dissipation will be strong or weak according to the position in which they are distributed over the domain. That is, we prove that if any elastic component (without dissipative mechanism) is next to a component with frictional dissipation, then the system is exponentially stable. Otherwise, when there is at least one component isolated between viscous components, then the system is no longer exponentially stable, but decays polynomially, that is we establish, \begin{theorem}\label{theo:total} The transmission problem \eqref{eq1}-\eqref{eq4} ($N \geq 2$) is exponentially stable if and only if any elastic part of the beam is connected with at least one component with frictional damping mechanisms. Otherwise the system is polynomially stable, with a rate of decay of the order $t^{-2}$. \end{theorem} This type of result is closely related to the {\it optimal design problem}. The main tool we use to show the exponential stability is the Pruess' characterization of exponentially stable semigroups. We prove the lack of exponential stability using the following new criterion that we show in this article \begin{theorem}\label{new} Let $\mathcal{H}_0$ be a closed subspace of a Hilbert space $\mathcal{H}$. Let $\mathcal{T}_0(t)$ be a group on $\mathcal{H}_0$ such that $\|\mathcal{T}_0(t)\|=1$ and $\mathcal{T}(t)$ be a contraction semigroup defined on $\mathcal{H}$. If the difference $\mathcal{T}(t)-\mathcal{T}_0(t)$ is compact from $\mathcal{H}_0$ to $\mathcal{H}$, then the semigroup $\mathcal{T}(t)$ is not exponentially stable. \end{theorem} The remaining part of the paper is organized as follows. In Section \ref{sec:semigroup} we show the well-posedness. In Section \ref{sec:expo-stab}, we show the exponential stability. In Section \ref{sec:lack-expo} the lack of exponential stability and Theorem \ref{new}. In Section \ref{sec:polin}, we complete the proof of Theorem \ref{theo:total} by showing the polynomial decay. Finally, we show the same result to semilinear models. \section{Well-posedness}\label{sec:semigroup} Let us introduce the phase space \[ \mathcal{H}=H_0^1(0,\ell)\times L^2(0,\ell)\times H_0^1(0,\ell)\times L^2(0,\ell). \] This is a Hilbert space with the norm \begin{equation}\label{def-normH} \|\mathbf{U}\|_{\mathcal{H}}^2=\int_{0}^{\ell}\varrho_1\,|\Phi|^2 + \varrho_2\,|\Psi|^2 + \kappa\,|\varphi_{x} + \psi|^2 + b\,|\psi_{x}|^2\ dx, \end{equation} for all $\mathbf{U}= (\varphi,\,\Phi,\psi,\Psi) \in \mathcal{H}$. Let $\mathcal{A}$ be the operator given by \begin{equation}\label{A1} \mathcal{A}\mathbf{U} = \begin{pmatrix} \Phi \\ \frac{1}{\varrho_1}[S_x-\gamma_1(x)\Phi ]\\ \Psi \\ \frac{1}{\varrho_2}[M_x-S-\gamma_2(x)\Psi ] \end{pmatrix}, \end{equation} where $S$ and $M$ are given in \eqref{def-SM}. The domain of $\mathcal{A}$ is given by \begin{equation}\label{A2} D(\mathcal{A})= \{\mathbf{U}\in \mathcal{H}: \Phi,\Psi\in H_0^1(0,\ell); S, M\in H^1(0,\ell)\}. \end{equation} A straightforward calculation gives \begin{equation}\label{energyvef} \operatorname{Re} \langle\mathcal{A}\mathbf{U},\, \mathbf{U}\rangle_{\mathcal{H}} = -\int_{0}^\ell \ \kappa_{0}\,|\Phi_{x} + \Psi|^2 + b_{0}\,|\Psi_{x}|^2+\gamma_1\,|\Phi|^2 + \gamma_2\,|\Psi|^2\ dx. \end{equation} Therefore $\mathcal{A}$ is a dissipative operator. Under the above conditions the transmission problem \eqref{eq1}-\eqref{ini} is equivalent to find $\mathbf{U}\in \mathcal{H}$, solution to \begin{equation}\label{general} \mathbf{U}_t = \mathcal{A} \mathbf{U}, \quad \mathbf{U}(0)=\mathbf{U}_0. \end{equation} where $\mathbf{U}_0 = (\varphi_0, \varphi_1, \psi_0, \psi_1)\in\mathcal{H}$ is the initial datum, defined by \eqref{ini}. Under the above notations the well posedness is a matter of routine. \begin{theorem} \label{theo2.2} For any $\mathbf{U}_{0}\in\mathcal{H}$ there exists a unique mild solution of \eqref{general}. Moreover if $\mathbf{U}_{0}\in{D}(\mathcal{A})$, then the solution is strong and $\mathbf{U}\in C^{1}([0,\,\infty[;\, \mathcal{H})\cap C([0,\,\infty[;\,{D}(\mathcal{A}))$. \end{theorem} \begin{proof} It is sufficient to show that $\mathcal{A}$ is the infinitesimal generator of a $C_0$ semigroup. Note that $\mathcal{A}$ is dissipative, closed and densely defined on $\mathcal{H}$. It is straightforward to prove that $0\in\varrho(\mathcal{A})$ (the resolvent set of $\mathcal{A}$). Our conclusion follows from Lummer Phillips's Theorem. \end{proof} We close this section by establishing the characterizations of the exponential and polynomial stabilization. due to Pr\"uss \cite{Pr84}-- Huang \cite{834231} and Borichev and Tomilov \cite{SeOptPo}. \begin{theorem}\label{Pruss} Let $ S(t)$ be a contraction $C_{0}$-semigroup, generated by $\mathcal{A}$ over a Hilbert space $\mathcal{H}$. Then, Pr\"uss \cite{Pr84}, Huang \cite{834231}, establish that there exists $C, \gamma>0$ satisfying \begin{equation}\label{ppp} \|S(t)\|\leq Ce^{-\gamma t}\,\Leftrightarrow\, i\mathbb{R}\subset\varrho(\mathcal{A}) \text{ and } \|(i\,\lambda I - \mathcal{A})^{-1}\|_{\mathcal{L}(\mathcal{H})} \leq M, \; \forall \lambda\in\mathbb{R}. \end{equation} For polynomial stability, Borichev and Tomilov \cite{SeOptPo} established the existence of $C>0$ such that \begin{equation}\label{bbb} \|\mathcal{S}(t)\mathcal{A}^{-1}\| \leq \frac{C}{t^{1/\alpha}}\, \Leftrightarrow\, i\mathbb{R}\subset\varrho(\mathcal{A}) \text{ and }\|(i\,\lambda I - \mathcal{A})^{-1}\| \leq M |\lambda|^{\alpha},\; \forall \lambda\in\mathbb{R} \end{equation} \end{theorem} \section{Exponential stability}\label{sec:expo-stab} For simplicity, we assume that if $I_{v_1}$ and $I_{v_2}$ are two viscoelastic components, then \begin{equation}\label{hipo} \overline{I_{v_1}}\cap\overline{I_{v_2}}=\emptyset. \end{equation} This hypothesis is only to simplify arguments, the result remains valid even when \eqref{hipo} fails. The resolvent equation $i\lambda \mathbf{U}-\mathcal{A} \mathbf{U}=\mathbf{F} $, in terms of its coordinates is given by \begin{gather} i\lambda \varphi - \Phi = F_1, \label{comp1e} \\ i\lambda \varrho_1\Phi - S_{x} + \gamma_1 \Phi = \varrho_1 F_2, \label{comp3e}\\ i\lambda \psi - \Psi = F_{3}, \label{comp2e}\\ i\lambda \varrho_2 \Psi - M_x + S + \gamma_2\Psi = \varrho_2 F_{4}, \label{comp4e} \end{gather} where $\mathbf{F} = (F_1, \dots, F_4) \in \mathcal{H}$ and $\varphi$ and $\psi$ verify Dirichlet boundary conditions \eqref{eq3}. \begin{lemma}\label{iR} The operator $\mathcal{A}$ defined by \eqref{A1} and \eqref{A2} satisfies $i \mathbb{R}\subset \varrho(\mathcal{A})$. \end{lemma} \begin{proof} We will reason by contradiction. Since $0 \in \varrho (\mathcal{A})$, the set $$ \mathcal{R} = \{ \beta > 0: [-i\beta, +i\beta] \subset \varrho(\mathcal{A})\} \neq \emptyset $$ Let $\overline{\lambda} : = \sup \mathcal{R}$. If $\overline{\lambda} = \infty$, then there is nothing to prove. Let us suppose that $\overline{\lambda}<\infty$. Hence, there exists a sequence $\{\beta_n\}_n \subset \mathbb{R}$ such that $\beta_n\to\overline{\lambda}$ and $\| (i \beta_n I - \mathcal{A})^{-1} \|\to\infty$, that is there exists a sequence $\{ \widetilde{F}_n \}_n$ of elements of $\mathcal{H}$ such that $$ \| \widetilde{F}_n\|_{\mathcal{H}} =1, \quad \text{and} \quad \| (\underbrace{i\beta_n I-\mathcal{A})^{-1} \widetilde{F}_n}_{:=W_n} \|_{\mathcal{H}} \underset{n \to \infty}{\longrightarrow} + \infty. $$ Letting $X_n = W_n/ \| W_n \|_{\mathcal{H}}$ and $F_n = \widetilde{F}_n/ \| W_n \|_{\mathcal{H}}$, we have \begin{equation}\label{contrad} \| X_n\|_{\mathcal{H}} =1, \quad\text{and} (i\beta_n I-\mathcal{A}) X_n = F_n \underset{n \to \infty}{\longrightarrow} 0 \, \text{ in } \, \mathcal{H} \end{equation} To arrive a contradiction it is enough to show $X_n \to 0$ as $n \to \infty$ strongly in $\mathcal{H}$. In fact, \eqref{energyvef} and \eqref{contrad} yield \begin{equation} \label{SC} \begin{aligned} &\operatorname{Re}\langle i\beta_n X_n-\mathcal{A}X_n, X_n \rangle \\ &= \int^L_0\kappa_0|\Phi_x^n+\Psi^n|^2+b_0|\Psi_x^n|^2+\gamma_1\,|\Phi^n|^2 + \gamma_2\,|\Psi^n|^2\, dx \to 0. \end{aligned} \end{equation} Since $\kappa_0$ and $b_0$ are positive over $\cup_{j=1}^m I_{v_j}$ we obtain \begin{equation}\label{fuerte} (\Phi_x^n+\Psi^n,\Psi_x^n)\to(0,0)\quad \text{strongly in } [L^2(\cup_{j=1}^m I_{v_j}) ]^2. \end{equation} Where $\cup_{j=1}^m I_{v_j}$ is the union of all the intervals with viscoelastic component. Using \eqref{comp1e}--\eqref{comp2e} we obtain $$ (\varphi_x^n+\psi^n,\psi_x^n)=\frac{1}{i\beta_n}\left[(\Phi_x^n+\Psi^n, \Psi_x^n)+(F_{1,x}^n+F_3^n, F_{3,x}^n)\right] \to (0,0) $$ strongly in $[L^2(\cup_{j=1}^m I_{v_j})]^2$. Using \eqref{contrad} once more we obtain $\|\mathcal{A}X_n\|\leq C$. Recalling the definition of $D(\mathcal{A})$ given in \eqref{A1}--\eqref{A2}, we have \begin{equation}\label{inq1} \int_0^\ell |\Phi_x^n|^2+|\Psi_x^n|^2+|S_x^n|^2+|M_x^n|^2\,dx \leq C \end{equation} which in particular implies the estimate \begin{equation}\label{inq2} \int_0^\ell |\Phi_x^n|^2+|\Psi_x^n|^2\,dx +\int_{[0,\ell]\setminus \cup_{j=1}^m I_{v_j}} |S_x^n|^2+|M_x^n|^2\,dx \leq C. \end{equation} Since $S_x^n=\kappa(\varphi_x^n+\psi^n)_x$ and $M_x^n=(b\psi_x^n)_x$ on $[0,\ell]\setminus \cup_{j=1}^m I_{v_j}$, there exists a subsequence of $X_n$, we still denote in the same way, such that \begin{gather*} (\Phi^n, \Psi^n) \to (\Phi, \Psi) \quad \text{strongly in } [L^2(0,\ell)]^2,\\ (\varphi_x^n+\psi^n, \psi_x^n)\quad \to\quad (\varphi_x+\psi, \psi_x) \quad \text{strongly in } [L^2([0,\ell] \setminus \cup_{j=1}^m I_{v_j})]^2. \end{gather*} The above convergence and \eqref{fuerte} imply $X_n\to X$ strongly in $\mathcal{H}$. Since $\gamma_1$ and $\gamma_2$ are positive over $\cup_{i=1}^r I_{f_i}$, relation \eqref{SC} implies $$ \varphi=\psi=\Phi=\Psi=0,\quad \text{on } (\cup_{i=1}^r I_{f_i}) \cup (\cup_{j=1}^m I_{v_j}) $$ Since any $I_e=]\alpha,\beta[$ is linked with $I_v$ or $I_f$, without loss of generality we can assume that $\{\alpha\}=\overline{I_v}\cap \overline{I_e}$. Since $\varphi=\psi=0$ in $I_v\cup I_f$, then system \eqref{comp1e}--\eqref{comp4e} over $I_e$ can be written as \begin{gather*} -\rho_1\lambda^2\varphi-(\kappa\varphi_x+\psi)_x=0,\quad -\rho_2\lambda^2\psi-(b\psi_x)_x+\kappa(\varphi_x+\psi)=0,\quad \text{in } [\alpha,\beta], \\ \varphi(\alpha)=\varphi_x(\alpha)=\psi(\alpha)=\psi_x(\alpha)=0. \end{gather*} By the uniqueness of ordinary differential equations we obtain $\overline{X} = 0$. The proof is now complete. \end{proof} Let us introduce the notation \begin{equation}\label{punto} \begin{gathered} \mathcal{I}_\varphi(s)={\varrho_1\kappa }|\Phi (s)|^2 +|S(s)|^2,\quad \mathcal{I}_\psi(s)={b\varrho_2}|\Psi(s)|^2 +|M(s)|^2,\\ \mathcal{I}(s)=\mathcal{I}_\varphi(s)+\mathcal{I}_\psi(s). \end{gathered} \end{equation} \begin{lemma}\label{chave} Let $]\alpha,\beta[$ any subinterval of $I_f$, then for $\lambda$ large enough, we have \begin{gather}\label{EstVisco} \int_{I_v}\varrho_1|\Phi|^2+\varrho_2|\Psi|^2+\kappa|\varphi_x +\psi|^2+b|\psi_x|^2\,dx \leq \frac{C}{|\lambda|}\left(\|U\|\|F\|+\|F\|_{\mathcal{H}}^2 \right), \\ \label{EstVisco2} \begin{aligned} &\int_{\alpha}^\beta\varrho_1|\Phi|^2+\varrho_2|\Psi|^2+\kappa|\varphi_x +\psi|^2+b|\psi_x|^2\,dx \\ &\leq C\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}}+c\|F\|_{\mathcal{H}}^2 +\frac{c}{|\lambda|}\left[\mathcal{I}(\alpha)+\mathcal{I}(\beta)\right] \end{aligned} \end{gather} \end{lemma} \begin{proof} Multiplying the resolvent system by $\overline{\mathbf{U}}$, integrating over all the beam's length $(0,\ell)$, and using the dissipation \eqref{energyvef} we obtain \begin{equation}\label{estimativa} \int_{0}^l \ \kappa_{0}\,|\Phi_{x} + \Psi|^2 + b_{0}\,|\Psi_{x}|^2+\gamma_1\,|\Phi|^2 + \gamma_2\,|\Psi|^2\ dx= \operatorname{Re}(\mathbf{F},\mathbf{U})_{\mathcal{H}} \end{equation} The above relation implies \begin{equation}\label{est2} \int_{I_v} |\Phi_{x} + \Psi|^2 + |\Psi_{x}|^2\,dx+\int_{I_f}|\Phi|^2 +|\Psi|^2\ dx\leq C\|\mathbf{F}\|_{\mathcal{H}}\|\mathbf{U}\|_{\mathcal{H}}. \end{equation} From equation \eqref{comp4e} we obtain \[ |\lambda|\|\Psi\|_{H^{-1}(I_v)} \leq C\|M\|_{L^2(I_v)}+C\|S\|_{L^2(I_v)}+C\|F\|_{\mathcal{H}} \] Therefore using \eqref{est2}, for $\lambda$ large enough, we obtain \begin{equation}\label{est3} |\lambda|^2\|\Psi\|_{H^{-1}(I_v)}^2\leq C\|U\|\|F\|+C\|F\|_{\mathcal{H}}^2 \end{equation} Then using interpolation and \eqref{est2} and \eqref{est3} we have \begin{align*} \|\Psi\|_{L^2(I_v)}^2 &\leq C\|\Psi\|_{H^{-1}(I_v)}\|\Psi\|_{H^{1}(I_v)}\\ &\leq \frac{C}{|\lambda|}\left(\|U\|\|F\|+\|F\|_{\mathcal{H}}^2 \right)^{1/2} \left(\|\Psi\|_{L^2(I_v)}+\|\Psi_x\|_{L^2(I_v)}\right)\\ &\leq \frac{C}{|\lambda|}\left(\|U\|\|F\|+\|F\|_{\mathcal{H}}^2 \right) +\frac 12 \|\Psi\|_{L^2(I_v)}^2. \end{align*} For $\lambda$ large enough. Therefore \begin{equation}\label{est7} \|\Psi\|_{L^2(I_v)}^2\leq \frac{C}{|\lambda|}\left(\|U\|\|F\|+\|F\|_{\mathcal{H}}^2 \right). \end{equation} Using \eqref{comp3e}, interpolation, and the above reasoning we obtain \begin{equation}\label{est8} \|\Phi\|_{L^2(I_v)}^2\leq \frac{C}{|\lambda|}\left(\|U\|\|F\|+\|F\|_{\mathcal{H}}^2 \right). \end{equation} Using \eqref{comp1e} and \eqref{est2} we obtain \begin{equation}\label{est9} \int_{I_v}\kappa|\varphi_x+\psi|^2+ b|\psi_x|^2\,dx \leq \frac{C}{|\lambda|^2}\left(\|U\|_{\mathcal{H}}\|F\|_{\mathcal{H}} +\|F\|_{\mathcal{H}}^2\right). \end{equation} For $\lambda $ large enough. From \eqref{est7}, \eqref{est8}, \eqref{est9}, the first part of the Lemma follows. Now, let us consider the interval $I_f=]\alpha,\beta[$. multiplying \eqref{comp3e} by $\overline{\varphi}$, \eqref{comp4e} by $\overline{\psi}$, integrating over $]\alpha,\beta[$ and taking the real part we obtain $$ \int_{I_f}\kappa |\varphi_x+\psi|^2 + b | \psi_x|^2\,dx = \left( S(s)\overline{\varphi}(s) + M(s) \overline{\psi} (s)\right) \big|_{\alpha}^{\beta} +\int_{I_f} \varrho_1\,|\Phi|^2 + \varrho_2\,|\Psi|^2\,dx +R, $$ with $| R | \leq C \| \mathbf{U} \|_{\mathcal{H}} \| \mathbf{F} \|_{\mathcal{H}}$. Using \eqref{comp1e} and \eqref{comp2e} we obtain $$ \Big| \left( S(s)\overline{\varphi}(s) + M(s) \overline{\psi} (s)\right)\big|_{\alpha}^{\beta}\Big| \leq \frac{c}{| \lambda |} \mathcal{I}(\alpha) +\frac{c}{| \lambda |} \mathcal{I}(\beta)+c\|F\|_\mathcal{H}^2. $$ Therefore, thanks to \eqref{est2} our conclusion follows. \end{proof} In what follows we will show the observability inequality. To do that, let us introduce the following notation. \begin{align*} \mathcal{L}(\alpha,\beta) &=\int_\alpha^\beta(b\varrho_2q)_x|\Psi|^2 +q_x|M|^2+(\kappa\varrho_1q)_x|\Phi|^2 +q_x|S|^2 \,dx\\ &\quad- \int_\alpha^\beta q \varrho_1\kappa\Phi \overline{\Psi}- qS\overline{M}\,dx, +\int_\alpha^\beta q \left(\gamma_1\Phi \overline{S} + \gamma_2\Psi\overline{M}\right)\,dx \end{align*} where \begin{equation}\label{def-q} q(x)= \frac{e^{nx}-e^{n\alpha}}{n}, \quad \text{or}\quad q(x)=\frac{e^{-n\beta}-e^{-nx}}{n}, \end{equation} Note that $q'(x)$ is large in comparison to $q$ for $n$ large, hence there exists positive constants $C_0$ and $C_1$ such that \begin{equation}\label{equivxx} C_0\int_\alpha^\beta \mathcal{I}(s)\,dx \leq \mathcal{L}(\alpha,\beta) \leq C_1\int_\alpha^\beta \mathcal{I}(s)\,dx \end{equation} \begin{lemma}\label{lemma:estim-bound} Let $\mathbf{U}$ be solution to the resolvent system \eqref{comp1e}-\eqref{comp4e}. Let $]\alpha,\beta[$ any subinterval of $I_e$, $I_f$ or $I_v$, then we have $$ \Big| q(s)\mathcal{I}(s)\big|_{\alpha}^{\beta} -\mathcal{L}(\alpha,\beta)\Big| \leq C\|\mathbf{U}\|\|\mathbf{F}\| +C\|\mathbf{F}\|^2,\quad ]\alpha,\beta[\subset I_f\quad\text{or}\quad ]\alpha,\beta[\subset I_e $$ and $$ \Big| q(s)\mathcal{I}(s)\big|_{\alpha}^{\beta}-\mathcal{L}(\alpha,\beta)\Big| \leq C|\lambda|^{1/2}\|\mathbf{U}\|\|\mathbf{F}\| +C\|\mathbf{F}\|^2,\quad ]\alpha,\beta[\subset I_v. $$ \end{lemma} \begin{proof} Multiply \eqref{comp3e} by $q\overline{S}$ and integrating over $[\alpha,\beta]$ we obtain $$ i\lambda \int_\alpha^\beta\varrho_1q\Phi \overline{S}\,dx- \int_\alpha^\beta qS_x \overline{S}- q\gamma_1\Phi \overline{S}\,dx =\int_\alpha^\beta \varrho_1qF_2\overline{S}\,dx $$ Recalling the definition of $S$ we obtain \begin{align*} &i\lambda \int_\alpha^\beta\varrho_1q\Phi \kappa\overline{[\varphi_x+\psi]}\,dx- \int_\alpha^\beta qS_x \overline{S}- q\gamma_1\Phi \overline{S}\,dx\\ &=\int_\alpha^\beta \varrho_1qF_2\overline{S}\,dx -i\lambda \int_\alpha^\beta\varrho_1q\Phi \kappa_0\overline{[\Phi_x+\Psi]}\,dx \end{align*} Using \eqref{comp1e} and recalling that $S=\kappa(\varphi_x+\psi)+\kappa_0(\Phi_x+\Psi)$ we obtain \begin{equation}\label{uno} -\frac{1}{2} \int_\alpha^\beta\kappa\varrho_1q\frac{d}{dx}|\Phi|^2 +q\frac{d}{dx}|S|^2 \,dx- \int_\alpha^\beta\varrho_1q \kappa\Phi \overline{\Psi}\,dx+\int_\alpha^\beta q\gamma_1\Phi \overline{S}\,dx =\mathcal{G} \end{equation} where $$ \mathcal{G}= \int_\alpha^\beta\varrho_1q\kappa\Phi (\overline{F_{1,x}+F_3})\,dx -i\lambda \int_\alpha^\beta\varrho_1q\Phi \kappa_0\overline{[\Phi_x+\Psi]}\,dx +\int_\alpha^\beta \varrho_1qF_2\overline{S}\,dx $$ Integrating by parts \eqref{uno} we obtain \begin{equation}\label{unox} \begin{aligned} &- q(s)\mathcal{I}_\varphi(s)\big|_{\alpha}^{\beta} +\int_\alpha^\beta(\kappa\varrho_1q)_x|\Phi|^2 +q_x|S|^2 \,dx \\ &- \int_\alpha^\beta\varrho_1q \kappa\Phi \overline{\Psi}\,dx +\int_\alpha^\beta q\gamma_1\Phi \overline{S}\,dx =2\mathcal{G} \end{aligned} \end{equation} Multiplying \eqref{comp4e} by $q\overline{M}$, integrating over $[\alpha,\beta]$, and using the same above arguments we obtain \begin{equation}\label{unoy} \begin{aligned} &- q(s)\mathcal{I}_\psi(s)\big|_{\alpha}^{\beta} +\int_\alpha^\beta(b\varrho_2q)_x|\Psi|^2 +q_x|M|^2 \,dx \\ &+\int_\alpha^\beta qS\overline{M}\,dx +\int_\alpha^\beta q\gamma_2\Psi \overline{M}\,dx =2\mathcal{F} \end{aligned} \end{equation} where $$ \mathcal{F}= -i\lambda \int_\alpha^\beta\varrho_2q\Psi b_0\overline{\Psi_x}\,dx +\int_\alpha^\beta \varrho_2qF_4\overline{M}\,dx. $$ Summing \eqref{unox}--\eqref{unoy} and recalling the definition of $\mathcal{L}$ we obtain \begin{equation} -q(s)\mathcal{I}\big|_{\alpha}^{\beta}+\mathcal{L}(\alpha,\beta) =2\mathcal{G}+2\mathcal{F}\label{iidd} \end{equation} Using \eqref{est2} and \eqref{est7} we obtain $$ |2\mathcal{G}|+|2\mathcal{F}|\leq C\|\mathbf{U}\|\|\mathbf{F}\| +C\|\mathbf{F}\|^2, \quad \forall \,]\alpha,\beta[\subset I_e\cup I_f $$ Similarly, using \eqref{est8} we obtain $$ |2\mathcal{G}|+|2\mathcal{F}|\leq C|\lambda|^{1/2}\|\mathbf{U}\|\|\mathbf{F}\| +C\|\mathbf{F}\|^2, \quad \forall \,]\alpha,\beta[\subset I_v $$ Therefore our conclusion follows. \end{proof} \begin{corollary}\label{coro} Assume \eqref{hipo} holds. Then for any $i=1,\dots, N-1$, there exists $C > 0$, such that $$ \mathcal{I}(\ell_i)\leq C \left( \|\mathbf{U}\|_{\mathcal{H}}^2 +\|\mathbf{U}\|_{\mathcal{H}}\|\mathbf{F}\|_{\mathcal{H}} \right). $$ \end{corollary} \begin{proof} From \eqref{hipo} we can assume that any $\ell_i$ belongs to the border of some elastic or frictional component, since $$ S(\ell_i^-)=S(\ell_i^+),\quad M(\ell_i^-)=M(\ell_i^+). $$ Therefore we can apply Lemma \ref{lemma:estim-bound} and inequalities \eqref{equivxx} we obtain $$ \mathcal{I}(\ell_i)\leq C \|\mathbf{U}\|_{\mathcal{H}}^2 +C\|\mathbf{U}\|_{\mathcal{H}}\|\mathbf{F}\|_{\mathcal{H}} $$ The conclusion follows. \end{proof} Now, we are in a position to prove the main result of this section. \subsection*{Proof of the necessary condition of Theorem \ref{theo:total}} \begin{figure}[ht] \begin{center} \setlength{\unitlength}{2.0pt} \begin{picture}(120,30) \put(-30,-40){\includegraphics[width=14cm]{fig5}} % Ncomp2.eps \put(3,4){$0$} \put(34,4){$\ell_1$} \put(47,4){$\ell_2$} \put(68,4){$\ell_3$} \put(94,4){$\ell_4$} \put(115,4){$\ell$} \put(16,22){$I_e$} \put(40,22){$I_f$} \put(58,22){$I_v$} \put(80,22){$I_e$} \put(104,22){$I_f$} \end{picture} \end{center} \caption{\it A five-components beam, exponentially stable.} \label{xx} \end{figure} From Lemma \ref{chave} we obtain for any interval $I_v$ and $I_f$ that \begin{align*} &\int_{I_v\cup I_f}\varrho_1|\Phi|^2+\varrho_2|\Psi|^2+\kappa|\varphi_x +\psi|^2+b|\psi_x|^2\,dx \\ &\leq C\|\mathbf{U}\|_{\mathcal{H}}\|\mathbf{F}\|_{\mathcal{H}} +c\|\mathbf{F}\|_{\mathcal{H}}^2+\frac{c}{|\lambda|} \sum_{i=1}^{N-1}\mathcal{I}(\ell_i). \end{align*} Using Corollary \ref{coro} we obtain \begin{equation}\label{energ0} \begin{aligned} &\int_{I_v\cup I_f}\varrho_1|\Phi|^2+\varrho_2|\Psi|^2+\kappa|\varphi_x+\psi|^2 +b|\psi_x|^2\,dx \\ &\leq C\|\mathbf{U}\|_{\mathcal{H}}\|\mathbf{F}\|_{\mathcal{H}} +c\|\mathbf{F}\|_{\mathcal{H}}^2+\epsilon\|\mathbf{U}\|_{\mathcal{H}}^2 \end{aligned} \end{equation} For $|\lambda|$ large enough. It remains to estimate the energy over intervals of type $I_e$. Let us denote $I_e = ]\alpha , \beta[$. From hypothesis, this interval is linked with an interval of type $I_f$, for example at the point $\{ \beta \}$. Using Lemma \ref{lemma:estim-bound}, over $I_e=]\alpha,\beta[$, we obtain \begin{equation}\label{energye} \int_{I_e}\mathcal{I}(s)\,ds \leq c\mathcal{I}(\beta)+ c\|\mathbf{U}\|_{\mathcal{H}}\|\mathbf{F}\|_{\mathcal{H}}. \end{equation} Since $\beta\in I_f$, we apply the transmission conditions and the observability estimate, Lemma \ref{lemma:estim-bound}, for the frictional part $$ \mathcal{I}(\beta)\leq c \int_{I_f}\mathcal{I}(s)\,ds +c\|\mathbf{U}\|_{\mathcal{H}}\|\mathbf{F}\|_{\mathcal{H}}. $$ Hence, from \eqref{energ0} and \eqref{energye}, we obtain \begin{equation}\label{energye2} \int_{I_e}\mathcal{I}(s)\,ds \leq C\|\mathbf{U}\|_{\mathcal{H}} \|\mathbf{F}\|_{\mathcal{H}} + \frac{C}{| \lambda|^2} \left( \| \mathbf{U} \|^2_{\mathcal{H}} + \| \mathbf{F} \|^2_{\mathcal{H}}\right). \end{equation} Therefore, adding all the energy over all interval $I_e$, $I_f$ and $I_v$ we obtain $$ \| \mathbf{U} \|^2 \leq C \| \mathbf{U}\|_{\mathcal{H}} \| \mathbf{F}\|_{\mathcal{H}} +\epsilon \| \mathbf{U} \|^2_{\mathcal{H}} + C\| \mathbf{F} \|^2_{\mathcal{H}}, $$ Which implies $\| \mathbf{U} \|\leq C\| \mathbf{F} \|_{\mathcal{H}} $, the result follows thanks to part \eqref{ppp} of Theorem \ref{Pruss}. \section{Lack of exponential stability}\label{sec:lack-expo} In this section we prove that system \eqref{eq1}--\eqref{ini} does not decays exponentially to zero when hypotheses of Theorem \ref{theo:total} fails. The proof is based on Theorem \ref{new}. Before going into the details, we recall some results on the Calkin Algebra (see \cite[pp. 248-250]{engelbook}, ). \subsection{Calkin algebra} Let $\mathcal{K}(\mathcal{H})$ be the set of all the compact operators over $\mathcal{H}$. It is a closed subspace and also a maximal ideal of $\mathcal{L}(\mathcal{H})$. The quotient space $\mathcal{C}(\mathcal{H}) : = \mathcal{L}(\mathcal{H})/\mathcal{K}(\mathcal{H})$, called the Calkin algebra, is a complete space with the norm $$ \|S\|_{\rm ess}:=\|\widetilde{S}\|_{\mathcal{C}(\mathcal{H})} :=\inf\{\|S-K\|_{\mathcal{L}(\mathcal{H})};\;\; K\in \mathcal{K}(\mathcal{H})\}. $$ So any operator of $S\in \mathcal{L}(\mathcal{H})$ can be projected onto $\mathcal{C}(\mathcal{H})$ in the following way $\mathcal{M}: \mathcal{L}(\mathcal{H}) \to \mathcal{C}(\mathcal{H})$ \[ \mathcal{M}(S)=\widetilde{S}=S+ \mathcal{K}(\mathcal{H}). \] Under the above notation we define the essential spectrum of $S$, $\sigma_{\rm ess}(S)$ as $\sigma(\widetilde{S})$ the spectrum $\widetilde{S}\in\mathcal{C}(\mathcal{H})$ and the essential spectral radius of an operator $S \in \mathcal{L}(\mathcal{H})$ as the spectral radios of $\widetilde{S}$, that is $r_{\rm ess}({S}) : = \, r(\widetilde{S})$. Note that from the definition of the essential norm, it holds: $$ \|S\|_{\rm ess}=\|S+K\|_{\rm ess},\quad \forall K\in \mathcal{K}(\mathcal{H}). $$ This implies the following result, due to Weyl. \begin{theorem}[Weyl] \label{weyl} The essential spectral radius is conserved under a relatively compact perturbation. That is to say, for any $S\in \mathcal{L}(X)$ and any $K \in \mathcal{K} (X)$, we have $$ r_{\rm ess}(S)= r_{\rm ess}(S + K). $$ \end{theorem} For an extension of this result, see \cite[Theorem 5.35]{katobook}. Let $S(t)$ be a semigroup. The type $\omega_0$ (or growth bound) and the essential type $\omega_{\rm ess}$ of the semigroup are defined as \begin{equation}\label{type} \omega_{0}(S) := \lim_{t\to \infty}\frac{\ln \|S(t)\|}{t},\quad \omega_{\rm ess}(S) = \lim_{t\to \infty}\frac{1}{t}\ln \|S(t)\|_{\rm ess}, \end{equation} Using the Gelfand Formula for the spectral radius of an operator, $$ r(S) = \underset{n \to \infty}{\lim} \left\| S^n\right\|^{1/n}. $$ Therefore, the spectral and the essential spectral radius of a semigroup $S(t)$ are given by $$ r(S(t))=e^{\omega_0 t},\quad r_{\rm ess}(S(t))=r(\widetilde{S}(t))=e^{\omega_{\rm ess} t} $$ \begin{proposition}\label{prop:type} Let $(T(t))_{t\geq 0}$ a $C_0-$semigroup on the Banach $X$ with generator $A$. Then $$ \omega_0=\max\{\omega_{\rm ess}, s(A)\}, $$ where $s(A)$ is the spectral bound of $A$. \end{proposition} For a proof of this result see \cite[Corollary 2.11]{engelbook}. We are now ready to prove our criterium for the lack of exponential stability of a $C_0$-semigroup. \subsection{Proof of Theorem \ref{new}} Since $\mathcal{T}_0(t)$, is a group satisfying $\|\mathcal{T}_0(t)\|=1$, we have that for all $\lambda\in \sigma(\mathcal{T} _0(t))$, $|\lambda|=1$. This implies that $r_{\rm ess}(\mathcal{T}_0(t))=1$. Let $P$ be the orthogonal projection operator of $\mathcal{H}$ onto $\mathcal{H}_0$. Then $\mathcal{T}_0(t)P\in \mathcal{L}(\mathcal{H})$. Moreover, we have that $$ r_{\rm ess}(\mathcal{T}_0(t)P)\geq 1. $$ Otherwise, if $r_{\rm ess}(\mathcal{T}_0(t)P)< 1$, from the Gelfand formula we obtain \begin{align*} 1&> \lim_{n\to \infty}\Big(\inf_{K\in\mathcal{K}(\mathcal{H})} \|[\mathcal{T}_0(t)P-K]^n\|_{\mathcal{H}}\Big)^{1/n}\\ &\geq \lim_{n\to \infty}\Big(\inf_{K\in\mathcal{K}(\mathcal{H})} \|\mathcal{T}_0(t)^nP-K\|_{\mathcal{H}}\Big)^{1/n}\\ &\geq \lim_{n\to \infty}\Big(\inf_{K\in\mathcal{K}(\mathcal{H})}\|P \big(\mathcal{T}_0(t)^nP-K \big)\|_{\mathcal{H}}\Big)^{1/n}\\ &\geq \lim_{n\to \infty}\Big(\inf_{K\in\mathcal{K}(\mathcal{H}_0)} \|\mathcal{T}_0(t)^n-K\|_{\mathcal{H}_0}\Big)^{1/n}. \end{align*} The last inequality holds because of the norm $1$ of the projection operator. But this would imply $r_{\rm ess}(\mathcal{T}_0(t))< 1$, which is a contradiction with the zero type of $\mathcal{T}_0 (t)$ by Proposition \ref{prop:type}. On the other hand, since $\mathcal{T}(t)-\mathcal{T}_0(t)$ is a compact operator from $\mathcal{H}_0$ to $\mathcal{H}$ the operator $[\mathcal{T}(t)-\mathcal{T}_0(t)]P$ is also compact operator over $\mathcal{H}$. Hence, from Theorem \ref{weyl}: $$ r_{\rm ess}(\mathcal{T}(t)P)=r_{\rm ess}(\mathcal{T}_0(t)P)\geq 1. $$ Using Gelfand's Formula once more, we have, for all $t > 0$: \[ 1 \leq r_{\rm ess}(\mathcal{T}(t)P) = \lim_{n\to \infty}\Big(\inf_{K\in\mathcal{K}(\mathcal{H})} \|[\mathcal{T}(t)-K]^n\|_{\mathcal{H}}\Big)^{1/n} \leq \|\mathcal{T}(t)\|, \] Therefore $\mathcal{T}(t)$ is not exponentially stable and the proof of Theorem \ref{new} is complete. \subsection{Lack of exponential stability} Here we assume that the elastic part is not linked with a frictional component as in Figure \ref{fig3}, we claim the following result. \begin{proposition}\label{prop-expo2} If there exists an elastic component not connected to a frictional component, then the transmission problem \eqref{eq1}--\eqref{ini} with $N \geq 2$ is not exponentially stable. \end{proposition} \begin{figure}[ht] \begin{center} \setlength{\unitlength}{2.0pt} \begin{picture}(120,30) \put(-30,-40){\includegraphics[width=14cm]{fig6}} % Ncomp3.eps \put(3,4){$0$} \put(34,4){$\ell_1$} \put(47,4){$\ell_2$} \put(68,4){$\ell_3$} \put(94,4){$\ell_4$} \put(115,4){$\ell$} \put(68,2){$\underbrace{\text{\hspace{1.9cm}}}$} \put(16,22){$I_v$} \put(40,22){$I_f$} \put(58,22){$I_v$} \put(80,22){$I_e$} \put(104,22){$I_v$} \end{picture} \end{center} \caption{A five-components beam, non exponentially stable.} \label{fig3} \end{figure} \begin{proof} Let us denote by $I_e=]\alpha,\beta[$ the elastic interval that does not have any frictional neighbor. In Figure $3$, the dissipative mechanisms are effective in all the components except in $I_4=]\ell_3,\ell_4[=]\alpha,\beta[$, this interval being isolated form the frictional ones. Let us define the space $\mathcal{H}_0$, as follows. $$ \mathcal{H}_0=\widetilde{H}_0^1(I_e)\times \widetilde{L}^2(I_e) \times \widetilde{H}_0^1(I_e)\times \widetilde{L}^2(I_e), $$ where \begin{gather*} \widetilde{L}^2(I_e)=\{g\in L^2(0,\ell): g(x)=0,\; \forall x\in ]0,\ell[\setminus I_e\},\\ \widetilde{H}_0^1(I_e)=\{ g \in H_0^1(0,\ell): g, g'\in L^2(I_e) \}. \end{gather*} Note that $\mathcal{H}_0$ is a closed subspace of $\mathcal{H}$, Denoting $\widehat{\mathbf{U}} = (\widehat{\varphi}, \widehat{\varphi}_t, \widehat{\psi}, \widehat{\psi}_t)$, \begin{equation}\label{transm-tild} \begin{gathered} \varrho_1\widehat{\varphi}_{tt}-[\kappa(\widehat{\varphi}_{x}+\widehat{\psi})]_x = 0,\quad \text{in } ]\alpha,\beta[\times \mathbb{R}_{+}, \\ \varrho_2\widehat{\psi}_{tt} -[b\widehat{\psi}_{x}]_x +\kappa(\widehat{\varphi}_{x}+\widehat{\psi}) = 0,\quad \text{in } ]\alpha,\beta[\times \mathbb{R}_{+}, \\ \widehat{\varphi}(\alpha,t)=\widehat{\varphi}(\beta,t) =0,\quad \widehat{\psi}(\alpha,t)=\widehat{\psi}(\beta,t)=0, \\ \widehat{\varphi}(x,0)=\varphi_0,\quad \widehat{\varphi}_t(x,0)=\varphi_1,\quad \widehat{\psi}(x,0)=\psi_0,\quad \widehat{\psi}_t(x,0)=\psi_1. \end{gathered} \end{equation} The elastic part being isolated from the rest of the components, this system is conservative, so it defines a group of isometries, with type $0$. Now we extend the solution to $]0,\ell[$ as $$ \widetilde{\varphi}(x,t)= \begin{cases} \widehat{\varphi}(x,t),&x\in I_e=]\alpha,\beta[,\\ 0,&x\in ]0,\ell[\setminus I_e, \end{cases} \quad \widetilde{\psi}(x,t)= \begin{cases} \widehat{\psi}(x,t), &x\in I_e=]\alpha,\beta[,\\ 0, &x\in ]0,\ell[\setminus I_e. \end{cases} $$ Under these conditions, for any $\mathbf{U}_0=({\varphi}_0,{\varphi}_1,{\psi}_0,{\psi}_1)\in\mathcal{H}_0$ we define the semigroup $\mathcal{T}_0(t)$ as $$ \mathcal{T}_0(t) \, \mathbf{U}_0=(\widetilde{\varphi}, \widetilde{\varphi}_t,\widetilde{\psi},\widetilde{\psi}_t). $$ Thus we have $\omega_0 (\mathcal{T}_0 (t)) = 0$ on $\mathcal{H}_0$. To apply Theorem \ref{new}, it remains to show that $\mathcal{T} (t) - \mathcal{T}_0 (t)$ is compact. Let $\mathbf{U}_0^n= (\varphi_0^n, \varphi_1^n,\psi_0^n,\psi_1^n) \in \mathcal{H}_0$ be a bounded sequence of $\mathcal{H}_0$. Denoting by $$ \mathbf{U}^n = (\varphi^n, \varphi_t^n, \psi^n, \psi_t^n) = \mathcal{T}(t) \mathbf{U}_0^n, $$ the solution to the original transmission problem with initial condition $\mathbf{U}_0^n$, and $$ \widetilde{\mathbf{U}}^n = (\widetilde{\varphi}^n,\widetilde{\varphi}_t^n, \widetilde{\psi}^n,\widetilde{\psi}_t^n)= \mathcal{T}_0(t) \, \mathbf{U}_0^n, $$ the solution to the modified problem. Let $$ \mathbf{Z}^n(t) : = \mathbf{U}^n - \widetilde{\mathbf{U}}^n = (\varphi^n, \varphi_t^n, \psi^n, \psi_t^n) -(\widetilde{\varphi}^n,\widetilde{\varphi}_t^n,\widetilde{\psi}^n, \widetilde{\psi}_t^n) =(W^n,W^n_t,V^n,V^n_t). $$ Recalling that $\widetilde{I} = \cup_{k=1}^N \, I_k$, the sequence $\mathbf{Z}^n$ satisfies \begin{gather} \label{WTimo1} \varrho_1\,W_{tt} - [\kappa(W_x+V)]_x - [\kappa_0(W_{xt}+V_t)]_x+\gamma_1W_t = 0\quad\text{in } \widetilde{I}\times \mathbb{R}^{+}, \\ \label{WTimo2} \varrho_2\,V_{tt} - [bV]_x - [b_0V_{xt}]_{x} + \kappa(W_x+V) +\gamma_2V_t = 0 \quad\text{in } \widetilde{I} \times \mathbb{R}^{+}. \end{gather} Let us introduce the energy of this problem, \begin{equation*}\label{def-EZ} E_{\mathbf{Z}^n} (t) : = \frac 1 2 \int_{0}^l \varrho_1|W_{t}^{n}|^2 + \varrho_2|V_{t}^{n}|^2 + \kappa |W_{x}^{n,i} + V^{n}|^2 + b |V_{x}^{n}|^2\ dx. \end{equation*} Since we are in a Hilbert space, it suffices to show that there exists a subsequence of $\{ \mathbf{Z}^n\}$ that converges in norm (or in energy). Multiplying equation \eqref{WTimo1} by $W_t^n$, \eqref{WTimo2} by $V_t^n$, and integrating on $\widetilde{I}$ we have \begin{align*} &\frac{d}{dt} E_{\mathbf{Z}^n}(t) +\int_0^l\kappa_0|W_{xt}^n+V_t^n|^2+b_0|V_{xt}^n|^2 +\gamma_1|W_t^n|^2+\gamma_2|V_t^n|^2\,dx\\ &=\kappa(W_x^n+V^n)W_t^n\big|_{\alpha}^{\beta} +\kappa_0(W_{xt}^n+V_t^n)W_t^n\big|_{\alpha}^{\beta} +bV_x^nV_t^n\big|_{\alpha}^{\beta}+b_0V_{xt}^nV_t^n\big|_{\alpha}^{\beta}\\ &=\kappa\widetilde{\varphi}_x^n\varphi_t^n\big|_{\alpha}^{\beta} + b\widetilde{\psi}_x^n\psi_t^n\big|_{\alpha}^{\beta}. \end{align*} Note that $\widetilde{\varphi}_x^n(\alpha,t)$ and $\widetilde{\varphi}_x^n(\beta,t)$ are bounded in $L^2(0,T)$. Since $E_{\mathbf{Z}^n} (0) = 0$, it follows that \begin{equation}\label{eq-energy} \begin{aligned} &E_{\mathbf{Z}^n} (t)+\int_0^T\int_0^l\kappa_0|W_{xt}^n+V_t^n|^2 +b_0|V_{xt}^n|^2+\gamma_1|W_t^n|^2+\gamma_2|V_t^n|^2\,dx \\ &= \kappa\int_0^T\widehat{\varphi}_x^n\varphi_t^n\big|_{\alpha}^{\beta}\,dt +b\int_0^T\widehat{\psi}_x^n\psi_t^n\big|_{\alpha}^{\beta}\,dt. \end{aligned} \end{equation} In the viscoelastic intervals $I_v$, the sequences $\varphi_{t}^{n},\ $ $\psi_{t}^{n}$, are bounded in the space $L^2(0,T;\ H^{1}(I_v))$ (from the energy dissipation estimate). Moreover, $\varphi_{tt}^{n}$, $\psi_{tt}^{n}$ are bounded in $L^2(0,T;\ H^{-1}(I_v))$. Hence, from compactness criterion of Aubin-Lions, we have, up to a subsequence, $$ (\varphi_t^n,\psi_t^n)\to (\varphi_t,\psi_t)\quad \text{strongly in } L^2(0,T;H^{1-\epsilon}(I_v)\times H^{1-\epsilon}(I_v)), $$ for all $0 < \epsilon < 1$. It yields $$ (\varphi_t^n(s,\cdot),\psi_t^n(s,\cdot))\to (\varphi_t(s,\cdot),\psi_t(s,\cdot))\quad \text{strongly in } L^2(0,T)\times L^2(0,T), $$ for $s = \alpha$ and $s = \beta$. Therefore we obtain, up to a subsequence, the strong convergence $E_{\mathbf{Z}^n} (t) \to E_{\mathbf{Z}} (t)$, where $\mathbf{Z} = \mathbf{U} - \widetilde{\mathbf{U}}$ is the difference of the weak limits. Therefore, since in a Hilbert space, the weak convergence and the convergence in norm imply the strong convergence, we conclude that $\mathcal{T}(t)-\mathcal{T}_0(t)$ is compact from $\mathcal{H}_0$ to $\mathcal{H}$. From Theorem \ref{new}, the semigroup $\mathcal{T} (t)$ is not exponentially stable and the proof of Proposition \ref{prop-expo2} is complete. \end{proof} \section{Polynomial decay}\label{sec:polin} To complete the proof of Theorem \ref{theo:total}, it remains to show the polynomial decay, under a non exponential configuration (as in Figure \ref{fig3} for example). \begin{proposition}\label{prop:polyn-stab} If there exists an elastic component not connected to a frictional component, then the semigroup $\mathcal{T} (t)$ defined by problem \eqref{eq1}--\eqref{ini} with $N \geq 2$ decays polynomially as $$ \|\mathcal{T}(t)\mathbf{U}_0\|_{\mathcal{H}} \leq \frac{c}{t^2}\|\mathbf{U}_0\|_{\mathcal{H}}. $$ \end{proposition} \begin{proof} As in the proof of the exponential stability we have \begin{equation}\label{poli1} \sum_{i=1}^N\int_{I_{v_i}\cup I_{f_i}}\mathcal{I}(s)\,ds \leq c \| \mathbf{U} \|_{\mathcal{H}} \| \mathbf{F} \|_{\mathcal{H}} + C\| \mathbf{F} \|_{\mathcal{H}}^2 , \end{equation} for $| \lambda | $ large enough. It remains to estimate the energy over the interval $I_e$we denote as $I_e = (\alpha, \beta)$. By the hypotheses, $\alpha\in I_v$ or $\beta\in I_v$. Using Lemma \ref{lemma:estim-bound} over $I_e$ we obtain \begin{equation}\label{energye3} \int_{I_e}\mathcal{I}(s)\,ds\leq C\mathcal{I}(\beta)+ C\|\mathbf{U}\|_{\mathcal{H}}\|\mathbf{F}\|_{\mathcal{H}}. \end{equation} Using Lemma \ref{lemma:estim-bound} over $I_v$, we have \begin{equation}\label{Ee} \mathcal{I}(\beta)\leq C|\lambda|^{1/2}\|\mathbf{U}\|_{\mathcal{H}} \|\mathbf{F}\|_{\mathcal{H}}+ C|\lambda|^{1/2}\|\mathbf{F}\|_{\mathcal{H}}^2. \end{equation} From inequality \eqref{EstVisco} of Lemma \ref{chave} and Lemma \ref{lemma:estim-bound} we have \begin{equation}\label{energye3b} \int_{I_e}\mathcal{I}(s)\,ds\leq C|\lambda|^{1/2}\|\mathbf{U}\|_{\mathcal{H}} \|\mathbf{F}\|_{\mathcal{H}}+ C|\lambda|^{1/2}\|\mathbf{F}\|_{\mathcal{H}}^2. \end{equation} From where it follows, with the Young inequality, that $\|\mathbf{U}\|_{\mathcal{H}}^2\leq c|\lambda| \|\mathbf{F}\|_{\mathcal{H}}^2$. Our conclusion follows thanks part \ref{bbb} of Theorem \ref{Pruss}. \end{proof} \section{Semi linear problem} Here we prove the exponential and polynomial stability for a long class of locally Lipschitz $ \mathcal {F} $ functions over a Hilbert space $ \mathcal {H} $. We consider are the following hypotheses: For any ball $ B_R=\{W\in\mathcal{H}: \|W\|_{\mathcal{H}}\leq R\} $, there exists a function $\widetilde{\mathcal{F}_R}$ globally of Lipschitz such that \begin{equation}\label{ff1} \mathcal{F}(0)=0,\quad \mathcal{F}(U)=\widetilde{\mathcal{F}_R}(U),\quad \forall U\in B_R; \end{equation} additionally, that there exists a positive constant $\kappa_0$ such that \begin{equation}\label{ff2} \int_0^t\widetilde{\mathcal{F}_R}(U(s))U(s)\,ds \leq \kappa_0\|U(0)\|_{\mathcal{H}}^2,\quad \forall U\in C([0,T];\mathcal{H})\,. \end{equation} Under these condition, we present the following result. \begin{theorem}\label{Lipschitz} Let $\{S(t)\}_{t\geq 0}$ be a contraction, exponentially or polynomially stable semigroup with infinitesimal generator $\mathbb{A}$ over the phase space $\mathcal{H}$. Let $\mathcal{F}$ locally Lipschitz on $\mathcal{H}$ satisfying conditions \eqref{ff1} and \eqref{ff2}. If there exists a global solution to \begin{equation}\label{absCE} U_{t}-\mathbb{A}U=\mathcal{F}(U),\quad U(0)=U_0\in\mathcal{H}, \end{equation} then the solution decays exponentially or polynomially respectively. \end{theorem} \begin{proof} By hypotheses, there exist positive constants $c_0$ and $\gamma$ such that $\|S(t)\|\leq c_0e^{-\gamma t}$, and $\widetilde{\mathcal{F}_R}$ is globally Lipschitz with Lipschitz constant $K_0$ satisfying \eqref{ff1} and \eqref{ff2}. Let us consider the space $$ E_{\mu}=\big\{V\in L^\infty(0,\infty;\mathcal{H}): t\mapsto e^{-\mu t}\|V(s)\|\in L^\infty(\mathbb{R})\big\} $$ Using standard fixed point arguments we can show that there exists only one global solution to \begin{equation}\label{absCE2} U_{t}^R-\mathbb{A}U^R=\widetilde{\mathcal{F}_R}(U^R),\quad U^R(0)=U_0\in\mathcal{H}, \end{equation} Multiplying the above equation by $U^R$ we obtain that $$ \frac 12 \frac{d}{dt}\|U^R(t)\|_{\mathcal{H}}^2-(\mathbb{A}U^R, U^R)_{\mathcal{H}} =(\widetilde{\mathcal{F}_R}(U^R),U^R)_{\mathcal{H}} $$ Since the semigroup is contractive, its infinitesimal generator is dissipative, therefore $$ \|U^R(t)\|_{\mathcal{H}}^2\leq \|U_0\|_{\mathcal{H}}^2 +2\int_0^t(\widetilde{\mathcal{F}_R}(U^R),U^R)_{\mathcal{H}}\,dt $$ Using \eqref{ff2} we obtain $$ \|U^R(t)\|_{\mathcal{H}}^2\leq (1+k_0)\|U_0\|_{\mathcal{H}}^2 $$ Nota that for $R> (1+k_0)\|U_0\|_{\mathcal{H}}^2$, we have that $$ \widetilde{\mathcal{F}_R}(V)=\mathcal{F}(V),\quad \forall \|V\|_{\mathcal{H}}\leq R $$ In particular we have $$ \widetilde{\mathcal{F}_R}(U^R(t))=\mathcal{F}(U^R(t)). $$ This means that $U^R$ is also solution of system \eqref{absCE} and because of the uniqueness we conclude that $U^R=U$. Therefore to show the exponential stability to system \eqref{absCE}, it is sufficient to show the exponential decay to system \eqref{absCE2}. To do that, we use fixed points arguments. $$ \mathcal{T}(V)=S(t)U_0+\int_0^{t}S(t-s)\widetilde{\mathcal{F}_R}(V(s))\,ds, $$ Note that $\mathcal{T}$ is invariant over $E_{\gamma-\delta}$ for $\delta$ small, ($\gamma-\delta>0$). In fact, for any $V\in E_{\gamma-\delta}$ we have \begin{align*} \|\mathcal{T}(V)\|_{\mathcal{H}} &\leq \|U_0\|_{\mathcal{H}}e^{-\gamma t} +\int_0^t\|\widetilde{\mathcal{F}_R}(V(s))\|_{\mathcal{H}}e^{-\gamma(t-s)}\,ds\\ &\leq \|U_0\|_{\mathcal{H}}e^{-\gamma t} +K_0\int_0^t\|V(s)\|_{\mathcal{H}} e^{-\gamma(t-s)}\,ds\\ &\leq \|U_0\|_{\mathcal{H}}e^{-\gamma t} +K_0e^{-\gamma t} \int_0^te^{\delta s}\,ds \sup_{s\in [0,t]} \{ e^{(\gamma-\delta) s}\|V(s)\|_{\mathcal{H}}\}\\ &\leq \|U_0\|_{\mathcal{H}}e^{-\gamma t} +\frac{K_0C}{\delta}e^{-(\gamma -\delta)t}. \end{align*} Therefore, $\mathcal{T}(V)\in E_{\gamma-\delta}$. Using standard arguments we can show that $\mathcal{T}^n$ satisfies $$ \|\mathcal{T}^n(W_1)-\mathcal{T}^n(W_2)\| \leq \frac{(k_1t)^n}{n!}\|W_1-W_2\|_{\mathcal{H}} $$ Therefore we have a unique fixed point satisfying $$ \mathcal{T}^n(U)=U=S(t)U_0+\int_0^{t}S(t-s)\widetilde{\mathcal{F}_R}(U(s))\,ds, $$ That is $U$ is a solution of \eqref{absCE2}, and since $\mathcal{T}$ is invariant over $E_{\gamma-\delta}$, then the solution decays exponentially. To show the polynomial stability we consider the space $$ E_{p}=\{V\in L^\infty(0,\infty;\mathcal{H}): t\mapsto (1+t)^p\|V(s)\|\in L^\infty(\mathbb{R})\} $$ To show the invariance we use $$ \sup_{t>0}(1+t)^p\int_0^t (1+t-s)^{-p}(1+s)^{-p}\,ds