\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{mathrsfs,amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 16, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2011/16\hfil Existence of pullback attractors]
{Existence of pullback attractors for the coupled suspension
 bridge equations}

\author[Q. Ma, B. Wang\hfil EJDE-2011/16\hfilneg]
{Qiaozhen Ma, Binli Wang}  % in alphabetical order

\address{Qiaozhen Ma \newline
College of Mathematics and Information Science,
 Northwest Normal University, Lanzhou, Gansu 730070, China}
\email{maqzh@nwnu.edu.cn} 

\address{Binli Wang \newline
College of Mathematics and Information Science,
 Northwest Normal University, Lanzhou, Gansu 730070, China}
\email{wangbinli8@126.com}

\thanks{Submitted July 12, 2010. Published January 28, 2011.}
\subjclass[2000]{35B40, 35L05, 58F12}
 \keywords{Coupled suspension
bridge equations; pullback $\mathscr{D}$-attractors}

\begin{abstract}
 In this article, we study the existence of pullback
 $\mathscr{D}$-attractors for the non-autonomous coupled suspension
 bridge equations with hinged ends and clamped ends, respectively.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this paper, we consider the following nonlinear problems which
describes a vibrating beam equation coupled with a vibrating string
equation
\begin{equation} \label{e1.1}
 \begin{gathered}
u_{tt}+\alpha u_{xxxx}+\delta_1u_{t}+{k(u-v)}^{+}
+f_B(u)=h_B(x,t),\quad x\in[0,L], \\
v_{tt}-\beta v_{xx}+\delta_2v_{t}-{k(u-v)}^{+}+f_{S}(v)=h_{S}(x,t),
\quad x\in[0,L]
\end{gathered}
\end{equation}
with the simply supported boundary-value conditions
\begin{equation}
u(0,t)=u(L,t)=u_{xx}(0,t)=u_{xx}(L,t)=0,\quad
v(0,t)=v(L,t)=0,\quad t\geq \tau, \label{e1.2}
\end{equation}
and the initial-value conditions
\begin{gather}
u(\tau,x)=u_0(x), u_{t}(\tau,x)=u_1(x),  x\in[0,L],\label{e1.3}\\
v(\tau,x)=v_0(x), v_{t}(\tau,x)=v_1(x),  x\in[0,L].\label{e1.4}
\end{gather}
 Where $k>0$ denotes the spring constant of
 the ties, $\alpha>0$ and $\beta>0$ are the flexural rigidity of the
 structure and coefficient of tensile strength of the
 cable, respectively. $\delta_1$, $\delta_2>0$ are
 constants, the force term $h_B$,
$h_{S}\in L^2_{\rm loc}(\mathbb{R};L^2(0,L))$. The nonlinear
 functions $f_B(u),f_{S}(v)\in C^2(\mathbb{R},\mathbb{R})$
satisfies the following assumptions:
 \begin{itemize}
\item[(F1)] $\liminf_{|r|\to\infty}\frac{F_B(r)}{r^2}\geq0$,
$\liminf_{|r|\to\infty}\frac{F_{S}(r)}{r^2}\geq0$, $\forall r\in
\mathbb{R}$;

\item[(F2)] $|f_B(r)|,|f_{S}(r)|\leq C_0(1+|r|^{p})$,
for all $p\geq 1$, $\forall r \in \mathbb{R}$;

\item[(F3)] $\liminf_{|r|\to\infty}\frac{r
f_B(r)-C_1F_B(r)}{r^2}\geq0$, $\liminf_{|r|\to\infty}\frac{r
f_{S}(r)-C_1F_{S}(r)}{r^2}\geq0$, $\forall r \in \mathbb{R}$,
\end{itemize}
where $C_0$, $C_1$ are positive constants,
$F_B(r)=\int_0^{r}f_B(\zeta)d\zeta$,
$F_{S}(r)=\int_0^{r}f_{S}(\zeta)d\zeta$.

For the mathematical model of suspension bridge, there are many
references to study the existence and asymptotic behavior of
solutions, see \cite{a1,h1,l1,l2,m1,m2,m3,m5,z1}
 and references therein.
For instance,  Lazer and McKenna studied the nonlinear
oscillation problems in a suspension bridge, and presented a
(one-dimensional) mathematical model for a suspension bridge as a
new problem of nonlinear analysis in \cite{l1}.
Ahmed and Harbi continued to discuss this problem in \cite{a1},
and pointed out that the
system \eqref{e1.1} is conservative and asymptotically stable
with respect
to the rest state for $k>0$, $f_B(u)\equiv 0\equiv f_S(v)$, and
furthermore showed that the corresponding Cauchy problem has at
least one weak solution. Holubov\'a and Matas considered
the more general nonlinear string-beam system in \cite{h1}
 and arrived at
the existence and uniqueness of the weak solution by the
Faedo-Galerkin methods. In 2004, Litcanu proved the existence of
weak $T$-periodic solutions of \eqref{e1.1} and obtained a
regularity result when $k(u-v)^+=\phi(u,v)$,
$f_B(u)\equiv  f_S(v)\equiv 0$ in \cite{l2}.
Similar models have also been investigated by Malik in \cite{m5}.
However, our aim is to study the longtime behavior of solutions for
the suspension bridge model. In 2005, we achieved first the
existence of global attractors of a weak solution for the autonomous
coupled suspension bridge equations in \cite{m1}; i.e., in \eqref{e1.1},
$h_B(x,t)$ and $h_S(x,t)$ do not depend on the time $t$ explicitly.
In the sequel, the existence of the strong solutions and the compact
global attractor have also been obtained for the autonomous coupled
suspension bridge equations and the single one which the motion of
the main cable is neglected, respectively, see \cite{m2,z1}.
For the limit of our knowledge, the existence of the pullback
attractors of \eqref{e1.1}
has no any results, while it is just our concern. For a good survey
of the literatures dedicated to the existence of attractors for the
dynamical systems we would like to mention some monograph
\cite{c2,r1,t1}
and so on.

 About the existence of pullback attractors for the dynamical systems,
 it has been developed for both non-autonomous and random dynamical
systems.
 In 2006, Caraballo et al. presented the concept of the pullback
$\mathscr{D}$-attractors in \cite{c1}, and obtained the abstract 
results verifying the existence
of pullback $\mathscr{D}$-attractors, moreover, they applied their
abstract results into the non-autonomous Navier-Stokes equation in
an unbounded domain. Zhong \cite{z1} and Wang \cite{w2} also established
some sufficient conditions for the existence of the pullback
$\mathscr{D}$-attractors by using the methods introduced in
\cite{m4}, and achieved the existence of pullback
$\mathscr{D}$-attractors for
non-autonomous Sine-Gordon equations and wave equations with
critical exponent, respectively. The existence of pullback
$\mathscr{D}-$ attractors for the single suspension bridge equation
was showed in \cite{p1}.  Motivated by \cite{c1,p1,w1,w2},
in this paper, we focus our attention on the existence of
pullback $\mathscr{D}$-attractors
for \eqref{e1.1}. Our main results are Theorem \ref{thm3.4} and
\ref{thm3.5}.

\section{Preliminaries}

 With the usual notation, let
$Y_0=L^2(0,L)$, $Y_1=H^1_0(0,L)$, $Y_2=D(A)=H^2(0,L)\cap
H^1_0(0,L)$,  where $A=-\frac{\partial^2}{\partial x^2},
A^2=\frac{\partial^4}{\partial x^4}$, and endow $Y_0$ with the
standard scalar product and norm $(\cdot,\cdot)$, $|\cdot|$.
Meanwhile, we denote $\|\cdot\|,|Au|$ be the norm of $Y_1,Y_2$,
respectively. In addition, let $\lambda_1$ be the first eigenvalue
of $Au=\lambda u,x\in[0,L]; u(0)=u(L)=0$, the corresponding
eigenfunctions $\phi_1(x)$ is positive on $[0,L]$. It's easy to
know that $\lambda_1^2$ is the first eigenvalue of $A^2
u=\lambda^2u, x\in[0,L], u(0)=u(L)=u_{xx}(0)=u_{xx}(L)=0$.
Choosing $\lambda=min\{\lambda_1,\lambda_1^2\}$, by the
Poincar\'{e} inequality, we
have
\begin{equation}
\|u\|^2\geq\lambda|u|^2,\quad \forall u\in Y_1;\quad
|Au|^2\geq \lambda \|u\|^2,\quad \forall u\in Y_2.\label{e2.1}
\end{equation}
Next we iterate some definitions and abstract results concerning
the pullback  attractor, which is necessary to obtain our main
results, please refer the reader to see \cite{c1,w1}
for more details.
 Let $(E,d)$ be a complete metric space, $(Q,\rho)$ be a metric
 space which will be called the parameter space. We define a
 non-autonomous dynamical system by a cocycle mapping
$\phi:\mathbb{R}_{+}\times Q\times  E$ which is driven by an
autonomous dynamical system  $\theta$ acting
 on a parameter space $Q$. Specifically,
$\theta=\{\theta_{t}\}_{t\in  \mathbb{R}}$ is a dynamical system
on  $Q$ with the properties that
\begin{itemize}
\item[(1)] $\theta_0(q)=q$, for all $q\in Q$;

\item[(2)] $\theta_{t+\tau}(q)=\theta_{t}(\theta_{\tau}(q))$,
for  all $q\in Q $, $t,\tau \in \mathbb{R}$;

\item[(3)] the mapping (t,q)$\to\theta_{t}(q)$ is continuous.
\end{itemize}

\begin{definition}\label{def2.1}\rm
A mapping  $\phi$ is said to be a cocycle on $E$ with
 respect to group $\theta$ if
\begin{itemize}
\item[(1)] $\phi(0,q,x)=x$, for all $(q,x)\in Q\times E$;

\item[(2)] $\phi(t+s,q,x)=\phi(s,\theta_{t}(q),\phi(t,q,x))$,
 for all $s,t\in \mathbb{R}_{+}$ and all $(q,x)\in Q\times E$.
\end{itemize}
\end{definition}

Let $\mathcal{P}(E)$ denote the family of all nonempty subsets of
 $E$, $\mathcal{B}(E)$ be the set of all bounded subsets of  $E$,
and $\mathcal{K}$ be the
 class of all families
$\hat D=\{D_{q}\}_{q\in Q}\subset \mathcal{P}(E)$. We
 consider a given  nonempty subclass
 $\mathscr{D}\subset \mathcal{K}.$

\begin{definition} \label{def2.2} \rm
A family $\hat{B}=\{B_{q}\}_{q\in Q}\in \mathcal{K}$
is said to be pullback $\mathscr{D}$-absorbing if for each $q\in Q$ and
$\hat{D}\in \mathscr{D}$, there exists $t_0(q,\hat{D})\geq0$ such that
$\phi(t,\theta_{-t}(q),D_{\theta_{-t}(q)})\subset B_{q}$, for all
 $t\geq t_0(q, \hat{D})$.
 \end{definition}

\begin{definition} \label{def2.3} \rm
Let $(\theta,\phi)$ be a non-autonomous dynamical system
on $Q\times E$. $(\theta,\phi)$ is said to be satisfying pullback
$\mathscr{D}$-condition (C) if for any $q\in Q$,
$\hat {C}\in \mathscr D$ and any $\epsilon>0$, there exists a
$t_0=t_0(q,\hat{C},\epsilon)\geq0$ and a finite dimensional
subspace $E_1$ of  $E$ such that
\begin{itemize}
\item[(i)] $P(\bigcup_{t\geq
t_0}\phi(t,\theta_{-t}(q),D_{\theta_{-t}(q)}))$ is bounded; and
\item[(ii)]
$\|(I-P)\phi(t,\theta_{-t}(q),D_{\theta_{-t}(q)})\|_{E}\leq\epsilon$,
where $P:E\to E_1$ is a bounded projector.
\end{itemize}
\end{definition}

\begin{theorem} \label{thm2.1}
Let $(\theta,\phi)$ be a non-autonomous dynamical system
on $Q\times E$. $(\theta,\phi)$ possesses a global pullback
$\mathscr{D}$-attractor $\hat{A}=\{A_{q}\}_{q\in Q}$ satisfying
 $A_{q}=\Lambda(\hat{D},q)=\bigcap_{s\geq0}\overline{\bigcup_{t\geq
s}\phi(t,\theta_{-t}(q),D_{\theta_{-t}(q)})}$ if
\begin{itemize}
\item[(1)] it has a pullback $\mathscr{D}$-absorbing set
$\hat{B}=\{B_{q}\}_{q\in Q}\in \mathscr{D}$;
\item[(2)] it satisfies pullback $\mathscr{D}$-Condition (C).
\end{itemize}
\end{theorem}


\section{Pullback $\mathscr{D}$-attractors in $E_0$}

 For brevity, we write $E_0=Y_2 \times Y_0 \times Y_1 \times Y_0$,
$y_0=(u_0,u_1,v_0,v_1)$,
 $y=y(t)=(u(t),u_{t}(t),v(t),v_{t}(t))$.
We need the following results.

\begin{theorem}[\cite{a1,m2,m3}] \label{thm3.1}
 Suppose that $y_0\in E_0$, $h_B,h_S\in L^2_{\rm loc}(\mathbb{R}, Y_0)$,
then \eqref{e1.1}-\eqref{e1.4} has a unique
solution
\begin{equation}
y\in C(\mathbb{R}_{\tau},E_0),\label{e3.1}
\end{equation}
where $\mathbb{R}_{\tau}=[\tau,+\infty)$. Furthermore,
$y_0 \mapsto y$ is continuous in $E_0$.
\end{theorem}

As in  \cite{c1,w1}, we denote by $E_0$ the space of vector function $y(x)$
with finite energy norm
$\|y\|_{E_0}^2=|Au|^2+\|v\|^2+|u_{t}|^2+|v_{t}|^2$. Then
we can construct the non-autonomous dynamical system generated by
problem $\eqref{e1.1}-\eqref{e1.4}$ in  $E_0$. We consider
 $Q=\mathbb{R}$, $\theta_{t}(\tau)=t+\tau$, and define
\begin{equation}
\phi(t,\tau,y_0)=y(t+\tau;\tau,y_0)
=\big(u(t+\tau),u_{t}(t+\tau),v(t+\tau),v_{t}(t+\tau)\big), \label{e3.2}
\end{equation}
$\tau\in \mathbb{R}$, $t\geq0,y_0\in E_0$.
Thus, thanks to  Theorem \ref{thm3.1}, we have
$\phi(t+s,\tau,y_0)=\phi(t,s+\tau,\phi(s,\tau,y_0))$,  for
$\tau\in \mathbb{R}, s, t\geq0$, and the
mapping $\phi(t,\tau,\cdot):E_0\to E_0$ defined by \eqref{e3.2}
is continuous. Therefore, the mapping $\phi$ defined by \eqref{e3.2}
is a continuous cocycle on $E_0$. Now we assume that
$h_B,h_{S}\in L_{\rm loc}^2(\mathbb{R};Y_0)$ with
\begin{gather}
\int^{t}_{-\infty}e^{\delta s}|h_B(x,s)|^2ds<\infty, \quad
 \forall  t\in \mathbb{R}, \label{e3.3}\\
\int^{t}_{-\infty}e^{\delta s}|h_{S}(x,s)|^2ds<\infty,\quad
 \forall  t\in \mathbb{R},  \label{e3.4}
\end{gather}
 where $\delta$ is a positive constant which will be characterized
later. Let $\mathcal{R}_{\delta}$ be the set
 of all function  $r:\mathbb{R}\to(0,+\infty)$ satisfying
\begin{equation}
\lim_{t\to-\infty}e^{\delta t}r^2(t)=0. \label{e3.5}
\end{equation}
Here $\mathscr{D}_{\delta,E_0}$ denotes the class of all families
$\hat{D}=\{D(t);t\in \mathbb{R}\}\subset \mathcal{ P}(E_0)$ with
$D(t)\subset \bar{B}(0,r_{\hat{D}}(t))$ for some
  $r_{\hat{D}}(t)\in \mathcal {R}_{\delta}$, where
  $\bar{B}(0,r_{\hat{D}}(t))$ is the closed ball in
  $E_0$ centered at 0 with radius  $r_{\hat{D}}(t)$.
 We also need the following lemmas.

\begin{lemma}[\cite{w1}] \label{lem3.2}
Suppose that the family $\{\omega_i\}_{i\in \mathbb N}$ and
 $\{\chi_i\}_{i\in \mathbb N}$ be an orthonormal  basis of
 $Y_2$ and $Y_1$, respectively, which consist of the
eigenvectors of $A^2$ and $A$, $h_B, h_{S}\in
L_{\rm loc}^2(\mathbb{R},Y_0)$  satisfy
\eqref{e3.3}-\eqref{e3.4}. Then
\begin{gather}
 \lim_{m\to\infty}\int_{-\infty}^{t}e^{\sigma
s}|(I-P_m)h_B(x,s)|^2ds=0, \quad \forall  t\in \mathbb{R},
\label{e3.6}\\
 \lim_{m\to\infty}\int_{-\infty}^{t}e^{\sigma
s}|(I-Q_m)h_{S}(x,s)|^2ds=0, \forall t\in
\mathbb{R},  \label{e3.7}
\end{gather}
where  $P_m:Y_2\to span\{\omega_1,\omega_2,\dots \omega_m\}$,
$Q_m:Y_1\to span\{\chi_1,\chi_2,\dots \chi_m\}$ are the orthogonal
projector.
\end{lemma}


\begin{lemma}[\cite{m2,m3}] \label{lem3.3}
 Assume that $f_B(u),f_{S}(v)\in C^2(\mathbb{R},\mathbb{R})$
satisfies {\rm (F2)}, moreover, $f_B(0)=f_{S}(0)=0$. Then
 $(f_B,f_{S}):Y_2\times Y_1\to Y_0\times Y_0$ are
continuous and compact.
\end{lemma}

\begin{theorem} \label{thm3.4}
Suppose that {\rm (F1)-(F3)} hold,
$h_B,h_{S}\in L_{\rm loc}^2(\mathbb{R},Y_0)$ with
\eqref{e3.3}--\eqref{e3.4}.
Then there exists a pullback  $\mathscr{D}_{\delta,E_0}$-absorbing
set in  $E_0$ for the non-autonomous dynamical
system $(\theta,\phi)$ associated with \eqref{e1.1}-\eqref{e1.4}.
\end{theorem}

\begin{proof}
 Let  $t\in \mathbb{R}$, $\tau\geq 0$, and  $y_0\in E_0$ be fixed.
Choose $0<\epsilon<\epsilon_0$, where
\begin{equation}
\epsilon_0=\min\{\frac{\delta_1}{4},\frac{\delta_2}{4},\frac{\alpha
\lambda^2}{2\delta_1},\frac{\beta
\lambda^2}{2\delta_2}\}.\label{e3.8}
\end{equation}
Taking the scalar product in $Y_0$ for the first equation
of \eqref{e1.1} with  $\phi=u_{t}+\epsilon u$  and the second
equation with $\varphi=v_{t}+\epsilon v$,  we have
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(|\phi|^2+\alpha|Au|^2+\beta\|v\|^2
 +|\varphi|^2)
 +\epsilon\alpha|Au|^2+(\delta_1-\epsilon)|\phi|^2\\
&-\epsilon(\delta_1-\epsilon)(u,\phi)
+\beta\epsilon\|v\|^2+(\delta_2-\epsilon)|\varphi|^2
 -\epsilon(\delta_2-\epsilon)(v,\varphi)\\
& +k((u-v)^{+},\phi-\varphi)
 +(f_B(u),\phi)+(f_{S}(v),\varphi)\\
&=(h_B(t),\phi)+(h_{S}(t),\varphi).
\end{aligned} \label{e3.9}
\end{equation}
Using H\"older and Young inequalities and \eqref{e3.8}, we conclude
\begin{gather}
\begin{aligned}
&\epsilon\alpha|Au|^2+(\delta_1-\epsilon)|\phi|^2
 -\epsilon(\delta_1-\epsilon)(u,\phi)+\beta\epsilon\|v\|^2
 +(\delta_2-\epsilon)|\varphi|^2
 -\epsilon(\delta_2-\epsilon)(v,\varphi)\\
&\geq\frac{\epsilon\alpha}{2}|Au|^2+\frac{\epsilon\beta}{2}\|v\|^2
 +\frac{\delta_1}{2}|\phi|^2+\frac{\delta_2}{2}|\varphi|^2;
\end{aligned} \label{e3.10} \\
k((u-v)^{+},\phi-\varphi)=\frac{1}{2}\frac{d}{dt}k|(u-v)^{+}|^2
+\epsilon k|(u-v)^{+}|^2.\label{e3.11}
\end{gather}
Due to (F3) it follows that
\begin{gather*}
\int_0^{L}uf_B(u)dx-C_1\int_0^{L}F_B(u)dx
 +\frac{\epsilon}{4}|Au|^2\geq-K_2,\quad \forall  u\in Y_2,\\
\int_0^{L}vf_{S}(v)dx-C_1\int_0^{L}F_{S}(v)dx
 +\frac{\epsilon}{4}\|v\|^2\geq-K_2,\quad \forall  v\in Y_1,
\end{gather*}
where $C_1,K_2$ are positive constants.
Then
\begin{gather}
(f_B(u),\phi)\geq\frac{d}{dt}\int_0^{L}F_B(u)dx
 +\epsilon C_1\int_0^{L}F_B(u)dx-\frac{\epsilon^2}{4}|Au|^2
 -\epsilon K_2,\label{e3.12}\\
(f_{S}(v),\varphi)\geq\frac{d}{dt}\int_0^{L}F_{S}(v)dx
 +\epsilon C_1\int_0^{L}F_{S}(v)dx-\frac{\epsilon^2}{4}\|v\|^2
 -\epsilon K_2.\label{e3.13}
\end{gather}
Together with \eqref{e3.10}-\eqref{e3.13}, from \eqref{e3.9}, it
leads to
\begin{equation}
\begin{aligned}
&\frac{d}{dt}(|\phi|^2+\alpha|Au|^2+\beta\|v\|^2+|\varphi|^2
+k|(u-v)^{+}|^2+2\int_0^{L}F_B(u)dx+2\int_0^{L}F_{S}(v)dx)\\
&+\epsilon(\alpha-\frac{\epsilon}{2})|Au|^2
 +\epsilon(\beta-\frac{\epsilon}{2})\|v\|^2+
\frac{\delta_1}{2}|\phi|^2+\frac{\delta_2}{2}|\varphi|^2+2\epsilon
k|(u-v)^{+}|^2\\
&+2\epsilon C_1\int_0^{L}F_B(u)dx+2\epsilon C_1\int_0^{L}F_{S}(v)dx\\
&\leq4\epsilon K_2+\frac{2}{\delta_1}|h_B(t)|^2
+\frac{2}{\delta_2}|h_{S}(t)|^2.
\end{aligned}\label{e3.14}
\end{equation}
 Provided $\epsilon$ is small enough such that
$\alpha-\frac{\epsilon}{2}>\frac{\alpha}{2}$,
$\beta-\frac{\epsilon}{2}>\frac{\beta}{2}$,
and set $\delta=\min\{\frac{\epsilon}{2},\epsilon C_1\}$, we deduce
\begin{align*}
&\frac{d}{dt}(|\phi|^2+\alpha|Au|^2+\beta\|v\|^2
 +|\varphi|^2+k|(u-v)^{+}|^2+2\int_0^{L}F_B(u)dx
 +2\int_0^{L}F_{S}(v)dx)\\
&+\delta(|\phi|^2+\alpha|Au|^2+\beta\|v\|^2+|\varphi|^2
 +k|(u-v)^{+}|^2+2\int_0^{L}F_B(u)dx+2\int_0^{L}\! F_{S}(v)dx)\\
&\leq4\epsilon K_2+\frac{2}{\delta_1}|h_B(t)|^2
 +\frac{2}{\delta_2}|h_{S}(t)|^2. %\eqref{e3.15}
\end{align*}

By (F1) we know that there exists a positive constant $K_1$
such that
\begin{gather}
\int_0^{L}F_B(u)dx+\frac{\alpha}{8}|Au|^2\geq -K_1,\quad \forall
  u\in Y_2,\label{e3.16}\\
\int_0^{L}F_{S}(v)dx+\frac{\beta}{8}\|v\|^2\geq -K_1,\quad \forall
 v\in Y_1.\label{e3.17}
\end{gather}
Therefore, by \eqref{e3.16}-\eqref{e3.17} it follows that
\begin{align*}
E(t)&=|\phi|^2+\alpha|Au|^2+\beta\|v\|^2+|\varphi|^2+k|(u-v)^{+}|^2\\
&\quad +2\int_0^{L}F_B(u)dx+2\int_0^{L}F_{S}(v)dx+4K_1\ge 0,
\end{align*}
and
$$
\frac{d}{dt}E(t)+\delta E(t)\leq4\epsilon K_2
+4\delta K_1+\frac{2}{\delta_1}|h_B(t)|^2
+\frac{2}{\delta_2}|h_{S}(t)|^2.
$$
Furthermore,
$$
\frac{d}{dt}(e^{\delta t}E(t))\leq e^{\delta t}(4\epsilon K_2
+4\delta K_1+\frac{2}{\delta_1}|h_B(t)|^2
+\frac{2}{\delta_2}|h_{S}(t)|^2).
$$
Integrating the above inequality from $t-\tau$ to $t$ yields
\begin{align*}
E(t)
&\leq e^{-\delta \tau} E(t-\tau)+\frac{4(\epsilon
 K_2+\delta K_1) }{\delta}+\frac{2}{\delta_1}e^{-\delta
 t}\int_{t-\tau}^{t}e^{\delta s}|h_B(s)|^2ds\\
&\quad +\frac{2}{\delta_2}e^{-\delta
 t}\int_{t-\tau}^{t}e^{\delta s}|h_{S}(s)|^2ds \\
&\leq e^{-\delta \tau} E(t-\tau)+\frac{4(\epsilon K_2
 +\delta K_1)}{\delta}
 +\frac{2}{\delta_1}e^{-\delta t}
 \int_{-\infty}^{t}e^{\delta s}|h_B(s)|^2ds\\
&\quad +\frac{2}{\delta_2}e^{-\delta t}
\int_{-\infty}^{t}e^{\delta s}|h_{S}(s)|^2ds.
\end{align*} %\eqref{e3.18}
For any $\hat{D}\in \mathscr{D}_{\delta,E_0},  y_0\in
D(t-\tau)$, by (F2), $\int_0^{L}F_B(u_0)dx$ and
$\int_0^{L}F_{S}(v_0)dx$ are bounded. Hence
\begin{align*}
&\sup _{y_0\in D(t-\tau)}E(t-\tau)\\
&=\sup _{y_0\in D(t-\tau)} \big\{|u_1+\epsilon
u_0|^2+\alpha|Au_0|^2+\beta\|v_0\|^2+|v_1+\epsilon
v_0|^2 +k|(u_0-v_0)^{+}|^2\\
&\quad +2\int_0^L F_B(u_0)dx+2\int_0^L F_{S}(v_0)dx\big\}
<\infty.
\end{align*} %\label{e3.19}
Using \eqref{e3.16} and \eqref{e3.17} again, we arrive at
\begin{align*}
E(t)&=|\phi|^2+\alpha|Au|^2+\beta\|v\|^2+|\varphi|^2+k|(u-v)^{+}|^2\\
&\quad +2\int_0^{L}F_B(u)dx+2\int_0^{L}F_{S}(v)dx+4K_1\\
&\geq|\phi|^2+\frac{3\alpha}{4}|Au|^2+\frac{3\beta}{4}\|v\|^2
 +|\varphi|^2. % \eqref{e3.20}
\end{align*}
Therefore, if we let
$\delta_0=\min\{\frac{3\alpha}{4}, \frac{3\beta}{4}, 1\}$,
$K=4(\epsilon K_2+\delta K_1)$, then
\begin{equation}
\begin{aligned}
&|\phi|^2+|Au|^2+|\varphi|^2+\|v\|^2\\
&\leq\frac{1}{\delta_0}(e^{-\delta \tau} E(t-\tau)+\frac{K}{\delta} 
+  \frac{2}{\delta_1}e^{-\delta
t}\int_{-\infty}^{t}e^{\delta s}|h_B(s)|^2ds\\
&\quad +\frac{2}{\delta_2}e^{-\delta
t}\int_{-\infty}^{t}e^{\delta s}|h_{S}(s)|^2ds),
\end{aligned} \label{e3.21}
\end{equation}
namely, $\|\phi(\tau,t-\tau,y_0)\|^2_{E_0}$
is bounded by the above expression %{e3.22}
for all $y_0\in D(t-\tau),t\in \mathbb{R}$  and $\tau\geq0$. Set
\begin{equation}
(R_{\delta}(t))^2=\frac{2}{\delta_0}
\Big(\frac{K}{\delta}+\frac{2}{\delta_1}e^{-\delta
t}\int_{-\infty}^{t}e^{\delta
s}|h_B(s)|^2ds+\frac{2}{\delta_2}e^{-\delta
t}\int_{-\infty}^{t}e^{\delta s}|h_{S}(s)|^2ds\Big),
\label{e3.23}
\end{equation}
and consider the family $\hat{B}_{\delta,E_0}$ of close balls in
 $E_0$ defined by
\begin{equation}
B_{\delta}(t)=\{ y\in E_0: \|y\|_{E_0}^2\leq(R_{\delta}(t))^2\}.
\label{e3.24}
\end{equation}
Thus from  \eqref{e3.3}, \eqref{e3.4} we know that
$\hat{B}_{\delta,E_0}$ is a pullback $\mathscr{D}_{\delta,E_0}$
absorbing for the cocyle $\phi$.
\end{proof}

\begin{theorem} \label{thm3.5}
Suppose that $h_B,h_{S}\in L_{\rm loc}^2(\mathbb{R},Y_0)$ and
 $f_B,f_{S}$ satisfies $(F1)-(F3)$, then there exists a
global pullback $\mathscr{D}_{\delta,E_0}$-attractor in
$E_0$ for the non-autonomous dynamical system
$(\theta,\phi)$ defined by \eqref{e3.2}.
\end{theorem}

\begin{proof}
Using Theorem \ref{thm2.1}, it is only enough to verify
pullback  $\mathscr{D}-$condition (C). If
 $\{\omega_i\}_{i=1}^{\infty}$ is orthonormal basis of
 $Y_2$, which consists of eigenvectors of  $A^2$, it is also
orthonormal basis of  $Y_1 ,Y_0$. The corresponding eigenvalues
are denoted by
$$
0<\nu_1<\nu_2\leq\nu_{3}\leq\dots, \quad
\nu_i\to\infty,   i\to\infty,
$$
and $A^2\omega_i=\nu_i\omega_i$ for all $i\in \mathbb N$. We
write $V_m=\operatorname{span}\{\omega_1,\omega_2,\dots,\omega_m\}$,
$P_m:Y_2\to V_m$ is orthogonal projector. In addition,
let  $\{\chi_i\}_{i=1}^{\infty}$ be an orthonormal basis of
 $Y_1$ which consists of eigenvectors of  $A$, the corresponding
eigenvalue are denoted by
$$
0<\lambda_1<\lambda_2\leq\lambda_{3}\leq\dots, \quad
\lambda_i\to\infty, \;  i\to\infty,
$$
and $A\chi_i=\lambda_i\chi_i$ for all  $i\in \mathbb N$. In
fact, by the boundary value conditions \eqref{e1.2},
$\omega_i=\chi_i$,
$\nu_i=\lambda_i^2$, $i=1,2, \dots$. We
write $G_m=\operatorname{span}\{\chi_1,\chi_2,\dots,\chi_m\}$,
$Q_m:Y_1\to G_m$ is orthogonal projector. Then
for all  $u \in Y_2$, $v \in Y_1$, we make the
decomposition
\begin{equation}
u=P_mu+(I-P_m)u\triangleq u_1+u_2, \quad
v=Q_mv+(I-Q_m)v\triangleq v_1+v_2.\label{e3.25}
\end{equation}
Taking the scalar product in  $Y_0$ for the first equation of
\eqref{e1.1} with $\phi_2=u_{2t}+\epsilon u_2$ and for the second
equation with $\varphi_2=v_{2t}+\epsilon v_2$, respectively,
after a computation, we find
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(|\phi_2|^2+\alpha|Au_2|^2+\beta\|v_2\|^2
 +|\varphi_2|^2) +\epsilon\alpha|Au_2|^2\\
&+(\delta_1-\epsilon)|\phi_2|^2-\epsilon(\delta_1-\epsilon)
 (u_2,\phi_2)+\beta\epsilon|\varphi_2|^2
 +(\delta_2-\epsilon)|\varphi_2|^2-\epsilon(\delta_2-\epsilon)
 (v_2,\varphi_2)\\
&+k((u-v)^{+},\phi_2-\varphi_2)+((I-P_m)f_B(u),\phi_2)
 +((I-Q_m)f_{S}(v),\varphi_2)\\
&=((I-P_m)h_B(t),\phi_2)+((I-Q_m)h_{S}(t),\varphi_2).
\end{aligned}   \label{e3.26}
\end{equation}
Similar to the estimates of \eqref{e3.10}, it follows that
\begin{equation}
\begin{aligned}
&\epsilon\alpha|Au_2|^2+(\delta_1-\epsilon)|\phi_2|^2
 -\epsilon(\delta_1-\epsilon)(u_2,\phi_2)
 +\beta\epsilon|\varphi_2|^2\\
&+(\delta_2-\epsilon)|\varphi_2|^2
 -\epsilon(\delta_2-\epsilon)(v_2,\varphi_2)\\
&\geq\frac{\epsilon\alpha}{2}|Au_2|^2
 +\frac{\epsilon\beta}{2}\|v_2\|^2
 +\frac{\delta_1}{2}|\phi_2|^2
 +\frac{\delta_2}{2}|\varphi_2|^2.
\end{aligned} \label{e3.27}
\end{equation}
Using the H\"older, Young  and Poincar\'e inequalities,
there exists a positive constant $c$, such that
\begin{equation}
\begin{aligned}
k((u-v)^{+},\phi_2-\varphi_2)
&\le k|(u-v)_2|\cdot|\phi_2-\varphi_2|
 \le k(|u_2|+|v_2|)(|\phi_2|+|\varphi_2|)\\
&\le \frac{ck^2}{\delta_1\nu_{m+1}}|Au_2|^2
 +\frac{ck^2}{\delta_2\lambda_{m+1}}\|v_2\|^2
 +\frac{\delta_1}{4}|\phi_2|^2+\frac{\delta_2}{4}|\varphi_2|^2.
\end{aligned} \label{e3.28}
\end{equation}
 Together with \eqref{e3.27}-\eqref{e3.28},
from \eqref{e3.26}, yields
\begin{equation}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}(|\phi_2|^2+\alpha|Au_2|^2
 +\beta\|v_2\|^2+|\varphi_2|^2)\\
&+(\frac{\epsilon\alpha}{2}-\frac{ck^2}{\delta_1\nu_{m+1}})|Au_2|^2
+(\frac{\epsilon\beta}{2}-\frac{ck^2}{\delta_2\lambda_{m+1}})\|v_2\|^2
+\frac{\delta_1}{4}|\phi_2|^2+\frac{\delta_2}{4}|\varphi_2|^2\\
&+((I-P_m)f_B(u),\phi_2)+((I-Q_m)f_{S}(v),\varphi_2)\\
&\leq((I-P_m)h_B(t),\phi_2)+((I-Q_m)h_{S}(t),\varphi_2).
\end{aligned} \label{e3.29}
\end{equation}
Taking $m$ large enough, such that
$\frac{\epsilon\alpha}{2}-\frac{ck^2}{\delta_1\nu_{m+1}}\ge
\frac{\epsilon\alpha}{4}$,
$\frac{\epsilon\beta}{2}-\frac{ck^2}{\delta_2\lambda_{m+1}} \ge
\frac{\epsilon\beta}{4}$, we have
\begin{align*}
&\frac 12\frac{d}{dt}(|\phi_2|^2+\alpha|Au_2|^2
 +\beta\|v_2\|^2+|\varphi_2|^2)
+\frac{\epsilon\alpha}{4}|Au_2|^2+\frac{\epsilon\beta}{4}\|v_2\|^2\\
&+\frac{\delta_1}{4}|\phi_2|^2
+\frac{\delta_2}{4}|\varphi_2|^2+((I-P_m)f_B(u),\phi_2)+((I-Q_m)f_{S}(v),\varphi_2)\\
&\leq((I-P_m)h_B(t),\phi_2)+((I-Q_m)h_{S}(t),\varphi_2).
\end{align*}%\label{e3.30}
Furthermore, there holds
\begin{align*}
&\frac{d}{dt}(|\phi_2|^2+\alpha|Au_2|^2+\beta\|v_2\|^2+|\varphi_2|^2)
+\frac{\epsilon\alpha}{2}|Au_2|^2+\frac{\epsilon\beta}{2}\|v_2\|^2
+\frac{\delta_1}{4}|\phi_2|^2+\frac{\delta_2}{4}|\varphi_2|^2\\
&\le\frac{8}{\delta_1}|(I-P_m)h_B(t)|^2+\frac{8}{\delta_2}|(I-Q_m)h_{S}(t)|^2
+\frac{8}{\delta_1}|(I-P_m)f_B(u)|^2\\
&+\frac{8}{\delta_2}|(I-Q_m)f_{S}(v)|^2
\end{align*} %\label{e3.31}
Set $\xi=\min\{\epsilon,\frac{\delta_1}{4},\frac{\delta_2}{4}\}$,
and
$\chi(t)=|\phi_2|^2+\alpha|Au_2|^2+\beta\|v_2\|^2+|\varphi_2|^2>0$.
Then
\begin{equation}
\begin{aligned}
\frac{d}{dt}\chi(t)+\xi\chi(t)
&\leq\frac{8}{\delta_1}|(I-P_m)h_B(t)|^2+\frac{8}{\delta_2}|(I-Q_m)h_{S}(t)|^2
+\frac{8}{\delta_1}|(I-P_m)f_B(u)|^2\\
&+\frac{8}{\delta_2}|(I-Q_m)f_{S}(v)|^2.
\end{aligned} \label{e3.32}
\end{equation}
Multiplying  both sides of \eqref{e3.21} with $e^{\xi t}$, we obtain
\begin{align*}
\frac{d}{dt}(e^{\xi t}\chi(t))
&\leq e^{\xi t}(\frac{8}{\delta_1}|(I-P_m)h_B(t)|^2
+\frac{8}{\delta_2}|(I-Q_m)h_{S}(t)|^2\\
&\quad +\frac{8}{\delta_1}|(I-P_m)f_B(u)|^2
+\frac{8}{\delta_2}|(I-Q_m)f_{S}(v)|^2).
\end{align*} %\label{e3.33}
Integrating  over $[t-\tau,t]$, it leads to
\begin{equation}
\begin{aligned}
\chi(t)
&\leq e^{-\xi\tau}\chi(t-\tau)
+\frac{8}{\delta_1} e^{-\xi t}\int_{t-\tau}^{t}e^{\xi s}
 |(I-P_m)h_B(s)|^2ds \\
&\quad +\frac{8}{\delta_2}e^{-\xi t}\int_{t-\tau}^{t}e^{\xi s}
 |(I-Q_m)h_{S}(s)|^2ds
 + \frac{8}{\delta_1}e^{-\xi t}\int_{t-\tau}^{t}e^{\xi
s}|(I-P_m)f_B(u)|^2ds\\
&\quad +\frac{8}{\delta_2}e^{-\xi t}\int_{t-\tau}^{t}e^{\xi
s}|(I-Q_m)f_{S}(v)|^2ds.
\end{aligned}  \label{e3.34}
\end{equation}

Firstly, for any  $t\in \mathbb{R}$, any $\epsilon>0$, there exist
$t_1\in(t-\tau,t)$ and $\tau_1>0$ such that
$u(s)=u(s,t-\tau,y_0)\in B_{\delta}(s),  v(s)=v(s,t-\tau,y_0)\in
B_{\delta}(s)$, for $\tau\geq\tau_1$, any $s\in[t-\tau,t_1]$,
any $y_0\in D(t-\tau)$.
Also for all $\tau\geq\tau_1$,
$$
\int_{t-\tau}^{t_1}e^{-\xi(t-s)}|(I-P_m)f_B(u)|^2ds\leq\frac{\delta_1\epsilon}{40},
 \int_{t-\tau}^{t_1}e^{-\xi(t-s)}|(I-Q_m)f_{S}(v)|^2ds
\leq\frac{\delta_1\epsilon}{40}.
$$ %\label{e3.35}
Secondly, we set
 $\hat{R}=\max_{s\in[t_1,t]}R_{\delta}(s)<\infty$, then
$$
|Au(s)|=|Au(s;t-\tau,u_0)|\leq\hat{R}, \|v(s)\|
=\|v(s;t-\tau,v_0)\|\leq\hat{R}
$$
for any  $s\in[t_1,t]$ and any  $y_0\in D(t-\tau)$. In line with
Lemma \ref{lem3.3},   for any  $\epsilon>0$,  any
$m\geq m_1, \tau\geq\tau_1$, we have
\begin{equation}
\int_{t_1}^{t}e^{-\xi(t-s)}|(I-P_m)f_B(u)|^2ds
 \leq\frac{\delta_1\epsilon}{40},
\int_{t_1}^{t}e^{-\xi(t-s)}|(I-Q_m)f_{S}(v)|^2ds
 \leq\frac{\delta_2\epsilon}{40}.\label{e3.36}
\end{equation}
Thirdly,  by Lemma \ref{lem3.2}, we can choose  $m$ larger enough, such that
\begin{gather*}
\int_{t-\tau}^{t}e^{-\xi(t-s)}|(I-P_m)h_B(s)|^2ds
\leq\frac{\delta_1\epsilon}{20}, %\label{e3.37}
\\
\int_{t-\tau}^{t}e^{-\xi(t-s)}|(I-Q_m)h_{S}(s)|^2ds
\leq\frac{\delta_2\epsilon}{20}. %\label{e3.38}
\end{gather*}
 Finally, using \eqref{e3.5}, there exists $\tau_2\geq0$ such that
\begin{equation}
e^{-\xi\tau}\chi(t-\tau)\leq\frac{\epsilon}{5}, \quad
 \forall \tau\geq\tau_2, \; y_0\in D(t-\tau). \label{e3.39}
\end{equation}
Now let $\tau_0=\max\{\tau_1,\tau_2\}$, from
\eqref{e3.34}-\eqref{e3.39}
yields $\chi(t)\leq\epsilon$.  Therefore, it is easy to see that
$$
\|\phi_2(\tau,t-\tau,y_0)\|_{E_0}^2\leq\epsilon, \quad
 \forall \tau\geq\tau_0,  y_0\in D(t-\tau).
$$
 The proof is complete.
\end{proof}

We remark that our main results is also true for \eqref{e1.1}
with fixed boundary-value conditions ends:
 $$
u(0,t)=u(L,t)=u_x(0,t)=u_x(L,t)=0, \quad v(0,t)=v(L,t)=0, \quad
 t\ge  \tau.
$$
 In this case, we put $Y_2=H_0^2(0,L)$. Theorem \ref{thm3.4} and
\ref{thm3.5}, as
 well as their proofs, are remain valid without any changes.

\subsection*{Acknowledgements}
The authors are grateful to the anonymous referees for their
valuable comments. This work was partly supported by grant 11061030
from the NSFC, grant 0801-02 from the Education Department
Foundation of Gansu Province,  and the Fundamental Research Funds
for the Gansu Universities.

\begin{thebibliography}{00}

\bibitem{a1} N. U. Ahmed, H. Harbi;
\emph{Mathematical analysis of dynamic models of suspension
bridges}, SIAM J. Appl. Math., 58(3)(1998): 853-874.

\bibitem{c1} T. Caraballo, G. Lukaszewicz, J. Real;
\emph{Pullback attractors for asymptotically compact nonautonomous
dynamical systems}, Nonlinear Analysis, 64(2)(2006): 484-498.

\bibitem{c2} V. V. Chepyzhov, M. I. Vishik;
 \emph{Attractors for Equations of Mathematical Physics},
 Volume 49 of American  Mathematical Society Colloquium Publications,
 American Mathematical Society, Providence, RI, 2002.

\bibitem{h1} G. Holubor\'a, A. Matas;
\emph{Initial-boundary value problem for the nonlinear string-beam
 system}, J. Math. Anal. Appl., 288(2003): 784-802.

\bibitem{l1} A. C. Lazer, P. J. McKenna;
\emph{Large-amplitude periodic oscillations in suspension bridges:
some new connection with nonlinear analysis, SIAM Review,}
32(4)(1990): 537-578.

\bibitem{l2} G. Litcanu;
\emph{A mathematical model of suspension bridges}, Appl.Math.,
49(1)(2004): 39-55.

\bibitem{m1} Q. Z. Ma, C. K. Zhong,
\emph{Existence of global attractors for the coupled system of
suspension bridge equations}, J. Math. Anal. Appl., 308(2005): 365-379.

\bibitem{m2} Q. Z. Ma, C. K. Zhong,
\emph{Existence of strong solutions and global attractors for the
coupled suspension bridge equations}, J. Differential Equations,
246(2009): 3755-3775.

\bibitem{m3} Q. Z. Ma, S. P. Wang, X. B. Cheng,
\emph{Uniform compact attractors for the coupled suspension bridge
equtions}, Applied Mathematics and Computation, in press.

\bibitem{m4} Q. F. Ma, S. H. Wang, C. K. Zhong,
\emph{Necessary and sufficient conditions for the existence of
global attractors for semigroups and applications}, J. Indiana
Univ. Math., 51(6)(2002): 1541-1557.

\bibitem{m5} J. Malik;
\emph{Mathematical modelling of cable-stayed bridges:
existence, uniqueness, continuous dependence on data, homogenization
of cable systems}, Appl.Math., 49(1)(2004): 1-38.

\bibitem{p1} J. Y. Park, Y. R. Kang,
\emph{Pullback $\mathscr {D}$-attractors for non-autonomous
suspension bridge equations},  Nonliear Analysis,
71(2009): 4618-4623.

\bibitem{r1} J. C. Robison;
 Infinite-dimensional dynamical systems, an
introduction to dissipative parabolic PDEs and theory of global
attractors, Cambridge University Press, 2001.

\bibitem{t1} R. Temam,
\emph{Infinite-dimensional dynamical systems in mechanics and
physics}, seccond edition, Springer-Verlag, 1997.

\bibitem{w1} Y. H. Wang, C. K. Zhong;
\emph{Pullback attractors $\mathscr{D}$-attractor for
nonautonomous Sine-Gordon equations}, Nonlinear Analysis,
67(2007): 2137-2148.

\bibitem{w2} Y. H. Wang,
\emph{Pullback attractors for nonautonomous wave equations with
critical exponent}, Nonlinear Analysis, 68(2008): 365-376.

\bibitem{z1} C. K. Zhong, Q. Z. Ma, C. Y. Sun,
\emph{Existence of strong solutions and global attractors for the
suspension bridge equations},  Nonlinear Analysis,
67(2007): 442-454.

\end{thebibliography}

\end{document}