\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 245, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/245\hfil Asymptotic stability of solutions] {Asymptotic stability of solutions to elastic systems with structural damping} \author[H. Fan, F. Gao \hfil EJDE-2014/245\hfilneg] {Hongxia Fan, Fei Gao} % in alphabetical order \address{Hongxia Fan \newline Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, 730070, China} \email{lzfanhongxia@163.com} \address{Fei Gao \newline Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, 730070, China} \email{11l111@sina.cn} \thanks{Submitted June 25, 2014. Published November 20, 2014.} \subjclass[2000]{35B40, 35G25, 47D03} \keywords{Asymptotic stability; elastic systems; structural damping; \hfill\break\indent exponential stability; sectorial operator} \begin{abstract} In this article, we study the asymptotic stability of solutions for the initial value problems of second order evolution equations in Banach spaces, which can model elastic systems with structural damping. The discussion is based on exponentially stable semigroups theory. Applications to the vibration equation of elastic beams with structural damping are also considered. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} The study of elastic systems with damping seems to have been initiated by Chen and Russell \cite{c1} in 1981. They considered the linear elastic systems with structural damping, \begin{equation} \label{e1.1} \begin{gathered} \ddot{u}(t)+B\dot{u}(t)+Au(t)=0,\\ u(0)=x_0,\quad \dot{u}(0)=y_0 \end{gathered} \end{equation} in a Hilbert space $\mathbb{H}$ with inner product $(\cdot,\, \cdot)$, where $A$ (the elastic operator) and $B$ (the damping operator) are unbounded positive definite self-adjoint operators in $\mathbb{H}$. Let $x_1=A^{1/2}u$, $x_2=\dot{u}$, we get the equivalent first-order linear systems \begin{gather*} \frac{d}{dt}\begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix} = \begin{pmatrix} 0 & A^{1/2} \\ -A^{1/2} & -B \\ \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix} =L_B \begin{pmatrix} x_1 \\ x_2 \\ \end{pmatrix},\\ x_1(0)=A^{1/2}x_0,\quad x_2(0)=y_0. \end{gather*} Chen and Russell \cite{c1} proved that $$ L_B= \begin{pmatrix} 0 & A^{1/2} \\ -A^{1/2} & -B \\ \end{pmatrix} $$ generates an analytic semigroup on $\mathbb{W}=\mathbb{H}\oplus \mathbb{H}$, if some additional conditions are satisfied. In the same paper, they pose the following conjecture proved by Huang \cite{h2,h3}: Let $D(B)\supset D(A^{1/2})$; then either of the following conditions (1) and (2) implies that $L_B$ generates an analytic semigroup on $\mathbb{W}$: \begin{itemize} \item[(1)] $\rho_1(A^{1/2}x,x)\leq(Bx,x)\leq\rho_2(A^{1/2}x,x)$ for all $x\in D(A^{1/2})$ or (not, in general, equivalent) \item[(2)] $\rho_1(Ax,x)\leq(B^2x,x)\leq\rho_2(Ax,x)$ for all $x\in D(A)$ \end{itemize} for some $\rho_1,\rho_2>0$ with $\rho_1\leq\rho_2$. In addition, the semigroup generated by $L_B$ is exponentially stable. But these results do not contain the case $B=\rho A$, which could possibly appear in engineering applications. For this situation, Massatt \cite{m1} shows that if $B=\rho A$ with $\rho>0$, then $$ \mathscr{A}_{\rho}= \begin{pmatrix} 0 & 1 \\ -A & -\rho A \\ \end{pmatrix} $$ generates an analytic semigroup which is exponentially stable. Huang \cite{h4} investigated the more widely used linear elastic systems \eqref{e1.1} with damping $B$ related in various ways to $A^{\alpha} (\frac{1}{2}\leq\alpha\leq 1)$, so that the $C_0$-semigroups associated with them are analytic and exponentially stable. Meanwhile, the spectral property and some fundamental results for the analytic property and the exponential stability of the semigroups associated with the systems were discussed. Then other sufficient conditions for $L_B$ generates an analytic semigroup were discussed in \cite{f1,f2,h1,h2,h3,h4,h5} and the references therein. Recently, the present authors \cite{f1} studied the linear second-order evolution equation \begin{equation} \begin{gathered} \ddot{u}(t)+\rho \mathscr{A}\dot{u}(t)+ \mathscr{A}^2u(t)=0, \quad t>0, \\ u(0)=x_0,\quad \dot{u}(0)=y_0, \end{gathered} \end{equation} in a frame of Banach spaces, which can model the elastic systems with structural damping. New forms of the corresponding first-order evolution equations were introduced and sufficient conditions for analyticity and exponential stability of the associated semigroups were given. In \cite{f3} and \cite{f2}, existence results of mild solutions for the elastic systems with structural damping were established by the fixed point theorems and monotone iterative technique in the presence of lower and upper solutions, respectively. However, the theory of the elastic systems with structural damping remains to be developed. In this paper, we concentrate on the asymptotic behavior of solutions for the linear elastic systems with structural damping \begin{equation} \label{e1.3} \begin{gathered} \ddot{u}(t)+\rho \mathscr{A}\dot{u}(t)+ \mathscr{A}^2u(t)=h(t), \quad t>0, \\ u(0)=x_0,\quad \dot{u}(0)=y_0 \end{gathered} \end{equation} and the semilinear elastic systems with structural damping \begin{equation} \label{e1.4} \begin{gathered} \ddot{u}(t)+\rho \mathscr{A}\dot{u}(t)+ \mathscr{A}^2u(t)=f(t,u(t)), \quad t>0,\\ u(0)=x_0,\quad \dot{u}(0)=y_0, \end{gathered} \end{equation} in a Banach space $\mathbb{X}$, where ``$\cdot$'' means $d/dt$, $\rho$ is the damping coefficient; $\mathscr{A}:\mathscr{D}(\mathscr{A})\subset\mathbb{X}\to\mathbb{X}$ is a sectorial operator and $-\mathscr{A}$ generates an analytic and exponentially stable semigroup $S(t)(t\geq0)$ on $\mathbb{X}$; $f\in C(J\times\mathbb{X},\mathbb{X})$, $x_0\in \mathscr{D}(\mathscr{A})$, $y_0\in\mathbb{X}$. \section{Preliminaries} \begin{definition}[\cite{e1}]\rm \label{def2.1} A semigroup $T(t)(t\geq0)$ on a Banach space $\mathbb{X}$ is called exponentially stable if there exist constants $\delta>0$, $M\geq1$ such that $$ \|T(t)\|\leq Me^{-\delta t},\quad t\geq 0. $$ \end{definition} First we present a simple result on the asymptotic behavior of mild solutions for the inhomogeneous initial value problem of the first-order linear evolution equation \begin{equation} \label{e2.1} \begin{gathered} u'(t)=Au(t)+h(t),\quad t>0,\\ u(0)=x. \end{gathered} \end{equation} \begin{lemma}[{\cite[Page 119, Theorem 4.4]{p1}}] \label{lem2.2} Let $\mu>0$ and let $A$ be the infinitesimal generator of a $C_0$-semigroup $T(t)(t\geq0)$ satisfying $\|T(t)\|\leq Me^{-\mu t}$. Let $h$ be bounded and measurable on $[0,+\infty)$. If $$ \lim_{t\to+\infty}h(t)=b, $$ then, $u(t)$, the mild solution of \eqref{e2.1} satisfies $$ \lim_{t\to+\infty}u(t)=-A^{-1}b.\quad\quad $$ \end{lemma} Next we recall some basic facts and conclusions on the elastic systems \eqref{e1.3} and \eqref{e1.4}, which can be found in \cite{f1,f3} in order to prove our main results. Since $\mathscr{A}$ is a sectorial operator on $\mathbb{X}$. It follows from the definition that there exist $\alpha\in(0,\frac{\pi}{2})$ and $K>0$ satisfying \begin{gather} \Sigma_{\alpha}:= \{\lambda||\arg\lambda|<\frac{\pi}{2}+\alpha\} \subset\rho(-\mathscr{A}), \label{e2.2}\\ \|(\lambda I+\mathscr{A})^{-1}\|\leq\frac{K}{1+|\lambda|},\quad \lambda\in \Sigma_{\alpha}. \label{e2.3} \end{gather} For the second-order equation $$ \ddot{u}(t)+\rho \mathscr{A}\dot{u}(t)+ \mathscr{A}^2u(t)=h(t),\quad t>0, $$ it has the decomposition $$ \quad\big(\frac{\partial}{\partial t}+\sigma_1\mathscr{A}\big) \big(\frac{\partial}{\partial t}+\sigma_2\mathscr{A}\big)u=h(t),\quad t>0. $$ Let $$ \frac{\partial u}{\partial t}+\sigma_2\mathscr{A}u=v(t),\quad\quad\quad\quad\quad $$ which means $v(0)=y_0+\sigma_2\mathscr{A}x_0:=v_0$. Then the elastic systems \eqref{e1.3} can be transformed into the following two abstract Cauchy problems in $\mathbb{X}$: \begin{equation} \label{e2.4} \begin{gathered} \frac{\partial v}{\partial t}+\sigma_1\mathscr{A}v=h(t),\quad t>0,\\ v(0)=v_0 \end{gathered} \end{equation} and \begin{equation} \label{e2.5} \begin{gathered} \frac{\partial u}{\partial t}+\sigma_2\mathscr{A}u=v(t),\quad t>0, \\ u(0)=x_0, \end{gathered} \end{equation} where \begin{equation} \label{e2.6} \sigma_1+\sigma_2=\rho,\quad \sigma_1\sigma_2=1. \end{equation} \begin{lemma}[\cite{f1}] \label{lem2.3} Let $\mathscr{A}:\mathscr{D}\mathscr(\mathscr{A})\subset\mathbb{X}\to\mathbb{X}$ be a sectorial operator, if the damping coefficient $\rho>2\cos\alpha$, then $-\sigma_1\mathscr{A}$, $-\sigma_2\mathscr{A}$ generate analytic and exponentially stable semigroups on $\mathbb{X}$, where $\alpha$ is defined in \eqref{e2.2} and $\sigma_1,\sigma_2$ are specified in \eqref{e2.6}. \end{lemma} For the convenience of the reader, throughout this paper we assume that $-\sigma_1\mathscr{A}$ and $-\sigma_2\mathscr{A}$ generate analytic and exponentially stable semigroups $S_1(t){(t\geq0)}$ and $S_2(t)(t\geq0)$ on $\mathbb{X}$, respectively. By Definition \ref{def2.1}, there exist constants $\delta_1>0,\delta_2>0$ and $M_1\geq1,M_2\geq1$ such that \begin{equation} \label{e2.7} \|S_1(t)\|\leq M_1e^{-\delta_1t},\quad \|S_2(t)\|\leq M_2e^{-\delta_2t},\quad t\geq0. \end{equation} \begin{definition}[\cite{f3}] \rm \label{def2.4} Let $\mathscr{A}:\mathscr{D}\mathscr(\mathscr{A})\subset\mathbb{X}\to\mathbb{X}$ be a sectorial operator, $\rho>2\cos\alpha$, and $f: J\times\mathbb{X}\to\mathbb{X}$ be a continuous function, $x_0\in \mathscr{D}(\mathscr{A})$, $y_0\in\mathbb{X}$. A continuous solution of the integral equation \begin{align*} u(t)&=S_2(t)x_0+\int_{0}^tS_2(t-s)S_1(s)v_0\,ds\\ &\quad +\int_{0}^t\int_{0}^sS_2(t-s)S_1(s-\tau)f(\tau,u(\tau))d\tau \,ds \end{align*} is said to be a mild solution of the initial-value problem \eqref{e1.4}, where $\alpha$ is defined in \eqref{e2.2}. \end{definition} \section{Main results} In this section it is our aim to introduce the asymptotic behavior of solutions for the elastic systems \eqref{e1.3} and \eqref{e1.4}, which can be given by the following theorems. \begin{theorem} \label{thm3.1} Let $\mathscr{A}:\mathscr{D}\mathscr(\mathscr{A})\subset\mathbb{X}\to\mathbb{X}$ be a sectorial operator, the damping coefficient $\rho>2\cos\alpha$, where $\alpha$ is defined in \eqref{e2.2}, $x_0\in \mathscr{D}(\mathscr{A})$, $y_0\in \mathbb{X}$, $h:[0,+\infty)\to\mathbb{X}$ is continuous. If $$ \lim_{t\to+\infty}h(t)=b, $$ then, the mild solution $u(t)$ of the initial value problem \eqref{e1.3} satisfies $$ \lim_{t\to+\infty}u(t)=\mathscr{A}^{-2}b. $$ \end{theorem} \begin{proof} Since $S_1(t)$ $(t\geq0)$ is exponentially stable on $\mathbb{X}$. By Definition \ref{def2.1} and Lemma \ref{lem2.2}, the mild solution $v(t)$ of the initial-value problem \eqref{e2.4} satisfies \begin{equation} \label{e3.1} \lim_{t\to+\infty}v(t)=(\sigma_1\mathscr{A})^{-1}b. \end{equation} Similarly, since $S_2(t)(t\geq0)$ is also exponentially stable on $\mathbb{X}$. By Definition \ref{def2.1}, Lemma \ref{lem2.2} and \eqref{e3.1}, the mild solution $u(t)$ of the initial value problem \eqref{e2.5} satisfies \begin{equation} \label{e3.2} \begin{aligned} \lim_{t\to+\infty}u(t) &=(\sigma_2\mathscr{A})^{-1}\lim_{t\to+\infty}v(t)\\ &=(\sigma_2\mathscr{A})^{-1}(\sigma_1\mathscr{A})^{-1}b \\ &=\frac{1}{\sigma_1\sigma_2}\mathscr{A}^{-2}b. \end{aligned} \end{equation} Combining this fact with \eqref{e2.6}, it follows that $\lim_{t\to+\infty}u(t) =\mathscr{A}^{-2}b$. \end{proof} We now show that if the semigroups $S_1(t)(t\geq0)$ and $S_2(t)(t\geq0)$ are exponentially stable on $\mathbb{X}$, then, we can choose the constants $\delta_1,\delta_2$ in \eqref{e2.7} satisfying $0<\delta_1<\delta_2$. If, on the contrary, let $\delta_1=\delta_2:=\delta$ and let $\delta=\delta'+\delta''$, where $\delta'>0$, $\delta''>0$, then for all $t\geq 0$, we have \begin{gather*} \|S_2(t)\|\leq M_2e^{-\delta t},\\ \|S_1(t)\|\leq M_1e^{-\delta t}=M_1e^{-(\delta'+\delta'')t} = M_1e^{-\delta't}e^{-\delta''t}\leq M_1e^{-\delta't}. \end{gather*} It is evident that $\delta>\delta'>0$. Hence, in what follows, we always assume that the constants $\delta_1$ and $\delta_2$ in \eqref{e2.7} satisfying $0<\delta_1<\delta_2$. Next we establish the globally asymptotic stability result of the zero solution for the initial value problem \eqref{e1.4}. \begin{theorem} \label{thm3.2} Let $\mathscr{A}:\mathscr{D}\mathscr(\mathscr{A})\subset\mathbb{X}\to\mathbb{X}$ be a sectorial operator, the damping coefficient $\rho>2\cos\alpha$, where $\alpha$ is defined in \eqref{e2.2}, $x_0\in \mathscr{D}(\mathscr{A})$, $y_0\in \mathbb{X}$, $f:[0,+\infty)\times\mathbb{X}\to\mathbb{X}$ is continuous and satisfies the following conditions: \begin{itemize} \item[(H1)] There exists $L>0$, such that $$ \|f(t,u_2)-f(t,u_1)\|\leq L\|u_2-u_1\|,\quad t\in[0,+\infty),\; u_1,u_2\in\mathbb{X}. $$ \item[(H2)] $f(t,\theta)=\theta$ ($\theta$ is the zero element of $\mathbb{X}$) for $t\geq 0$. \item[(H3)] $00,\\ u(0,t)=u(1,t)=0, \quad t>0,\\ u_{xx}(0,t)=u_{xx}(1,t)=0,\quad t>0,\\ u(x,0)=\varphi(x),\quad u_{t}(x,0)=\psi(x),\quad x\in (0,1), \end{gathered} \end{equation} where $u_{xxxx}$ denotes the elastic effect, $u_{xxt}$ is the damping term, $\rho=4$ is the damping coefficient and the non-homogeneous term $h(x,t)$ be defined by \begin{equation} \label{e4.2} h(x,t)=\begin{cases} \frac{2x^2t^2}{1+3x^2t^2}, & x\in (0,1),\; t\geq0,\\[4pt] 2/3, & x=0,\; t\geq0. \end{cases} \end{equation} Let $I=[0,1]$ and choose $\mathbb{X}=L^{p}(I) (2\leq p<+\infty)$. Define a linear operator $\mathscr{A}:\mathscr{D}(\mathscr{A})\subset\mathbb{X}\to\mathbb{X}$ by \begin{equation} \label{e4.3} \mathscr{D}(\mathscr{A})=W^{2,p}(I)\cap W_0^{1,p}(I),\quad \mathscr{A}u=-\Delta u, \end{equation} where $\Delta$ is the Laplace operator acting on functions on the interval $I$. Choosing $\alpha=\arccos 2/5\in (0,\pi/2)$, by \cite{f1}, $\mathscr{A}$ is a sectorial operator for the region $\sum_{\alpha}$ defined by \eqref{e2.2}. Let $h(t)=h(\cdot,t)$, then the problem \eqref{e4.1} can be rewritten into the abstract form \begin{equation} \begin{gathered} \ddot{u}(t)+4\mathscr{A}\dot{u}(t)+ \mathscr{A}^2u(t)=h(t), \quad t>0,\\ u(0)=\varphi,\quad \dot{u}(0)=\psi. \end{gathered} \end{equation} \begin{theorem} \label{thm4.1} Let $2\leq p<+\infty$, for every $\varphi\in W^{2,p}(I)\cap W_0^{1,p}$ and $\psi\in L^p(I)$, the mild solution $u(t)$ of the equation \eqref{e4.1} satisfying $\lim_{t\to +\infty}u(t)=\Delta^{-2}\frac{2}{3}$. \end{theorem} \begin{proof} By setting $\rho=4$ and $\alpha=\arccos 2/5$, it is easy to verify that the damping coefficient $\rho$ satisfies $\rho>2\cos\alpha$. From \eqref{e4.2}, it follows that $h(t)$ is continuous on $[0,+\infty)$ and $\lim_{t\to +\infty}h(t)=\frac{2}{3}$. Hence by Theorem \ref{thm3.1}, the mild solution $u(t)$ of the equation \eqref{e4.1} satisfying $\lim_{t\to +\infty}u(t)=\Delta^{-2}\frac{2}{3}$. \end{proof} In what follows, we consider the nonlinear vibration equation of elastic beams with structural damping, namely the following initial-boundary value problem \begin{equation} \label{e4.5} \begin{gathered} u_{tt}-4u_{xxt}+u_{xxxx}=\frac{1}{2}\sin u(x,t), \quad x\in (0,1),\; t>0,\\ u(0,t)=u(1,t)=0,\quad t>0,\\ u_{xx}(0,t)=u_{xx}(1,t)=0,\quad t>0,\\ u(x,0)=\varphi(x),\quad u_{t}(x,0)=\psi(x),\quad x\in (0,1), \end{gathered} \end{equation} Let $u(t)=u(\cdot,t)$, $f(t,u(t))=\frac{1}{2}\sin u(\cdot,t)$. Then the initial-boundary value problem \eqref{e4.5} can be rewritten to the Cauchy problem of the second order evolution equation in the Banach space $\mathbb{X}$ \begin{equation} \label{e4.6} \begin{gathered} \ddot{u}(t)+4\mathscr{A}\dot{u}(t)+ \mathscr{A}^2u(t)=f(t,u(t)), \quad t>0, \\ u(0)=\varphi,\quad \dot{u}(0)=\psi, \end{gathered} \end{equation} where $\mathscr{A}$ is defined in \eqref{e4.3} and $\mathscr{A}$ is a sectorial operator for the region $\Sigma_{\alpha} (\alpha=\arccos 2/5)$ defined by \eqref{e2.2}. We assume that $\varphi\in\mathscr{D}(\mathscr{A})$ and $\psi\in \mathbb{X}$, Then the equation \eqref{e4.6} has the following decomposition form \begin{equation} \begin{gathered} (\frac{\partial}{\partial t}+\sigma_1\mathscr{A})(\frac{\partial}{\partial t} +\sigma_2\mathscr{A})u=f(t,u(t)), \quad t>0,\\ u(0)=\varphi,\quad \dot{u}(0)=\psi, \end{gathered} \end{equation} where $\sigma_1=2-\sqrt{3}$, $\sigma_2=2+\sqrt{3}$ are defined by \eqref{e2.6}. It is well-known \cite{h1,p1}, $-\mathscr{A}$ generates an analytic and exponentially stable semigroup $S(t)(t\geq 0)$ satisfying $$ \|S(t)\|\leq e^{-t},\quad t\geq0. $$ By Lemma \ref{lem2.3} and the characterization of the infinitesimal generators of $C_0$-semigroups, $-\sigma_1\mathscr{A}$ and $-\sigma_2\mathscr{A}$ generate analytic and exponentially stable semigroups $S_1(t)(t\geq 0)$ and $S_2(t)(t\geq 0)$ respectively, which satisfy $$ \|S_i(t)\|=\|S(\sigma_it)\|\leq e^{-\sigma_i t},\quad t\geq 0,\quad i=1,2. $$ Now take $ M_1=M_2=1$, $\delta_1=\sigma_1=2-\sqrt{3}$ and $\delta_2=\sigma_2=2+\sqrt{3}$, we obtain that \begin{equation} \label{e4.8} \frac{1}{2}<\frac{\delta_1(\delta_2-\delta_1)}{M_1M_2}=4\sqrt{3}-6. \end{equation} \begin{theorem} \label{thm4.2} Let $2\leq p<+\infty$, for every $\varphi\in W^{2,p}(I)\cap W_0^{1,p}$ and $\psi\in L^p(I)$, the mild solution $u(t)$ of the equation \eqref{e4.5} satisfying $\|u(t)\|_p\to 0$ as $t\to\infty$. \end{theorem} \begin{proof} By $\rho=4$ and $\alpha=\arccos 2/5$, we can easily obtain that the damping coefficient $\rho$ satisfies $\rho>2\cos\alpha$. Since $f(x,t,u(x,t))=\frac{1}{2}\sin u(x,t)$ is continuous on $[0,1]\times [0,+\infty)\times \mathbb{X}$ and satisfying \begin{gather} |f'_u(x,t,u)|=\frac{1}{2}|\cos u(x,t)|\leq\frac{1}{2},\quad (x,t,u)\in[0,1]\times[0,+\infty)\times\mathbb{X}; \label{e4.9} \\ f(x,t,0)=\sin 0=0,\quad (x,t)\in[0,1]\times[0,+\infty). \label{e4.10} \end{gather} From \eqref{e4.9}, for $u_1,u_2\in\mathbb{X}$, we have \begin{equation} |f(x,t,u_2)-f(x,t,u_1)| \leq\frac{1}{2}|u_2-u_1|,\quad (x,t)\in [0,1]\times [0,+\infty). \end{equation} Which implies \begin{equation} \label{e4.12} \|f(t,u_2)-f(t,u_1)\|_p\leq\frac{1}{2}\|u_2-u_1\|_p,\quad t\in [0,+\infty),\; u_1,u_2\in \mathbb{X}. \end{equation} Then assumptions (H1) and (H2) hold. According to \eqref{e4.8} and \eqref{e4.12}, we obtain that $(H3)$ is satisfied. Hence by Theorem \ref{thm3.2}, we conclude that the mild solution $u(t)$ of \eqref{e4.5} satisfying $\lim_{t\to+\infty}u(t)=0$, which implies $\|u(t)\|_p\to 0$ as $t\to+\infty$. \end{proof} \subsection*{Acknowledgments} This work was supported by the National Natural Science Foundation of China (No. 11361032) and the Youth Science Foundation of Lanzhou Jiaotong University (No. 2013025). \begin{thebibliography}{99} \bibitem{c1} G. Chen, D. L. Russell; \emph{A mathematical model for linear elastic systems with structural damping}, Quart. Appl. Math. 39 (1981/1982) 433-454. \bibitem{c2} S. Chen, R. Triggiani; \emph{Proof of extensions of two conjectures on structural damping for elastic systems: the case $1/2\leq\alpha\leq1$}, Pasific J. Math. 39 (1989) 15-55. \bibitem{c3} S. Chen, R. Triggiani; \emph{Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0<\alpha<1/2$}, Proc. AMS 110. No. 2 (1990) 401-415. \bibitem{e1} K.J. Engel, R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. \bibitem{f1} H. Fan, Y. Li; \emph{Analyticity and exponential stability of semigroups for the elastic systems with structural damping in Banach spaces}, J. Math. Anal. Appl. 410 (2014) 316-322. \bibitem{f2} H. Fan, Y. Li; \emph{Monotone iterative technique for the elastic systems with structural damping in Banach spaces}, Comput. Math. Appl. 68 (2014) 384-391. \bibitem{f3} H. Fan, Y. Li, P. Chen; \emph{Existence of mild solutions for the elastic systems with structural damping in Banach spaces}, Abstract and Applied Analysis, Volume 2013, Artical ID 746893, 6 pages. \bibitem{h1} D. Henry; \emph{Geometric Theory of Semilinear Parabolic Equations}, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981. \bibitem{h2} F. Huang; \emph{On the holomorphic property of the semigroup associated with linear elastic systems with structural damping}, Acta Math. Sci.(Chinese) 5 (1985) 271-277. \bibitem{h3} F. Huang; \emph{A problem for linear elastic systems with structural damping}, Acta Math. Sci. Sinic. 6 (1986) 107-113. \bibitem{h4} F. Huang; \emph{On the mathematical model for linear elastic systems with analytic damping}, SIAM, J. Cont, Opt. 26 (1988) 714-724. \bibitem{h5} F. Huang, K. Liu; \emph{Holomiphic property and exponential stability of the semigroup associated with linear elastic systems with damping}, Ann. Diff. Eqs. 4(4) (1988) 411-424. \bibitem{h6} F. Huang, K. Liu, G. Chen; \emph{Differentiability of the semigroup associated with a structural damping model}, in ``Proc. 28th IEEE CDC, Tampa, Dec., 1989,'' 2034-2038. \bibitem{h7} F. Huang, Y. Huang, F. Guo; \emph{Analyticity and differentiability of the $C_0$-semigroup associated with Euler-Bernoulli beam equations with structural damping}, Sinica Math. Sci.(Chinese) 2 (1992) 122-133. \bibitem{l1} K. Liu, Z. Liu; \emph{Analyticity and differentiability of semigroups associated with elastic systems with damping and gyroscopic forces}, J. Differential Equations. 141 (1997) 340-355. \bibitem{m1} P. Massat; \emph{Limiting behavior for strong damped non-linear wave equations}, J. Differential Equations, 48 (1983) 334-349. \bibitem{p1} A. Pazy; \emph{Semigroups of Linear Operators and Applications to Partial Differential Equations}, Springer-Verlag, New York, 1983. \bibitem{r1} D. L. Russell; \emph{On mathematical models for the elastic beam with frequency-proportional damping}, in ``Control and Estimation of Distributed Parameter Systems" (H. T. Banks, Ed.), SIAM, Philadephia, 1992. \end{thebibliography} \end{document}