\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 01, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/01\hfil Behavior of the energy] {Behavior of the energy for Lam\'e systems in bounded domains with nonlinear damping and external force} \author[A. Bchatnia, M. Daoulatli \hfil EJDE-2013/01\hfilneg] {Ahmed Bchatnia, Moez Daoulatli} % in alphabetical order \address{Ahmed Bchatnia \newline Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Campus Universitaire 2092 - El Manar 2, Tunis, Tunisia} \email{ahmed.bchatnia@fst.rnu.tn} \address{Moez Daoulatli \newline Department of Mathematics, Faculty of Sciences of Bizerte, University of Carthage, 7021, Jarzouna, Bizerte, Tunisia} \email{moez.daoulatli@infcom.rnu.tn} \thanks{Submitted November 8, 2012. Published January 7, 2013.} \subjclass[2000]{35L05, 35B40} \keywords{Lam\'e system; nonlinear damping; bounded domain; external force} \begin{abstract} We study behavior of the energy for solutions to a Lam\'e system on a bounded domain, with localized nonlinear damping and external force. The equation is set up in three dimensions and under a microlocal geometric condition. More precisely, we prove that the behavior of the energy is determined by a solution to a forced differential equation, an it depends on the $L^2$ norm of the force. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of the problem} Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^3$. Let us consider the Lam\'e system with localized nonlinear damping and external force, $$\begin{gathered} \partial _t^2u-\Delta _eu+a(x) g(\partial _tu) =f(t,x), \quad \text{in } \mathbb{R}_{+}\times \Omega , \\ u=0 \quad \text{on }\mathbb{R}_{+}\times \partial \Omega , \\ u(0,x) =\varphi _1(x), \quad \partial _tu(0,x) =\varphi _2(x) \quad \text{in }\Omega . \end{gathered} \label{systnon}$$ Here $\Delta _e$ denotes the elasticity operator, which is the $3\times 3$ matrix-valued differential operator defined by $\Delta _eu=\mu \Delta u+(\lambda +\mu ) \nabla \operatorname{div}u, \quad u=(u_1,u_2,u_3),$ and we assume that the Lam\'e constants $\lambda$ and $\mu$ satisfy the conditions $$\mu >0,\quad \lambda +2\mu >0. \label{lame coefficient}$$ Moreover, $a(x) \in L^{\infty }(\Omega )$ is a nonnegative real function, $f$ is in $(L_{\rm loc}^2(\mathbb{R}_{+},L^2(\Omega ) ) ) ^3$ and $g(\partial _tu)=(g_1(\partial _tu_1),g_2(\partial _tu_2), g_3(\partial _tu_3)),$ where $g_i:\mathbb{R}\to \mathbb{R}$ is a continuous monotone increasing function satisfying $g_i(0)=0$ and the following growth assumption: $$c_1s^2\leq g_i(s) s\leq c_2s^2,\quad | s| \geq 1,\quad \text{for }i=1,2,3, \label{g near infinity}$$ with $c_1,c_2>0$. We can find applications for this system in geophysics and seismic waves propagation. In the case $\lambda +\mu =0$ we obtain a vector wave equation and we aim in this article to generalize some well known results for the wave equation. In this framework, due to the nonlinear semi-group theory, it is well known that, for every $\varphi =(\varphi _1,\varphi _2)\in \mathcal{H}=( H_0^1(\Omega ) ) ^3\times (L^2(\Omega ) ) ^3$, the system \eqref{systnon} admits a unique global solution $u(t,x)$ such that $$u\in C^{0}(\mathbb{R}_{+},(H_0^1(\Omega ) ) ^3) \cap C^1(\mathbb{R}_{+},(L^2(\Omega )) ^3) .$$ The energy of $u$ at time $t$ is defined by $$E_{u}(t) =\frac{1}{2}\int_{\Omega }(\mu | \nabla u| ^2+(\lambda +\mu ) | \operatorname{div} u| ^2+| \partial _tu| ^2)(t,x)dx,$$ and the following energy functional law holds \begin{aligned} &E_{u}(t) +\int_s^t\int_{\Omega }a(x) g(\partial _tu(\sigma ,x) ) \cdot \partial _tu(\sigma ,x) \,dx\,d\sigma \\ &=E_{u}(s)+\int_s^t\int_{\Omega }f(t,x)\cdot \partial _tu(\sigma ,x) \,dx\,d\sigma , \end{aligned} \label{energy identity} for every $t\geq s\geq 0$. For the literature we quote essentially the result of Bisognin et al \cite{biso} which established that the solutions of a system in elasticity theory with a nonlinear localized dissipation decay in an algebraic rate to zero using some energy identities associated with localized multipliers. For more results on the energy decay for the Lam\'e system with linear or nonlinear damping we refer the reader to Alabau and Komornik \cite{alab, alab1}, Alabau \cite{alab2}, Guesmia \cite {gues}, Horn \cite{horn1, horn2} and references therein. We note that the method used in these papers is based on technical multipliers. In the same spirit, we can also quote the work of Guesmia \cite{gues1} for the observability, exact controllability and internal or boundary stabilization of general elasticity systems with variable coefficients depending on both time and space variables. See also the work of Bellassoued \cite{bella} which investigate the decay property of the solutions to the initial-boundary value problem for the elastic wave equation with a local time-dependent nonlinear damping. We note moreover that Burq and Lebeau \cite{ledBurq} introduced the microlocal defect measures attached to sequences of solutions of the Lam\'e system and proved a propagation result when the energy of the longitudinal component goes to zero. Finally, Daoulatli et al \cite{ddk} adapted the Lax-Philips theory, and under the assumption (GC), gave the rate of decay of the local energy for solutions of the Lam\'e system on exterior domain with nonlinear localized damping. Let us indicated that all the result above are without external force and no result seems to be known when $f\neq 0$. We specially mention the result of Daoulatli \cite{daou1}, which study the behavior of the energy of solutions of the wave equation with localized damping and an external force on compact Riemannian manifold with boundary. The main purpose of this work is to give the behavior of the energy of solutions of \eqref{systnon}. First we recall the following definition. \begin{definition} \label{def1.1} \rm We will call \textit{generalized bicharacteristic path} any curve which consists of generalized bicharacteristics of the principal symbol $p$ (where $p(t,x;\tau ,\xi )=(\mu | \xi | ^2-\tau ^2) ^2((\lambda +2\mu ) | \xi | ^2-\tau ^2)$), with possibility of moving from a characteristic manifold to another, at each point of $T^{\ast }(\partial \Omega )$, in the way indicated in \cite{ddk}. \end{definition} \begin{remark} \label{rmk1.1} \rm A \textit{generalized} geodesic path is constituted of segments living in $\Omega$, that intersect the boundary transversally (at hyperbolic points for $p_{L}(t,x;\tau ,\xi )=c_{L}^2| \xi | ^2-\tau ^2$ or $p_{T}(t,x;\tau ,\xi )=c_{T}^2| \xi | ^2-\tau ^2$ (where $c_{L}=\sqrt{\lambda +2\mu }$ and $c_{T}=\sqrt{\mu }$ ), or tangentially (at diffractive points). These segments may be connected to arcs of curves living on $\partial \Omega$ which are projections of glancing rays associated to $p_{L}$ or $p_{T}$. The projection of such a generalized bicharacteristic path on $\bar{\Omega}$ will be called a \textit{generalized geodesic path}. \end{remark} \begin{definition} \label{def1.2} \rm Let $\omega$ be an open subset of $\Omega$, $T>0$ and consider the following assumption: \begin{itemize} \item[(GC)] every generalized geodesic path of $\Omega$, issued at $t=0$, meets $\mathbb{R}_{+}\times \omega$ between the limits $0$ and $T$. \end{itemize} We shall relate the open subset $\omega$ with the damper $a$ by $\omega =\{ x\in \Omega:a(x)>\mu >0\}$. \end{definition} Before stating the main result of this paper, we will define some functions. According to \cite{las-tat} there exists a concave continuous, strictly increasing functions $h_i$ $(i=1,2,3)$, linear at infinity with $h_i(0)=0$ such that $$h_i(g_i(s)s)\geq \epsilon _0(| s| ^2+| g_i(s)| ^2) ,\quad |s| \leq \eta , \label{hi inequality}$$ for some $\epsilon _0$, $\eta >0$. For example when $g_i$ is superlinear, odd and the function $s\longmapsto \sqrt{s}g_i(\sqrt{s})$ is convex, then $h_i^{-1}(s)=\sqrt{s}g_i(\sqrt{s})$ when $| s| \leq \eta$. For further information on the construction of a such function we refer the interested reader to \cite{daou2, mid, las-tat}. With this function, we define $$h(s)=s+h_0(s),\quad \text{where }h_0(s) =\overset{3}{ \underset{i=1}{\sum }}m_a(\Omega _{T})h_i(\frac{s}{m_a(\Omega _{T})}), \label{fonction h}$$ for $s\geq 0$, $dm_a=a(x)\,dx\,dt$ and $\Omega _{T}=(0,T)\times \Omega$. In this article, we show that under the assumption (GC) we obtain the following observability inequality: Non-autonomous observability inequality: There exists a constant $T>0$ such that the solution $u(t,x)$ to the nonlinear problem \eqref{systnon} with initial data $\varphi =(\varphi _0,\varphi_1)$ satisfies $E_{u}(t)\leq C_{T}h\Big(\int_t^{t+T}\int_{\Omega }a(x)g(\partial _tu)\cdot \partial _tu+| f(\sigma ,x)| ^2\,dx\,d\sigma\Big) ,$ for every $t\geq 0$. From the observability inequality above, we infer that the behavior of the energy depends on $\| f(t,x) \| _{L^2(\Omega ) }$. More precisely, we will prove that this behavior is governed by a forced differential equation and depends on $\Gamma (t) =2\Big(\| f(t,.) \| _{L^2(\Omega ) }^2+\psi ^{\ast }(\| f( t,.) \| _{L^2(\Omega ) }) \Big) ,$ where $\psi ^{\ast }$ is the convex conjugate of the function $\psi$, defined by $\psi (s) =\begin{cases} \frac{1}{2T}h^{-1}(\frac{s^2}{8C_{T}e^{T}}) & s\in \mathbb{R}_{+}, \\ +\infty & s\in \mathbb{R}_{-}^{\ast }, \end{cases}$ with $C_{T}\geq 1$ and $T>0$. More precisely we have the following theorem. \begin{theorem} \label{thm1} Let the function $h$ be defined by \eqref{fonction h}. We assume that the assumption {\rm (GC)} holds and $\Gamma (t) =2\Big(\| f(t,.) \| _{L^2(\Omega ) }^2+\psi ^{\ast }(\| f( t,.) \| _{L^2(\Omega ) }) \Big) \in L_{\rm loc}^1(\mathbb{R}_{+}) .$ Let $u(t)$ be the solution to \eqref{systnon} with initial condition $(\varphi _0,\varphi _1) \in \mathcal{H}$. Then $$E_{u}(t) \leq 2e^{T}(S(t-T)+\int_{t-T}^t\Gamma (s) ds) ,\quad t\geq T, \label{energy bound theorem}$$ where $S(t)$ is the positive solution of the ordinary differential equation $$\frac{dS}{dt}+\frac{1}{4T}h^{-1}(\frac{1}{K}S) =\Gamma ( t) ,\quad S(0) =E_{u}(0) , \label{sharp ODE}$$ with $K\geq 2C_{T}$. Moreover, \begin{itemize} \item If there exists $C>0$, such that $\int_{t-T}^t\Gamma (\tau ) d\tau \leq C$, for every $t\geq T$. Then $E_{u}( t)$ is bounded. \item If $\int_{t-T}^t\Gamma (\tau ) d\tau \to 0$ as $t\to +\infty$, and if $E_{u}(t)$ admits a limit at infinity, then the limit is zero. \item If $\Gamma \in L^1(\mathbb{R}_{+})$, then $E_{u}(t) \to 0$ as $t\to +\infty$. \item If $\int_{t-T}^t\Gamma (\tau ) d\tau \to +\infty$ as $t\to +\infty$, then $S(t)\to +\infty$ as $t\to +\infty$. \end{itemize} \end{theorem} We discuss now the methods used for establishing the main result. We note that the present work is compared to the work of \cite{daou1} and \cite{ddk}. Here, we follow the same program and we study the behavior of the energy for the Lam\'e system with Dirichlet boundary condition in a bounded domain and by adding the external force. We consider the notion of bicharacteristic path and we adapt for our context a propagation result for the microlocal defect measures attached to sequences of solutions of \eqref{systnon}. We deduce then a nonlinear observability estimate which is needed to prove Theorem \ref{thm1}. \section{Proof of the main result} Before presenting the proof of our main theorem, we introduce some notation and recall some results from the literature. \begin{proposition} \label{prop2.1} Let $u$ be a solution of \eqref{systnon} with initial data in the energy space. Then $$E_{u}(t) \leq (1+\frac{1}{\epsilon })e^{\epsilon ( t-s) }\Big(E_{u}(s) +\frac{1}{\epsilon } \int_s^t\int_{\Omega }| f(\sigma ,x)| ^2\,dx\,d\sigma \Big) , \label{energy bound}$$ for every $\epsilon >0$ and for every $t\geq s\geq 0$. \end{proposition} \begin{proof} Let $t\geq s\geq 0$. From the energy identity \eqref{energy identity}, we infer that $E_{u}(t) \leq E_{u}(s) +\int_s^t\int_{\Omega }f(t,x)\cdot \partial _tu(\sigma ,x) \,dx\,d\sigma .$ Using Young's inequality, we obtain $E_{u}(t) \leq E_{u}(s) +\frac{1}{\epsilon } \int_s^t\int_{\Omega }| f(\sigma ,x)|^2\,dx\,d\sigma +\epsilon \int_s^tE_{u}(\sigma )d\sigma ,$ for every $\epsilon >0$. Now Gronwall's inequality gives $E_{u}(t) \leq e^{\epsilon (t-s) }\Big(E_{u}( s) +\frac{1}{\epsilon }\int_s^t\int_{\Omega }| f(\sigma ,x)| ^2\,dx\,d\sigma \Big) .$ \end{proof} By analogy with \cite[Proposition 5.1]{ddk}, we obtain the following result. \begin{proposition}\label{propagation} Let $(u_n)$ be a bounded sequence of solutions of the linear Lam\'e system $$\begin{gathered} \partial _t^2u_n-\Delta _eu_n=0 \quad \text{in }\mathbb{R}_{+}\times \Omega , \\ u_n=0 \quad \text{on }\mathbb{R}_{+}\times \partial \Omega , \\ (u_n(0,x) ,\partial _tu_n(0,x) ) =\varphi _n(x) \quad \text{in }\Omega . \end{gathered} \label{linear un}$$ with initial data in $\mathcal{H}$, weakly converging to $0$ in $\mathcal{H}$. We assume that {\rm (GC)} holds and that $\partial _tu_n\to 0$ in $(L_{\rm loc}^2(]0,T[\times \omega ) ) ^3$. Then there exists a subsequence (still denoted $(u_n) )$ such that $u_n\to 0$ in $( H_{\rm loc}^1(]0,T[,H^1(\Omega ) )) ^3$. \end{proposition} Before giving the proof of Proposition \ref{propagation}, we recall some facts on microlocal defect measures associated to bounded sequences of solutions to the linear Lam\'e system with Dirichlet boundary conditions. We give them within their original statement \cite{ddk}, and we note that (with obvious modifications of their proofs) all these results remain valid in our situation. We consider the linear Lam\'e system on $\mathbb{R}\times \Omega$. $$\begin{gathered} \partial _t^2u-\Delta _eu=0, \quad \text{in }\mathbb{R}\times \Omega , \\ u=0 \quad \text{on }\mathbb{R}\times \partial \Omega , \\ (u(0,x) ,\partial _tu(0,x) =(\varphi _1(x),\varphi _2(x)) \in (H_0^1(\Omega )) ^3\times (L^2(\Omega ))^3. \end{gathered} \label{linear lame}$$ We decompose first the solution of system \eqref{linear lame} into $$u=u_{L}+u_{T}, \label{ul+ut}$$ where the longitudinal wave $u_{L}$ and the transversal wave $u_{T}$, respectively, satisfies the wave system $$\begin{gathered} (\partial _t^2-c_{L}^2\Delta )u_{L}=0,\quad \operatorname{rot}u_{L}=0, \\ (\partial _t^2-c_{T}^2\Delta )u_{T}=0,\quad \operatorname{div}u_{T}=0, \\ u=u_{L}+u_{T}=0\quad \text{on } \mathbb{R}\times \partial \Omega , \end{gathered}$$ with $c_{L}=\sqrt{\lambda +2\mu }$ and $c_{T}=\sqrt{\mu }$. Moreover, if $(u_n) _n$ is a bounded sequence of solutions of \eqref{linear lame} weakly converging to $0$ in $(H_{\rm loc}^1(\mathbb{R} _t,H^1(\Omega ) ) ) ^3$, the sequences $(u_{n_{L}})$ and $(u_{n_{T}})$ are also of bounded energy and weakly converging to $0$ in $(H_{\rm loc}^1(\mathbb{R}_t,H^1(\Omega ) ) ) ^3$. In this way, according to \cite{ledBurq}, we can attach to $(u_{n_{L}})$ (resp. $(u_{n_{T}})$) a microlocal defect measure $\nu _{L}$ (resp. $\nu _{T}$). These measures are orthogonal in the measure theory sense (see \cite[Proposition 4.4]{ledBurq} or \cite[Lemme 3.30]{duy}). In addition, $\nu _{L}$ is supported in the characteristic set \begin{align*} \operatorname{Char}\mathcal{L} &= (\operatorname{Char}\mathcal{L})_{\Omega } \cup (\operatorname{Char}\mathcal{L})_{\partial \Omega } \\ &=\{(t,x,\tau ,\xi ):x\in \Omega ,\,t>0,\,c_{L}^2|\xi |^2-\tau ^2=0\} \\ &\quad\cup \{(t,y,\tau ,\eta ):y\in \partial \Omega ,\,t>0,\,r_{L}:=\tau ^2-c_{L}^2| \eta | ^2\geq 0\}, \end{align*} and $\nu _{T}$ is supported in $\operatorname{Char}\mathcal{T=}\{(t,x,\tau ,\xi );\,x\in \Omega ,\,t>0,\, c_{T}^2|\xi |^2-\tau ^2=0\}.$ This fact is known as the elliptic regularity theorem for the m.d.m's. Let us now analyze the propagation properties of the measures $\nu _{L}$ and $\nu _{T}$. $\$In the interior, i.e. in $T^{\ast }(\mathbb{R}\times \Omega )$, we are in presence of two waves which propagate independently, so we have at our disposal the classical measures propagation theorem of \cite{ge1}. Near the boundary $\partial \Omega$, we have to take into account, the nature of the bicharacteristics hitting $\partial \Omega$. Take $\rho$ in $\operatorname{Char}P_{\partial \Omega }=\{(t,y,\tau ,\eta );\,y\in \partial \Omega ,\,t>0,\,r_{T}:=\tau ^2-c_{T}^2| \eta | ^2\geq 0\}$; for $r_{L,T}=r_{L,T}(\rho )\geq 0$, we denote $\gamma _{L,T}^{-}$ (resp. $\gamma _{L,T}^{+}$) the (longitudinal/transversal) incoming (resp. outgoing) bicharacteristic to (resp. from) $\rho$ (this half bicharacteristic does not contain $\rho )$. Following then word by word the argument developed in \cite[proof of Theorem 4]{ledBurq}, we have \begin{proposition} \label{pro-rl} With the notation above, we have \begin{enumerate} \item $r_{L}<0$, $\rho$ is an elliptic point for the longitudinal wave. Hence, $\nu _{L}=0$ near $\rho$ and \begin{itemize} \item[(a)] $\nu _{T}=0$ near $\rho$ if $r_{T}<0$, \ \item[(b)] $\nu _{T}$ propagates from $\gamma _{T}^{-}$ to $\gamma _{T}^{+}$ if $0\leq r_{T}$. \end{itemize} \item $00$, such that the following inequality holds: $$E_{u}(t)\leq C_{T}h\Big(\int_t^{t+T}\int_{\Omega }a(x)g(\partial _tu)\cdot \partial _tu+| f(\sigma ,x)| ^2\,dx\,d\sigma \Big) , \label{observability}$$ for every $t\geq 0,$for every solution $u$ of \eqref{systnon} with initial data in the energy space $\mathcal{H}$, and for every $f$in $(L_{\rm loc}^2(\mathbb{R}_{+},L^2(\Omega ) ) ) ^3$. \end{proposition} \begin{proof} To prove this result we argue by contradiction. We assume that there exist a sequence $(u_n) _n$ solution of \eqref{systnon} with initial data in the energy space, a non-negative sequence $(t_n)_n$and $f_n$ in $(L_{\rm loc}^2(\mathbb{R}_{+},L^2(\Omega ) ) ) ^3$, such that $E_{u_n}(t_n)\geq nh\Big(\int_{t_n}^{t_n+T}\int_{\Omega}a(x)g(\partial _tu_n) \cdot \partial _tu_n+| f_n(\sigma,x)| ^2\,dx\,d\sigma \Big) .$ Moreover, $u_n$ has the following regularity: $u_n\in C\big(\mathbb{R}_{+},(H_0^1(\Omega ) ) ^3\big) \cap C^1\big(\mathbb{R}_{+},(L^2(\Omega ) ) ^3\big) .$ Setting $\alpha _n=(E_{u_n}(t_n) ) ^{1/2}>0$ and $v_n(t,x) =\frac{u_n(t_n+t,x) }{\alpha _n}$. Then $v_n$ satisfies $$\begin{gathered} \partial _t^2v_n-\Delta _ev_n+\frac{1}{\alpha _n}a( x) g(\alpha _n\partial _tv_n) =\frac{1}{\alpha _n} f_n(t_n+t,x), \quad \text{in }\mathbb{R}_{+}\times \Omega , \\ v_n=0 \quad \text{on }\mathbb{R}_{+}\times \partial \Omega , \\ (v_n(0,x) ,\quad \partial _tv_n(0,x) )=\frac{1}{\alpha _n}(u_n(t_n,x) ,\partial _tu_n( t_n,x) ) , \quad \text{in }\Omega . \end{gathered} \label{systemvn}$$ Moreover $E_{v_n}(0) =1$ and $1\geq \frac{n}{\alpha _n^2}h\Big(\int_0^{T}\int_{\Omega }a(x)g(\alpha _n\partial _tv_n)\cdot \alpha _n\partial _tv_n+| f_n(t_n+t,x)| ^2\,dx\,dt\Big) .$ Since $h=I+h_0$ and $h_0$ is non-negative and increasing function and from the inequality above, we infer that $$\int_0^{T}\int_{\Omega }| \frac{1}{\alpha _n} f_n(t_n+t,x)| ^2\,dx\,dt\leq \frac{1}{n}\underset{n\to +\infty }{\longrightarrow }0 \label{estimate1}$$ and $$\Big[ I+\overset{3}{\underset{i=1}{\sum }}m_a(\Omega _{T}) h_i\circ \frac{I}{m_a(\Omega _{T}) }\Big] \Big( \int_0^{T}\int_{\Omega }a(x)g(\alpha _n\partial _tv_n)\cdot \alpha _n\partial _tv_n\,dx\,dt\Big) \leq \frac{\alpha _n^2}{n}.$$ Re-using the fact that the function $h_0$ is non-negative gives $$\alpha _n^{-1}\int_0^{T}\int_{\Omega }a(x)g(\alpha _n\partial _tv_n)\cdot \partial _tv_n\,dx\,dt\underset{n\to +\infty }{ \longrightarrow }0 \label{ag tend}$$ and $$h_i\Big(\frac{1}{m_a(\Omega _{T}) }\int_0^{T} \int_{\Omega }a(x)g_i(\alpha _n\partial _tv_n)\alpha _n( \partial _tv_n) _i\,dx\,dt\Big) \leq \frac{\alpha _n^2}{ nm_a(\Omega _{T}) },\quad i=1,2,3. \label{h0ag}$$ Denote $\Omega _{1,i}=\{ (t,x) \in [ 0,T] \times \Omega :| \alpha _n(\partial _tv_n) _i( t,x) | <\mu \}$ and $\Omega _{2,i}=\Omega _{T}\backslash \Omega _{1,i}$. Since $g_i$ has a linear behavior on $\{ | s| \geq \eta \}$, using \eqref{ag tend}, we infer that $$\| a(x)(\partial _tv_n) _i\| _{L^2(\Omega _{2,i}) }^2\leq c_1\alpha _n^{-1}\int_0^{T}\int_{\Omega }a(x) g(\alpha _n\partial _tv_n) \cdot \partial _tv_n\,dx\,d\tau \underset{ n\to +\infty }{\longrightarrow }0. \label{omega2i}$$ Moreover, $h_i$ is concave, then using (the reverse) Jensen's inequality \begin{align*} &h_i\Big(\frac{1}{\mathfrak{m}_a(\Omega _{T}) } \int_0^{T}\int_{\Omega }a(x) g_i(\alpha _n\partial _tv_n) \alpha _n(\partial _tv_n) _i\,dx\,d\tau \Big) \\ &\geq \frac{1}{\mathfrak{m}_a(\Omega _{T}) }\int_{\Omega _{T}}h_i(g_i(\alpha _n\partial _tv_n) \alpha _n(\partial _tv_n) _i) d\mathfrak{m}_a, \end{align*} which gives $\alpha _n^{-2}\int_{\Omega _{1,i}}h_i(g_i(\alpha _n\partial _tv_n) \alpha _n(\partial _tv_n) _i) d\mathfrak{m}_a\leq \frac{1}{n}.$ Therefore, from \eqref{hi inequality} we obtain $\int_{\Omega _{1,i}}a(x) [ \alpha _n^{-2}| g_i(\alpha _n(\partial _tv_n) _i) | ^2+| (\partial _tv_n) _i|^2] \,dx\,dt\underset{n\to +\infty }{\longrightarrow }0.$ Combining the estimate above with \eqref{omega2i} we obtain $$\| a(x) \partial _tv_n\| _{( L^2(\Omega _{T}) ) ^3}\underset{n\to +\infty }{ \longrightarrow }0 \label{adtvn}$$ and we conclude that $$\| \frac{1}{\alpha _n}a(x) g(\alpha _n\partial _tv_n) \| _{(L^2(\Omega _{T}) ) ^3}\underset{n\to +\infty }{\longrightarrow }0. \label{estimate2}$$ Hence, passing to the limit in \eqref{systemvn}, we see that the weak limit $v\in (H^1([ 0,T] \times \Omega ) ) ^3$ satisfies the system $$\begin{gathered} \partial _t^2v-\Delta _ev=0 \quad \text{in }] 0,T[ \times \Omega , \\ v=0\quad \text{on }] 0,T[ \times \Omega , \\ (v(0,x) ,\partial _tv(0,x) )=\psi (x), \quad \text{in }\Omega\,. \end{gathered}$$ Moreover, we obtain $$a(x)\partial _tv=0,\quad \text{on }\Omega _{T}. \label{v nul}$$ Now, let $w_n$ be the solution of the system $$\begin{gathered} \partial _t^2w_n-\Delta _ew_n=0, \quad \text{in }\mathbb{R}_{+}\times \Omega , \\ w_n=0,\quad \text{on }\mathbb{R}_{+}\times \Omega , \\ (w_n(0,x) ,\quad \partial _tw_n(0,x)) =\frac{1}{\alpha _n}(u_n(t_n,x) ,\partial _tu_n(t_n,x) ) ,\quad \text{in }\mathbb{R}_{+}\times \Omega . \end{gathered}$$ It is clear that the sequence $(w_n) _n$ is bounded in $(H_{\rm loc}^1([ 0,T] \times \Omega ) )^3$; moreover, by the hyperbolic energy inequality, \eqref{estimate1} and \eqref{estimate2} we infer that $$\sup_{0\leq t\leq T} E_{v_n-w_n}(t) \leq C(T) \| \frac{1}{\alpha _n}a(x) g(\partial_tv_n) -\frac{1}{\alpha _n}f_n(t_n+t,x)\| _{L^2(\Omega _{T}) }^2\underset{n\to +\infty }{ \longrightarrow }0. \label{linearisable}$$ Consequently, thanks to \eqref{adtvn}, we deduce that $$\| a(x) \partial _tw_n\| _{(L^2(\Omega _{T}) ) ^3}\underset{n\to +\infty }{ \to }0, \label{wn nul}$$ to obtain a contradiction we use the following result for which we postpone its proof. \begin{proposition}\label{observability conservative} We assume that the assumption {\rm (GC)} holds. Then there exists $\alpha _{T}>0$, such that the inequality $$E_{w}(0) \leq \alpha _{T}\Big(\int_0^{T}\int_{\omega }| \partial _tw| ^2\,dx\,ds\Big) \label{observability consevative formula}$$ holds for every solution $w$ of $$\begin{gathered} \partial _t^2w-\Delta _ew=0, \quad \text{in }\mathbb{R}_{+}\times \Omega , \\ w=0, \quad \text{on }\mathbb{R}_{+}\times \partial \Omega , \\ (w(0,x) ,\partial _tw(0,x) ) =(w_0(x),w_1(x)) , \quad \text{in }\Omega \end{gathered} \label{lame linear}$$ with initial data in the energy space $\mathcal{H}$. \end{proposition} Now, using \eqref{wn nul} and Proposition \ref{observability conservative}, we obtain $1=E_{v_n}(0) =E_{w_n}(0) \leq \alpha _{T}\int_0^{T}\int_{\omega }| \partial _tw_n| ^2\,dx\,dt\underset{n\to +\infty }{\longrightarrow }0\,,$ and this concludes the Proof of Proposition \ref{propo observability}. \end{proof} \begin{proof}[Proof of Proposition \ref{observability conservative}] We argue by contradiction: we suppose the existence of a sequence $(w_n)$, solutions of \eqref{lame linear} such that $\int_0^{T}\int_{\omega }| \partial _tw_n| ^2\,dx\,dt\leq \frac{E_{w_n}(0) }{n}.$ Denote $\alpha _n=E_{w_n}(0) ^{1/2}$ and $z_n=\frac{w_n}{\alpha _n}$. Moreover $z_n$ satisfies $$\begin{gathered} \partial _t^2z_n-\Delta _ez_n=0, \quad \text{in }\mathbb{R}_{+}\times \Omega , \\ z_n=0, \quad \text{in }\mathbb{R}_{+}\times \partial \Omega , \\ E_{z_n}(0)=1, \quad \int_0^{T}\int_{\omega }| \partial _tz_n| ^2\,dx\,dt\leq \frac{1}{n}. \end{gathered} \label{zn}$$ The sequence $z_n$ is bounded in $C^{0}([ 0,T] ,(H^1(\Omega ) ) ^3)\cap C^1([ 0,T] ,( L^2(\Omega ) ) ^3)$, then, it admits a subsequence, still denoted by $z_n$, that is weakly-* convergent in the space $L^{\infty }([ 0,T] ,(H^1(\Omega ) ) ^3) \cap W^{1,\infty}((0,T) ,(L^2(\Omega ) ) ^3)$. In this way, $z_n\rightharpoonup z$ in $(H^1([ 0,T]\times \Omega ) ) ^3$. Passing to the limit in the equation satisfied by $z_n$ we obtain $$\begin{gathered} \partial _t^2z-\Delta _ez=0, \quad \text{in }] 0,T[ \times \Omega , \\ z=0 \quad \text{in }] 0,T[ \times \partial \Omega , \\ \partial _tz=0 \quad \text{on }] 0,T[ \times \omega . \end{gathered} \label{prolongement unique}$$ We need to check that the trivial solution, $v=0$, is the only solution of \eqref{prolongement unique} in $C^{0}([ 0,T] ,(H^1(\Omega ) ) ^3)\cap C^1([ 0,T] ,(L^2(\Omega ) )^3)$. For this, we identify the function $z$ solution of \eqref{prolongement unique} with its initial data $\phi \in \mathcal{H}$, and we consider the space $G=\{\phi \in\mathcal{H},z\text{ is a solution of \eqref{prolongement unique}}\}$. Every $z$ in $G$ is smooth on $]0,T[\times \omega$; therefore, according to the geometric control condition and the result of \cite{yama} on propagation of singularities, $G$\ is constituted of smooth functions. Moreover, $G$ is obviously closed in $\mathcal{H}$, and we deduce that it is of finite dimension. On the other hand, $\partial /\partial t$ operates on $G$, so it admits an eigenvalue $\lambda$, and there exists a nonzero function $z_0(x)$ on $\Omega$ such that $\Delta _ez_0=\lambda z_0$, $z_0\equiv 0$ on $\omega$, $z_0=0$ on $\partial \Omega$; and this is impossible by unique continuation property of $\Delta _e$ (see, for instance, \cite{dehrob}). Now, we multiply $E_{z_n}(s)$ by $\varphi (s)$, with $\varphi \in C_0^{\infty }(] 0,T[ )$, $\varphi =1$ on $] \varepsilon ,T-\varepsilon [$, $\varphi \geq 0$, and we integrate. This gives \begin{align*} &\int_0^{T}\varphi (s) E_{z_n}(s)ds \\ &= \frac{1}{2}\int_0^{T}\int_{\Omega }(\mu \varphi (s) | \nabla z_n| ^2+(\lambda +\mu ) \varphi (s) | \operatorname{div}z_n| ^2+\varphi ( s) | \partial _tz_n| ^2)(s,x)\,dx\,ds. \end{align*} Proposition \ref{propagation} and \eqref{zn} imply that the second member approaches $0$ as $n\to +\infty$. Using the fact that $E_{z_n}(s)=1$, we obtain $T-2\varepsilon \to 0$ as $n\to +\infty$ and this gives a contradiction. \end{proof} We recall now the following lemma due to \cite{daou1} which is useful to determine the behavior of the energy. \begin{lemma}\label{lemma las tat} Let $T>0$ and \begin{itemize} \item $\Gamma \in L_{\rm loc}^1(\mathbb{R}_{+})$ and non-negative. Setting $\delta (t)=\int_t^{t+T}\Gamma (s) ds$, for $t\geq 0$. \item $W(t)$ be a non-negative function for $t\in \mathbb{R}_{+}$. Moreover we assume that there exists a positive, monotone, increasing function $\alpha$ with $\alpha (0) \geq 1$, such that $W(t) \leq \alpha (t-s) \Big[ W(s)+\int_s^t\Gamma (\sigma ) d\sigma \Big] , \quad\text{for every }t\geq s\geq 0.$ \item Suppose that $\ell$ and $I-\ell :\mathbb{R}_{+}\to \mathbb{R}$ are increasing functions with $\ell (0)=0$ and $$W((m+1) T) +\ell \{ W(mT) +\delta (mT) \} \leq W(mT) +\delta (mT) , \label{lemma las tat inequality}$$ for $m=0,1,2,\dots$, where $\ell (s)$ does not depend on $m$. \end{itemize} Then $W(t) \leq \alpha (T) \Big(S(t-T) +\int_{t-T}^t\Gamma (s) ds\Big) ,\quad \forall t\geq T,$ where $S(t)$ is the non negative solution of the differential equation $$\frac{dS}{dt}+\frac{1}{T}\ell (S) =\Gamma (t) ;\quad S(0)=W(0). \label{Ode lema}$$ Moreover, we assume that $\ell$ is continuous, strictly increasing and $\lim_{s\to +\infty } \ell (s) =+\infty$ \begin{itemize} \item If there exists $C>0$, such that $\int_{t-T}^t\Gamma (\tau ) d\tau \leq C$, for every $t\geq T$. Then $S(t)$ is bounded. \item If $\int_{t-T}^t\Gamma (\tau ) d\tau \to 0$ as $t\to +\infty$, and if $S(t)$ admits a limit at infinity, then this limit is zero. \item If $\Gamma \in L^1(\mathbb{R}_{+})$, then $S(t)\to 0$ as $t\to+\infty$. \item We assume that $\lim_{s\to +\infty }(I-\ell) (s) =+\infty$, then if $\int_{t-T}^t\Gamma (\tau ) d\tau \to +\infty$ as $t\to +\infty$, we have $S(t)\to +\infty$ as $t\to +\infty$. \end{itemize} \end{lemma} We can now proceed the proof of the main result of this article. \begin{proof}[Proof of Theorem \ref{thm1}] We assume that the assumption {\rm (GC)} holds. Let $u$ be a solution of \eqref{systnon} with initial data in the energy space. Then according to Proposition \ref{propo observability}, we have $$E_{u}(t) \leq C_{T}h\Big(\int_t^{t+T}\int_{\Omega }a( x) g(\partial _tu) \cdot \partial _tu\,dx\,d\sigma +\int_t^{t+T}\int_{\Omega }| f(s,x) | ^2\,dx\,ds\Big) , \label{proof 1}$$ for some $C_{T}\geq 1$. The energy identity \eqref{energy identity} gives $$\int_t^{t+T}\int_{\Omega }a(x) g(\partial _tu) \cdot \partial _tu\,dx\,d\sigma \leq E_{u}(t) -E_{u}( t+T) +\int_t^{t+T}\int_{\Omega }| f(\sigma ,x) \cdot \partial _tu| \,dx\,d\sigma . \label{proof 2}$$ Let $\psi$ be defined by $\psi (s) =\begin{cases} \frac{1}{2T}h^{-1}(\frac{s^2}{8C_{T}e^{T}}) & s\in \mathbb{R}_{+}, \\ +\infty & s\in \mathbb{R}_{-}^{\ast }. \end{cases}$ It is clear that $\psi$ convex is and proper function. Hence, we can apply Young's inequality \cite{rockfellar} \begin{align*} \int_t^{t+T}\int_{\Omega }| f(\sigma ,x) \cdot \partial _tu| \,dx\,d\sigma &\leq \int_t^{t+T}\| f( \sigma ,.) \| _{L^2}\| \partial _tu(\sigma ,.) \| _{L^2}d\sigma \\ &\leq \int_t^{t+T}\psi ^{\ast }(\| f(\sigma ,.) \| _{L^2}) +\psi (\| \partial _tu(\sigma ,.) \| _{L^2}) d\sigma , \end{align*} where $\psi ^{\ast }$ is the convex conjugate of the function $\psi$, defined by $\psi ^{\ast }(s) =\sup_{y\in \mathbb{R}} [ sy-\psi (y) ]$ Using the energy inequality \eqref{energy bound} and the observability estimate \eqref{proof 1}, we infer that $\int_t^{t+T}\psi (\| \partial _tu(\sigma ,.) \| _{L^2}) d\sigma \leq \frac{1}{2}\Big( \int_t^{t+T}\int_{\Omega }g(\partial _su) \cdot \partial _sud\mathfrak{m}_a+\int_t^{t+T}\int_{\Omega }| f( s,x) | ^2\,dx\,ds\Big)$ then \eqref{proof 2} gives \begin{align*} &\int_t^{t+T}\int_{\Omega }a(x) g(\partial _tu) \cdot \partial _tu\,dx\,d\sigma \\ &\leq 2\Big(E_{u}(t) -E_{u}(t+T) +\int_t^{t+T}\int_{\Omega }| f( s,x) | ^2\,dx\,ds +\int_t^{t+T}\psi ^{\ast }(\| f(\sigma ,.) \| _{L^2}) d\sigma \Big)\,. \end{align*} The inequality above combined with the observability estimate \eqref{proof 1} and the fact $h=I+\mathfrak{m}_a(\Omega _{T}) h_0\circ {\frac{I}{\mathfrak{m}_a(\Omega _{T}) }}$ is increasing, gives $E_{u}(t) \leq C_{T}h\Big(4\Big(E_{u}(t) -E_{u}(t+T) +2\int_t^{t+T}\| f(\sigma ,.) \| _{L^2}^2+\psi ^{\ast }(\| f(\sigma,.) \| _{L^2}) d\sigma \Big) \Big) .$ Setting $\Gamma (s) =2(\| f(\sigma ,.)\| _{L^2}^2+\psi ^{\ast }(\| f(s,.) \| _{L^2}) ) .$ Therefore, $E_{u}(t) +\int_t^{t+T}\Gamma (s) ds\leq Kh\Big( 4\Big(E_{u}(t) -E_{u}(t+T) +\int_t^{t+T}\Gamma (s) \,dx\,ds\Big) \Big) ,$ with $K\geq 2C_{T}$. Setting $\theta (t) =\int_t^{t+T}\Gamma (s) ds$. Thus $$E_{u}(t+T) +\frac{1}{4}h^{-1}\Big(\frac{1}{K}( E_{u}(t) +\theta (t) ) \Big) \leq E_{u}(t) +\theta (t) , \label{observability final}$$ for every $t\geq 0$. Take $t=mt$, $m\in\mathbb{N}$, $E_{u}((m+1) T) +\frac{1}{4}h^{-1}\Big(\frac{1}{K} (E_{u}(mT) +\theta (mT) ) \Big) \leq E_{u}(mT) +\theta (mT) .$ Setting $W(t) =E_{u}(t)$, $\ell (s) =\frac{1}{4}h^{-1}\circ \frac{I}{K}$ and $$\Gamma (s) =2(\| f(s,.) \|_{L^2}^2+\psi ^{\ast }(\| f(s,.) \| _{L^2}) ) .$$ It is clear that the functions $\ell$ and $I-\ell$ are increasing on the positive axis and $\ell (0) =0$. The function $\Gamma \in L_{\rm loc}^1(\mathbb{R}_{+})$ and non-negative on $\mathbb{R}_{+}$. According to lemma \ref{lemma las tat}, we obtain $E_{u}(t) \leq 2e^{T}\Big(S(t-T) +\int_{t-T}^t\Gamma (s) ds\Big) ,\quad \forall t\geq T,$ where $S(t)$ is the solution of the following differential equation $\frac{dS}{dt}+\frac{1}{T}\ell (S) =\Gamma (t), \quad S(0)=W(0).$ The function $\ell$ is continuous, strictly increasing and $\lim_{s\to +\infty } \ell (s) =+\infty$, therefore using Lemma \ref{lemma las tat}, we infer that \begin{itemize} \item If there exists $C>0$, such that $\int_{t-T}^t\Gamma (\tau ) d\tau \leq C$ for every $t\geq T$. Then $S(t)$ is bounded, which gives $E_{u}(t)$ is bounded. \item We assume that $E_{u}(t) \to \alpha \geq 0$ as $t\to +\infty$ and $\int_{t-T}^t\Gamma (\tau ) d\tau \to 0$ as $t\to +\infty$. Consequently \eqref{observability final} gives $$E_{u}(t) +\ell \Big(E_{u}(t-T) +\int_{t-T}^t\Gamma (\tau ) d\tau \Big) \leq E_{u}(t-T) +\int_{t-T}^t\Gamma (\tau ) d\tau ,$$ for every $t\geq T$. Passing to the limit in the inequality above, we infer that $\ell (\alpha ) =0$. Which means $\alpha =0$. Therefore, if $E_{u}(t)$ admits a limit at infinity, then the limit is zero. \item If $\Gamma \in L^1(\mathbb{R}_{+})$, then $S(t)\to 0$ as $t\to +\infty$, which gives $E_{u}(t)\to 0$ as $t\to +\infty$. \item Since $h^{-1}$ is linear at infinity, therefore $(I-\ell )$ is positive and linear at infinity, which gives $\lim_{s\to +\infty } (I-\ell ) (s) =+\infty$. Thus, if $\int_{t-T}^t\Gamma (\tau ) d\tau \to +\infty$ as $t\to +\infty$, we obtain $S(t) \to +\infty$ as $t\to +\infty$. \end{itemize} \end{proof} \section{Applications} \subsection*{Preliminary results} In the following proposition we give a result on the behavior of the solutions of \eqref{sharp ODE} due to \cite{daou1}. \begin{proposition}\label{lemma ode} Let $p$ a differentiable, strictly increasing function on $\mathbb{R}_{+}$ with $p(0) =0$. We assume that there exists $m_1>0$ such that, $p(x) \leq m_1x$ for every $x\in [ 0,\eta ]$ for some $0<\eta <<1$ and that the property $$p(Kx) \geq mp(K) p(x) , \label{Lem:p lower bound}$$ holds, for some $m>0$ and for every $(K,x) \in [ 1,+\infty[ \times\mathbb{R}_{+}$. We suppose that $\Gamma \in C^1(\mathbb{R}_{+})$ and non-negative. (1) Let $\tilde{p}$ be a increasing function vanishing at the origin. Let $S$ satisfy the differential equation $$\frac{dS}{dt}+\tilde{p}(S) =\Gamma (t) ,\quad S(0) \geq 0. \label{ode positivity}$$ Then $S(t) \geq 0$ for every $t\geq 0$. (2) Let $S$ be a non-negative function, satisfying the differential inequality $\frac{dS}{dt}+p(S) \leq \Gamma (t) ,\quad S(0) \geq 0.$ \begin{itemize} \item[(a)] If $\Gamma (t) =0$, for every $t\geq 0$, then $S(t) \leq \psi ^{-1}(t)$, for every $t\geq 0$ where $\psi (x) =\int_{x}^{S(0) }\frac{ds}{p(s) }$, $x\in ] 0,S(0)]$. \item[(b)] If $\Gamma (t) >0$, for every $t\geq 0$, and \begin{itemize} \item[(i)] There exist $c>0$ and $\kappa \geq 1$ such that \begin{gather} \frac{d}{dt}p^{-1}(\Gamma (t) ) +c\Gamma ( t) \leq 0,\text{ for every }t\geq 0 , \label{application lemma 1}\\ mp(\kappa ) -\kappa c-1\geq 0,\quad \kappa p^{-1}\circ \Gamma (0) \geq S(0) , \label{application assumption} \end{gather} then $S(t) \leq \kappa \psi ^{-1}(ct)$ for every $t\geq 0$, where $\psi (x) =\int_{x}^{p^{-1}\circ \Gamma (0) }\frac{ds }{p(s) },\quad x\in ] 0,p^{-1}\circ \Gamma (0) ] .$ Noting that in this case we have $p^{-1}\circ \Gamma (t) \leq \psi ^{-1}(ct)$, for every $t\geq 0$. \item[(ii)] There exist $c>0$ and $\kappa \geq 1$ such that $\frac{d}{dt}p^{-1}(\Gamma (t) ) +c\Gamma (t) \geq0$, for every $t\geq 0$ and $mp(\kappa ) -c\kappa -1\geq 0,\quad \kappa p^{-1}\circ \Gamma (0) \geq S(0) ,$ then $S(t) \leq \kappa p^{-1}\circ \Gamma (t)$, for every $t\geq 0$. Noting that in this case we have $p^{-1}\circ \Gamma (t) \geq \psi ^{-1}(ct)$ for every $t\geq 0$, where $\psi (x) =\int_{x}^{p^{-1}\circ \Gamma (0) }\frac{ds }{p(s) },\quad x\in ] 0,p^{-1}\circ \Gamma (0) ] .$ \end{itemize} \end{itemize} \end{proposition} \subsection*{Examples} Setting $\Gamma (t) =2\Big(\| f(t,.) \| _{L^2(\Omega ) }^2+\psi ^{\ast }(\| f( t,.) \| _{L^2(\Omega ) }) \Big) ,$ where $\psi ^{\ast }$ is the convex conjugate of the function $\psi$, defined by $\psi (s) =\begin{cases} \frac{1}{2T}h^{-1}(\frac{s^2}{8C_{T}e^{T}}) & s\in \mathbb{R}_{+} \\ +\infty & s\in \mathbb{R}_{-}^{\ast }, \end{cases}$ and $\psi ^{\ast }(s) =\underset{y\in \mathbb{R}}{\sup }[ sy-\psi (y) ]$. To obtain the rate of decay, we use proposition \ref{lemma ode}. \subsection*{$g_i$ is linearly bounded} We have $h(s) =2s$, then $\psi ^{\ast }\Big(\| f(t,.) \| _{L^2( M) }\Big) \leq C_1\| f(t,.) \|_{L^2(M) }^2,$ for some $C_1>0$. The ODE \eqref{sharp ODE} governing the energy bound reduces to $$\frac{dS}{dt}+CS=\Gamma (t) , \label{equation linear}$$ where the constant $C>0$ and does not depend on $E_{u}(0)$. (1) If there are constants $C_0>0$ and $\theta \in \mathbb{R}$, such that $\Gamma (t) \leq C_0e^{-\theta t}$. We have $\int_{t-T}^te^{-\theta s}ds\leq \begin{cases} | \frac{1}{\theta }| [ e^{| \theta | T}-1] e^{-\theta t} & \theta \neq 0 \\ T & \theta =0 \end{cases}$ for $t\geq T$. Multiply both sides of \eqref{equation linear} by $\exp (Ct)$ and integrate from $0$ to $t$, to obtain \begin{itemize} \item[(a)] $C>\theta$, $E_{u}(t) \leq c(1+E_{u}(0) ) e^{-\theta t}$ for $t\geq 0$, \item[(b)] $C=\theta$, $E_{u}(t) \leq c(1+E_{u}(0) ) (1+t) e^{-\theta t}$ for $t\geq 0$, \item[(c)] $C<\theta$, $E_{u}(t) \leq c(1+E_{u}(0) ) e^{-Ct}$ for $t\geq 0$. \end{itemize} (2) If there are constants $C_0>0$ and $\theta \in\mathbb{R}$, such that $\Gamma (t) \leq C_0(1+t) ^{-\theta }$, then we have $\int_{t-T}^t(1+s) ^{-\theta }ds\leq \begin{cases} T(1+t-T) ^{-\theta } & \theta >0 \\ T(1+t) ^{-\theta } & \theta \leq 0 \end{cases}$ for $t\geq T$. Therefore, $$E_{u}(t) \leq \begin{cases} c(1+E_{u}(0) ) (1+t-T) ^{-\theta } & \theta >0 \\ c(1+E_{u}(0) ) T(1+t) ^{-\theta } & \theta \leq 0 \end{cases} \label{example 2}$$ for $t\geq T$, where $c>0$. \subsection*{The nonlinear case} The rate of decay of the energy depends only on the behavior of $h^{-1}$ near zero. To determine it, we have only to find $00$ such that $S(t) \leq A$, for every $t\geq 0$. We choose $K\geq \max (C_{T},\frac{A}{N_0})$. The ODE \eqref{sharp ODE} governing the energy bound reduces to $\frac{dS}{dt}+C_1h_i^{-1}(\frac{S}{2KC_2}) \leq \Gamma (t) ,$ with $S(0) =E_{u}(0)$. (3) If $\Gamma \in L_{\rm loc}^1(\mathbb{R}_{+})$ and $\int_{t-T}^t\Gamma (\tau ) d\tau \underset{t\to +\infty }{\longrightarrow }+\infty ,$ then $S(t)\to +\infty$ as $t\to +\infty$. Therefore, there exists $t_0>0$ such that $\frac{S(t) }{K}>>1$ for $t\geq t_0$. Since the function $h$ is strictly increasing and linear at infinity, then the ODE \eqref{sharp ODE} governing the energy bound reduces to $\frac{dS}{dt}+\frac{C}{K}S\leq \Gamma (t) \quad \text{on }[t_0,+\infty [ ,$ with $S(t_0) \leq E_{u}(0)+\int_0^{t_0}\Gamma (s) ds$. \subsection*{Example 1: Sublinear near the origin} Assume $g_i(s) =s| s| ^{r_0-1}$, $| s| <1$, $r_0\in (0,1)$. We choose $h_i^{-1}(s) =\sqrt{s}g_i^{-1}(\sqrt{s}) =s^{ \frac{1+r_0}{2r_0}}$, for $0\leq s\leq 1$. We have $\psi ^{\ast }\Big(\| f(t,.) \| _{L^2(\Omega ) }\Big) \leq \tilde{C}\Big(\| f(t,.) \| _{L^2(\Omega ) }^{r_0+1}+\| f( t,.) \| _{L^2(\Omega ) }^2\Big) ,$ for some $\tilde{C}>0$. The ODE \eqref{sharp ODE} governing the energy bound reduces to $\frac{dS}{dt}+CS^{(1+r_0) /2r_0}\leq \Gamma (t) ,$ where $C$ is positive and depends on $K$. (1) If there are constants $C_0>0$ and $\theta >0$ such that $\Gamma (t) \leq C_0(1+t) ^{-\theta }$, then \begin{enumerate} \item $\theta \in ] 0,\frac{1+r_0}{1-r_0}]$ implies $E_{u}(t) \leq c(1+t-T) ^{-\frac{2r_0\theta }{ 1+r_0}},\text{ }t\geq T,$ where $c>0$. \item $\theta \geq \frac{1+r_0}{1-r_0}$ implies $E_{u}(t) \leq c(1+t-T) ^{-\frac{2r_0}{1-r_0}}, \text{ }t>T,$ with $c>0$ and depend on $E_{u}(0)$. \end{enumerate} (2) If there are constants $C_0>0$ and $\theta >0$, such that $\Gamma (t) \leq C_0e^{-\theta t}$, then $E_{u}(t) \leq c(t-T+1) ^{-\frac{2r_0}{1-r_0}}, \quad t>T,$ where $c$ is positive and depends on $E_{u}(0)$. \subsection*{Example 2: Different behavior} Assume \begin{gather*} g_1(s) =\begin{cases} s^2e^{-1/s^2} & 0\leq s<1 \\ -s^2e^{-1/s^2} & -11 \\ g_3(s) =s| s| ^{r_0-1},\quad | s| <1,\; r_0\in (0,1). \end{gather*} We choose \begin{gather*} h_1^{-1}(s) =\sqrt{s}g_1(\sqrt{s})=s^{3/2}e^{-1/s}, \quad 00$and$s>0$. 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