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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 96, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/96\hfil Existence and asymptotic behavior]
{Existence and asymptotic behavior of solutions to the generalized
damped Boussinesq equation}

\author[Y. Wang\hfil EJDE-2012/96\hfilneg]
{Yinxia Wang} 

\address{Yinxia Wang \newline
 School of Mathematics and Information Sciences,
 North China University of Water Resources and Electric
Power,  Zhengzhou 450011, China}
\email{yinxia117@126.com}

\thanks{Submitted May 31, 2012. Published June 10, 2012.}
\subjclass[2000]{35L30, 35L75}
\keywords{Generalized damped equation; global solution;
asymptotic behavior}

\begin{abstract}
 We consider the Cauchy problem for the  $n$-dimensional generalized
 damped Boussinesq equation. Based on decay estimates of solutions to
 the corresponding linear equation, we define a solution space with
 time weighted norms.  Under small condition on the initial value,
 the existence and asymptotic behavior  of global solutions
 in the corresponding Sobolev spaces are established
 by the contraction mapping principle.
\end{abstract}

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


\section{Introduction}

We study the Cauchy problem of the generalized damped Boussinesq equation 
in $n$ space dimensions
\begin{equation}
u_{tt}-a\Delta u_{tt}-2b\Delta u_t-\alpha\Delta^3u+\beta\Delta^2 u-\Delta u
=\Delta f(u)\label{e1.1}
\end{equation}
with the initial value
\begin{equation}
t=0:\quad  u=u_0(x),\quad u_t=u_1(x). \label{e1.2}
\end{equation}
Here $u = u(x, t) $ is the unknown function of
$ x = (x_1, \cdots , x_n)\in \mathbb{R}^n$ and $ t > 0$, $a, b, \alpha, \beta$
are positive constants.
The nonlinear term $ f(u) = O(u^{1+\theta})$ and $\theta$ is a positive integer.

 The first initial boundary value problem for
\begin{equation}
u_{tt}-a\Delta u_{tt}-2b\Delta u_t-\alpha\Delta^3u+\beta\Delta^2 u-\Delta u
=\gamma\Delta( u^2)\label{e1.3}
\end{equation}
in a unit circle was investigated in \cite{zll}, where $a, b, \alpha, \beta$
 are positive constants and $\gamma$ is a constant.
 The existence and the uniqueness of strong solution was established and the
solutions were constructed in the form of series in the small parameter present
in the initial conditions. The long-time asymptotics was also obtained in
the explicit form. In \cite{lwwl}, the authors considered  the  initial-boundary
value problem for \eqref{e1.3} in the unit ball $ B \subset \mathbb{R}^3$,
similar results were established.
It is well-known that the equation \eqref{e1.3} is closely contacted with
many wave equations.
For example, the equation (which we call the Bq equation)
$$
u_{tt}-u_{xx}+u_{xxxx}=(u^2)_{xx},
$$
which was derived by Boussinesq in 1872 to describe shallow water waves. The improved
Bq equation(which we call IBq equation) is
$$
u_{tt}-u_{xx}-u_{xxtt}=(u^2)_{xx}.
$$
A modification of the IBq equation analogous of the MKdV equation yields
$$
u_{tt}-u_{xx}-u_{xxtt}=(u^3)_{xx},
$$
which we call the IMBq equation (see \cite{m1}). \eqref{e1.1} is a
higher order wave equation. In \cite{WW}, we considered the Cauchy
problem for the Cahn-Hilliard equation with inertial term. Combining high
frequency, low frequency technique and energy methods,
 we obtained global existence and asymptotic behavior of solutions.
Wang, Liu and Zhang \cite{W5} investigated a fourth wave equation that
 is of the regularity-loss type. Based on the  decay property of the solution
operators,  global existence and asymptotic behavior of solutions are obtained.
 For  global existence and asymptotic behavior of solutions to higher order wave
equations, we refer to \cite{LK1}-\cite{LK2} and \cite{WX}-\cite{Y1} and
references therein.

 The main purpose of this paper is to establish
global existence  and asymptotic behavior of solutions to 
\eqref{e1.1}, \eqref{e1.2}  by using the contraction mapping
principle. Firstly, we consider the decay property of the following
linear equation
\begin{equation}
u_{tt}-a\Delta u_{tt}-2b\Delta u_t-\alpha\Delta^3u+\beta\Delta^2 u-\Delta u=0.
 \label{e1.4}
\end{equation}
We obtain the following decay estimate of solutions to \eqref{e1.4} associated
with initial condition \eqref{e1.2},
\begin{equation}
 \|\partial^k_xu(t)\|_{L^2}\leq  C(1+t)^{-\frac{n}{4}-\frac{k}{2}
-\frac{1}{2}}(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1}
+ \|u_0\|_{H^{s+2}}+\|u_1\|_{H^{s}}) \label{e1.5}
\end{equation}
$(k\leq s+2)$,
\begin{equation}
\|\partial^h_x u_t(t)\|_{L^2}\leq  C(1+t)^{-\frac{n}{4}
-\frac{h}{2}-1}(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1}
+ \|u_0\|_{H^{s+2}}+\|u_1\|_{H^{s}})  \label{e1.6}
\end{equation}
$(h \leq s)$
Based on the  estimates \eqref{e1.5} and \eqref{e1.6}, we define
a solution space with time weighted norms. 
Then global existence and asymptotic
behavior of  classical solutions to \eqref{e1.1}, \eqref{e1.2} are obtained
by using the contraction mapping principle.

 We give notation which is used in this paper.
Let $\mathscr{F}[u]$ denote the Fourier transform of $u$ defined by
$$
\hat{u}(\xi)=\mathscr{F}[u]=\int_{\mathbb{R}^n}e^{-i\xi\cdot x}u(x)dx, 
$$
and we denote its inverse transform by $\mathscr{F}^{-1}$.

For $1\leq p\leq \infty$, $L^p=L^p(\mathbb{R}^n)$ denotes the usual Lebesgue 
space with the norm $\|\cdot\|_{L^p}$. The usual Sobolev space of $s$ is
 defined by $H^s_p=(I-\Delta)^{-s/2}L^p$ with  the norm
 $\|f\|_{H^s_p}=\|(I-\Delta)^{s/2}f\|_{L^p}$; the homogeneous Sobolev 
space of $s$ is defined by $\dot{H}^s_p=(-\Delta)^{-s/2}L^p$ with 
 the norm $\|f\|_{H^s_p}=\|(-\Delta)^{s/2}f\|_{L^p}$;
 especially $H^s=H^s_2, \dot{H}^s=\dot{H}^s_2$. Moreover, we know that
 $H^s_p=L^p\cap \dot{H}^s_p$ for $s\geq 0$.

Finally, in this paper, we denote every positive constant by the same symbol $C$
or $ c$ without confusion. $[\cdot] $ is the Gauss symbol.

 The  article is organized as follows.  In Section 2 we derive the solution formula
of our semi-linear problem.  We study the decay property of the solution
operators appearing in the solution formula  in section 3. 
Then, in Section 4, we discuss the
linear problem and show the decay estimates. Finally, we prove  global existence 
 and  asymptotic behavior of solutions for the Cauchy problem 
\eqref{e1.1}, \eqref{e1.2} in Section 5.

\section{Solution formula}
The aim of this section is to derive the solution formula for
 problem \eqref{e1.1}, \eqref{e1.2}. We first investigate the equation \eqref{e1.4}. 
Taking the Fourier transform, we have
\begin{equation}
(1+a|\xi|^2)\hat{u}_{tt}+2b|\xi|^2\hat{u}_t+(\alpha|\xi|^6
+\beta|\xi|^4+|\xi|^2)\hat{u}=0. \label{e2.1}
\end{equation}
The corresponding  initial value are
\begin{equation}
t=0: \quad \hat{u}=\hat{u}_0(\xi), \quad \hat{u}_t=\hat{u}_1(\xi). \label{e2.2}
\end{equation}
The characteristic equation of \eqref{e2.1} is
\begin{equation}
(1+a|\xi|^2)\lambda^2+2b|\xi|^2\lambda+\alpha|\xi|^6+\beta|\xi|^4+|\xi|^2=0.
 \label{e2.3}
\end{equation}
Let $\lambda=\lambda_{\pm}(\xi)$ be the corresponding eigenvalues of \eqref{e2.3},
we obtain
\begin{equation}
\lambda_{\pm}(\xi)=\frac{-b|\xi|^2\pm|\xi|
\sqrt{-1-(a+\beta-b^2)|\xi|^2-(\alpha+a\beta)|\xi|^4-a\alpha|\xi|^6}}{1+a|\xi|^2}.
\label{e2.4}
\end{equation}
The solution to the problem \eqref{e2.1}-\eqref{e2.2} is given in the form
\begin{equation}
\hat{u}(\xi, t)=\hat{G}(\xi, t)\hat{u}_1(\xi)+\hat{H}(\xi, t)\hat{u}_0(\xi),
\label{e2.5}
\end{equation}
where
\begin{equation}
\hat{G}(\xi, t)=\frac{1}{\lambda_{+}(\xi)-\lambda_{-}(\xi)}
(e^{\lambda_{+}(\xi)t}-e^{\lambda_{-}(\xi)t}) \label{e2.6}
\end{equation}
and
\begin{equation}
\hat{H}(\xi, t)=\frac{1}{\lambda_{+}(\xi)
-\lambda_{-}(\xi)}(\lambda_{+}(\xi)e^{\lambda_{-}(\xi)t}-
\lambda_{-}(\xi)e^{\lambda_{+}(\xi)t}). \label{e2.7}
\end{equation}
We define $G(x, t)$  and $H(x, t)$ by
$$
 G(x, t) = \mathscr{F}^{-1}[\hat{G}(\xi, t)](x), \quad
 H(x, t) = \mathscr{F}^{-1}[\hat{H}(\xi, t)](x),
$$
respectively, where
$\mathscr{F}^{-1}$ denotes the inverse Fourier transform.
Then, applying $\mathscr{F}^{-1}$ to \eqref{e2.5}, we obtain
\begin{equation}
u(t)=G(t)*u_1+H(t)*u_0. \label{e2.8}
\end{equation}
By the Duhamel principle, we obtain the solution formula to \eqref{e1.1},
 \eqref{e1.2},
\begin{equation}
u(t)=G(t)*u_1+H(t)*u_0+\int^t_0G(t-\tau)*(I-a\Delta)^{-1}\Delta f(u)(\tau)d\tau.
 \label{e2.9}
\end{equation}


\section{Decay Property}
The aim of this section is to establish  decay estimates of the solution
 operators $G(t)$ and $H(t)$
appearing in the solution formula \eqref{e2.8}.

\begin{lemma} \label{lem3.1} 
The solution of  problem \eqref{e2.1}, \eqref{e2.2} satisfies
\begin{equation}
|\xi|^2(1+|\xi|^2)|\hat{u}(\xi, t)|^2+|\hat{u}_t(\xi, t)|^2
\leq Ce^{-c\omega(\xi)t}(|\xi|^2(1+|\xi|^2)|\hat{u}_0(\xi)|^2+|\hat{u}_1(\xi)|^2),
\label{e3.1}
\end{equation}
for $\xi \in \mathbb{R}^n $ and $t\geq 0$, where
 $\omega(\xi)=\frac{|\xi|^2}{1+|\xi|^2}$.
\end{lemma}

\begin{proof}
 Multiplying \eqref{e2.1} by $\bar{\hat{u}}_t$ and taking the real part yields
\begin{equation}
\frac{1}{2}\frac{d}{dt}\{(1+a|\xi|^2)|\hat{u}_t|^2
+(\alpha|\xi|^6+\beta|\xi|^4+|\xi|^2)|\hat{u}|^2\}+2b|\xi|^2|\hat{u}_t|^2=0.
 \label{e3.2}
\end{equation}
 Multiplying \eqref{e2.1} by $\bar{\hat{u}}$ and taking the real part, we obtain
 \begin{equation}
\frac{1}{2}\frac{d}{dt}\{b|\xi|^2|\hat{u}|^2+2(1+a|\xi|^2)
Re(\hat{u}_t\bar{\hat{u}})\}+(\alpha|\xi|^6+\beta|\xi|^4+|\xi|^2)|\hat{u}|^2
 -(1+a|\xi|^2)|\hat{u}_t|^2=0.
\label{e3.3}
\end{equation}
Multiplying both sides of \eqref{e3.2} and  \eqref{e3.3} by
 $(1+a|\xi|^2)$ and $b|\xi|^2$
respectively, summing up the products yields
\begin{equation}
\frac{d}{dt}E+F=0, \label{e3.4}
\end{equation}
where
\begin{align*}
E&=\frac{1}{2}(1+a|\xi|^2)^2|\hat{u}_t|^2+(1+a|\xi|^2)(\alpha|\xi|^6
+\beta|\xi|^4+|\xi|^2)|\hat{u}|^2
+b^2|\xi|^4|\hat{u}|^2\\
&\quad +b|\xi|^2(1+a|\xi|^2)
\operatorname{Re}(\hat{u}_t\bar{\hat{u}})
\end{align*}
and
$$
F=b|\xi|^2(\alpha|\xi|^6+\beta|\xi|^4+|\xi|^2)|\hat{u}|^2
+b|\xi|^2(1+a|\xi|^2)|\hat{u}_t|^2.
$$
A simple computation implies that
\begin{equation}
C(1+|\xi|^2)^2E_0\leq E\leq C(1+|\xi|^2)^2E_0, \label{e3.5}
\end{equation}
where
$$
E_0=|\xi|^2(1+|\xi|^2)|\hat{u}|^2+|\hat{u}_t|^2.
$$
Note that
$F \geq c|\xi|^2E_0$.
It follows from \eqref{e3.5} that
\begin{equation}
F\geq c\omega(\xi)E, \label{e3.6}
\end{equation}
where
$$
\omega(\xi)=\frac{|\xi|^2}{1+|\xi|^2}.
$$
Using \eqref{e3.4} and \eqref{e3.6}, we obtain
$$
\frac{d}{dt}E+c\omega(\xi)E\leq 0.
$$
Thus
$E(\xi, t)\leq e^{-cw(\xi)t}E(\xi, 0)$,
which together with \eqref{e3.5} proves the desired estimates \eqref{e3.1}.
 Then the proof is  complete.
\end{proof}

\begin{lemma} \label{lem3.2} 
Let $\hat{G}(\xi, t)$ and $\hat{H}(\xi, t)$ be the fundamental solution of  
\eqref{e1.4} in the Fourier space,  which are given
in \eqref{e2.6} and \eqref{e2.7}, respectively. Then we have the estimates
\begin{equation}
|\xi|^2(1+|\xi|^2)|\hat{G}(\xi, t)|^2+|\hat{G}_t(\xi, t)|^2
\leq Ce^{-c\omega(\xi)t} \label{e3.7}
\end{equation}
and
\begin{equation}
|\xi|^2(1+|\xi|^2)|\hat{H}(\xi, t)|^2+|\hat{H}_t(\xi, t)|^2
\leq C|\xi|^2(1+|\xi|^2)e^{-c\omega(\xi)t} \label{e3.8}
\end{equation}
for $\xi \in \mathbb{R}^n $ and $t\geq 0$, where
 $\omega(\xi)=\frac{|\xi|^2}{1+|\xi|^2}$.
\end{lemma}

\begin{proof} If $\hat{u}_0(\xi)=0$, from \eqref{e2.5}, we obtain
$$
\hat{u}(\xi, t)=\hat{G}(\xi, t)\hat{u}_1(\xi), \quad
 \hat{u}_t(\xi, t)=\hat{G}_t(\xi, t)\hat{u}_1(\xi).
$$
Substituting the equalities into \eqref{e3.1} with  $\hat{u}_0(\xi)=0$,
 we obtain \eqref{e3.7}.
In what follows, we consider  $\hat{u}_1(\xi)=0$, it follows from \eqref{e2.5} 
that
$$
\hat{u}(\xi, t)=\hat{H}(\xi, t)\hat{u}_0(\xi),\hat{u}_t(\xi, t)
=\hat{H}_t(\xi, t)\hat{u}_0(\xi).
$$
Substituting the equalities into \eqref{e3.1} with  $\hat{u}_1(\xi)=0$, we obtain
the desired estimate \eqref{e3.8}. The Lemma is proved.
\end{proof}


\begin{lemma} \label{lem3.3} Let $k\geq 0$ and $1\leq p\leq 2$. Then  we have
\begin{gather}
\|\partial^k_xG(t)*\phi\|_{L^2}\leq
 C(1+t)^{-(\frac{n}{2}(\frac{1}{p}-\frac{1}{2})+\frac{k}{2}
+\frac{l}{2}-\frac{1}{2})}\|\phi\|_{\dot{H}^{-l}_p}
+Ce^{-ct}\|\partial^{(k-2)_{+}}_x\phi\|_{L^2}, \label{e3.9}
\\
\|\partial^k_xH(t)*\phi\|_{L^2}\leq C(1+t)^{-(\frac{n}{2}
 (\frac{1}{p}-\frac{1}{2})+\frac{k}{2}+\frac{l}{2})}\|\phi\|_{\dot{H}^{-l}_p}
 + Ce^{-ct}\|\partial^{k}_x\phi\|_{L^2}\label{e3.10}
\\
\|\partial^k_xG_t(t)*\phi\|_{L^2}\leq C(1+t)^{-(\frac{n}{2}(\frac{1}{p}-\frac{1}{2})
+\frac{k}{2}+\frac{l}{2})}\|\phi\|_{\dot{H}^{-l}_p}
+Ce^{-ct}\|\partial^{k}_x\phi\|_{L^2}, \label{e3.11}
\\
\|\partial^k_xH_t(t)*\phi\|_{L^2}\leq C(1+t)^{-(\frac{n}{2}(\frac{1}{p}-\frac{1}{2})
+\frac{k}{2}+\frac{l}{2}+\frac{1}{2})}\|\phi\|_{\dot{H}^{-l}_p}
+ Ce^{-ct}\|\partial^{k+2}_x\phi\|_{L^2}\label{e3.12}
\\
 \|\partial^k_xG(t)*(I-a\Delta)^{-1}\Delta g\|_{L^{2}}\leq C(1+t)^{-(\frac{n}{4}
+\frac{k}{2}+\frac{1}{2})}\|g\|_{L^1}+Ce^{-ct}\|\partial^{k}_x g\|_{L^2},
\label{e3.13}
\\
 \|\partial^k_xG_t(t)*(I-a\Delta)^{-1}\Delta g\|_{L^{2}}\leq C(1+t)^{-(\frac{n}{4}
+\frac{k}{2}+1)}\|g\|_{L^1}+Ce^{-ct}\|\partial^{k}_x g\|_{L^2},
\label{e3.14}
\end{gather}
where $(k-2)_{+}= \max\{0, k-2\}$.
\end{lemma}

\begin{proof}
Firstly, we prove \eqref{e3.9}. By the Plancherel theorem and \eqref{e3.7},
 we obtain
\begin{equation}
\begin{aligned}
& \|\partial^k_xG(t)*\phi\|^2_{L^2} \\
&= { \int_{|\xi|\leq R_0}|\xi|^{2k}|\hat{G}(\xi, t)|^2|\hat{\phi}(\xi)|^2d\xi
 + \int_{|\xi|\geq R_0}|\xi|^{2k}|\hat{G}(\xi, t)|^2|\hat{\phi}(\xi)|^2d\xi}
\\
&\leq  C\int_{|\xi|\leq  R_0}|\xi|^{2k-2}e ^{-c|\xi|^2t}|\hat{\phi}(\xi)|^2d\xi\\
&\quad +{  Ce^{-ct}\int_{|\xi|\geq R_0}|\xi|^{2k}(|\xi|^2(1+|\xi|^2))^{-1}|
 \hat{\phi}(\xi)|^2d\xi}\\
&\leq    C\||\xi|^{-l}\hat{\phi}(\xi)\|^2_{L^{p'}}
\Big(\int_{|\xi|\leq R_0}|\xi|^{(2k-2+2l)q}e ^{-cq|\xi|^2t}d\xi\Big)^{1/q}\\
&\quad+ Ce^{-ct}\|\partial^{(k-2)_{+}}_x\phi\|^2_{L^2},
\end{aligned}
\label{e3.15}
\end{equation}
 where
$R_0$ is a small positive constant and $\frac{1}{p}+\frac{1}{p'}=1$,
$\frac{2}{p'}+\frac{1}{q}=1$.
It follows from Hausdorff-Young inequality that
\begin{equation}
\|\,|\xi|^{-l}\hat{\phi}(\xi)\|_{L^{p'}}\leq C\|(-\Delta)
^{-\frac{l}{2}}\phi\|_{L^p}. \label{e3.16}
\end{equation}
By a straight computation, we obtain
\begin{equation}
\begin{aligned}
{ \Big(\int_{|\xi|\leq R_0}|\xi|^{(2k-2+2l)q}e ^{-cq|\xi|^2t}d\xi\Big)
^{1/q}}&\leq {  C(1+t)^{-(\frac{n}{2q}+k-1+l)}}\\
&\leq {  C(1+t)^{-(n(\frac{1}{p}-\frac{1}{2})+k-1+l)}.}
\end{aligned}
\label{e3.17}
\end{equation}
Combining \eqref{e3.15}, \eqref{e3.16} and \eqref{e3.17} yields \eqref{e3.9}.

\quad  Similarly, using \eqref{e3.7} and \eqref{e3.8}, respectively, we can prove \eqref{e3.10}-\eqref{e3.12}.

\quad  In what follows, we prove \eqref{e3.13}.
 By the Plancherel theorem,  \eqref{e3.7},  and Hausdorff-Young inequality, we have
\begin{align*}
& \|\partial^k_xG(t)*(I-a\Delta)^{-1}\Delta g\|^2_{L^2}\\
&= { \int_{|\xi|\leq R_0}|\xi|^{2k}|\hat{G}(\xi, t)|^2|\xi|^4
 (1+a|\xi|^2)^{-2}|\hat{g}(\xi)|^2d\xi}\\
&\quad { +\int_{|\xi|\geq R_0}|\xi|^{2k}|\hat{G}(\xi, t)|^2|\xi|^4
 (1+|\xi|^2)^{-2}|\hat{g}(\xi)|^2d\xi}\\
&\leq { C\int_{|\xi|\leq  R_0}|\xi|^{2k+2}e ^{-c|\xi|^2t}|\hat{g}(\xi)|^2d\xi}
 +Ce^{-ct}\int_{|\xi|\geq  R_0}|\xi|^{2k}|\hat{g}(\xi)|^2d\xi \\
&\leq    C\|\hat{g}(\xi)\|^2_{L^\infty}\int_{|\xi|\leq R_0}|\xi|^{2k+2}
 e ^{-c|\xi|^2t}d\xi
 +Ce^{-ct}\|\partial^k_xg\|^2_{L^2} \\
&\leq    C (1+t)^{-(\frac{n}{2}+k+1)}\|g\|^2_{L^1}
+Ce^{-ct}\|\partial^k_xg\|^2_{L^2}. 
\end{align*}
 where $R_0$ is a small positive constant.
Thus \eqref{e3.13} follows. Similarly, we can prove \eqref{e3.14}.
Thus we have completed the proof of lemma.
\end{proof}

\section{Decay estimate for solutions to the linear equation}

\begin{theorem} \label{thm4.1}
   Assume that $u_0  \in H^{s+2}(\mathbb{R}^n)\cap \dot{H}^{-1}_1(\mathbb{R}^n)$,
 $u_1  \in H^{s}(\mathbb{R}^n)\cap \dot{H}^{-2}_1(\mathbb{R}^n)$
$(s\geq [\frac{n}{2}]+5)$. Then the classical solution $ u(x, t)$ to \eqref{e1.4}  
associated with initial condition \eqref{e1.2}, which is given by the formula 
\eqref{e2.8}, satisfies the decay estimates
\begin{equation}
\|\partial^k_x u(t)\|_{L^2}\leq  C(1+t)^{-\frac{n}{4}-\frac{k}{2}
-\frac{1}{2}}(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1}
+ \|u_0\|_{H^{s+2}}+\|u_1\|_{H^{s}})  \label{e4.1}
\end{equation}
for $k \leq s+2$,
\begin{equation}
\|\partial^h_x u_t(t)\|_{L^2}\leq  C(1+t)^{-\frac{n}{4}
-\frac{h}{2}-1}(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1}
+ \|u_0\|_{H^{s+2}}+\|u_1\|_{H^{s}})  \label{e4.2} 
\end{equation}
for $h\leq s$,
\begin{equation}
\|\partial^m_x u(t)\|_{L^\infty}\leq C(1+t)^{-\frac{n}{2}-\frac{m}{2}
-\frac{1}{2}}(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1}
+\|u_0\|_{H^{s+2}}+\|u_1\|_{H^{s}}) \label{e4.3}
\end{equation}
for $m \leq s+1-[\frac{n}{2}].$
\end{theorem}

\begin{proof}
 Firstly, we prove \eqref{e4.1}. Using \eqref{e3.9} and \eqref{e3.10}, we obtain
\begin{align*}
&{ \|\partial^k_x u(t)\|_{L^2}}\\
 &\leq { \|\partial^k_xG(t)*u_1\|_{L^2}+
 C\|\partial^h_xH(t)*u_0\|_{L^2}}\\
& \leq {  C(1+t)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}(\|u_0\|_{\dot{H}^{-1}_1}
 + \|u_1\|_{\dot{H}^{-2}_1})+Ce^{-ct}(\|u_0\|_{H^{s+2}}+\|u_1\|_{H^{s}} )}\\
& \leq   C(1+t)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}(\|u_0\|_{\dot{H}^{-1}_1}
+ \|u_1\|_{\dot{H}^{-2}_1}+ \|u_0\|_{H^{s+2}}+\|u_1\|_{H^{s}}).
\end{align*}
Similar to the proof of \eqref{e4.1}, using \eqref{e3.11} and \eqref{e3.12},
 we can prove \eqref{e4.2}.  In what follows, we prove \eqref{e4.3}.  
Using  \eqref{e4.1} and Gagliardo-Nirenberg inequality, it is not difficult
 to get \eqref{e4.3}.
 The Lemma is proved. 
\end{proof}

\section{Existence of global solution and asymptotic behavior}

 The purpose of this section is to prove the existence and asymptotic 
behavior of global solutions to the Cauchy problem \eqref{e1.1}, \eqref{e1.2}. 
We need the following Lemma, which come from \cite{lc} (see also \cite{zsm}).

 \begin{lemma} \label{lem5.1}
 Let $s$ and $\theta$ be positive integers, $\delta>0$, $p, q, r\in [1, \infty]$ 
satisfy $\frac{1}{r}  =\frac{1}{p}+\frac{1}{r}$, and let
$k\in \{0, 1, 2, \cdots, s\}$. Assume that $F(v)$ is a class of $C^s$ and satisfies
 $$
|\partial^l_vF(v)|\leq C_{l, \delta}|v|^{\theta+1-l}, \quad
|v|\leq \delta, \quad 0\leq l\leq s, l<\theta+1
$$
 and
$$
|\partial^l_vF(v)|\leq C_{l, \delta}, \;\; |v|\leq \delta,  l\leq s, \theta+1\leq l.
$$
 If $v\in L^p\cap W^{k, q}\cap L^\infty$ and $\|v\|_{L^\infty}\leq \delta$, 
then
\begin{gather*}
\|F(v)\|_{W^{k, r}} \leq C_{k, \delta}\|v\|_{W^{k, q}}\|v\|_{L^p}
\|v\|^{\theta-1}_{L^\infty},\\
\|\partial^\alpha_xF(v)\|_{L^r}\leq C_{k, \delta}
\|\partial^\alpha_x v\|_{L^{ q}}\|v\|_{L^p}\|v\|^{\theta-1}_{L^\infty},
 \quad |\alpha|\leq k.
\end{gather*}
\end{lemma}

 \begin{lemma} \label{lem5.2} 
Let $s$ and $\theta$ be positive integers, $\delta>0$, $p, q, r\in [1, \infty]$ 
satisfy $\frac{1}{r}  =\frac{1}{p}+\frac{1}{r}$, and let 
$k\in \{0, 1, 2, \cdots, s\}$. Let  $F(v)$ be a function that satisfies the
 assumptions of Lemma \ref{lem5.1}. Moreover, assume that
 $$
|\partial^s_vF(v_1)-\partial^s_vF(v_2)|\leq C_\delta(|v_1|+|v_2|)
^{\max\{\theta-s, \theta\}}|v_1-v_2|, \quad
 |v_1|\leq \delta, \quad |v_2|\leq \delta.
$$
 If $v_1, v_2\in L^p\cap W^{k, q}\cap L^\infty$ and 
$\|v_1\|_{L^\infty}\leq \delta, \|v_2\|_{L^\infty}\leq \delta$, then 
for $|\alpha|\leq k$, we have
 \begin{align*}
&\|\partial^\alpha_x(F(v_1)-F(v_2))\|_{L^r}\\
&\leq   C_{k, \delta}\{(\|\partial^\alpha_xv_1\|_{L^q}
 +\|\partial^\alpha_xv_2\|_{L^q})\|v_1-v_2\|_{L^p}\\
&\quad +   (\|v_1\|_{L^p}+\|v_2\|_{L^p})\|\partial^\alpha_x(v_1-v_2)
\|_{L^ q}\}(\|v_1\|_{L^\infty}+\|v_2\|_{L^\infty})^{\theta-1}.
\end{align*}
\end{lemma}

 Based on the estimates \eqref{e4.1}-\eqref{e4.3} of solutions to  \eqref{e1.4} 
associated with initial condition \eqref{e1.2}, we define the following 
solution space
$$
X=\{u\in C([0, \infty); H^{s+2}(\mathbb{R}^n))\cap C^1([0, \infty);
 H^{s}(\mathbb{R}^n)) : \|u\|_X<\infty\},
$$
 where
$$
{ \|u\|_{X}}=  \sup_{t\geq 0}\big\{\sum_{k \leq  s+2}(1+t)^{\frac{n}{4}
+\frac{k}{2}+\frac{1}{2}}\|\partial^k_xu(t)\|_{L^{2}}
+\sum_{h\leq s}(1+t)^{\frac{n}{4}
+\frac{h}{2}+1}\|\partial^h_xu_t(t)\|_{L^{2}}\},
$$
For $R>0$, we define
$X_R=\{u\in  X: \|u\|_X\leq R\}$.
For $m \leq s+1-[\frac{n}{2}]$, using Gagliardo-Nirenberg inequality, we obtain
\begin{equation}
\|\partial^m_xu(t)\|_{L^{\infty}}\leq C (1+t)^{-(\frac{n}{2}
+\frac{m}{2}+\frac{1}{2})}\|u\|_{X}. \label{e5.1}
\end{equation}

\begin{theorem} \label{thm5.1}  
 Assume that $u_0  \in H^{s+2}(\mathbb{R}^n)\cap \dot{H}^{-1}_1(\mathbb{R}^n)$, 
$u_1  \in H^{s}(\mathbb{R}^n)\cap \dot{H}^{-2}_1(\mathbb{R}^n)$
$(s\geq [\frac{n}{2}]+5)$ 
and integer $\theta\geq 2$. Let
$f(u)$ be a function of class $C^{s+2}$  and satisfy Lemmas \ref{lem5.1}
 and \ref{lem5.2}. 
Put
$$
E_0=\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1}
+\|u_0\|_{H^{s+2}}+\|u_1\|_{H^{s}}.
$$
If $E_0$ is suitably small, the Cauchy problem \eqref{e1.1}-\eqref{e1.2} has 
a unique global classical solution $u(x, t)$ satisfying
$u\in C([0, \infty); H^{s+2}(\mathbb{R}^n))$, 
$u_t\in C([0, \infty); H^{s}(\mathbb{R}^n))$,
$u_{tt}\in L^\infty([0, \infty); H^{s-2}(\mathbb{R}^n))$. 
 Moreover, the solution satisfies the decay estimate
\begin{gather}
\|\partial^k_xu(t)\|_{L^2}\leq CE_0 (1+t)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}},
 \label{e5.2}\\
\|\partial^h_xu_t(t)\|_{L^2}\leq CE_0(1+t)^{-\frac{n}{4}-\frac{h}{2}-1}\label{e5.3}
\end{gather}
for $k \leq s+2$ and $h\leq s$.
\end{theorem}

\begin{proof} 
Define the mapping
\begin{equation}
\Psi(u)=G(t)*u_1+H(t)*u_0+\int^t_0G(t-\tau)*(I-a\Delta)^{-1}\Delta f(u(\tau))d\tau.
 \label{e5.4}
\end{equation}
Using \eqref{e3.9}-\eqref{e3.10}, \eqref{e3.13}, Lemma \ref{lem5.1} and \eqref{e5.1},
for $k\leq s+2$ we obtain
\begin{align*}
&{ \|\partial^k_x \Psi(u)\|_{L^2}}\\
&\leq { C\|\partial^k_xG(t)*u_1\|_{L^2}+   C\|\partial^k_xH(t)*u_0\|_{L^2} }\\
 &\quad{ +C\int^{t}_0\|\partial^k_xG(t-\tau)*(I-a\Delta)^{-1}\Delta f(u(\tau))\|_{L^2}d\tau}\\
&\leq { C(1+t)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}(\|u_0\|_{\dot{H}^{-1}_1}}+ \|u_1\|_{\dot{H}^{-2}_1})+ Ce^{-ct}(\|u_0\|_{H^{s+2}+\|u_1\|_{H^s})}\\
&\quad{ +C\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}} \|f(u)\|_{L^1}d\tau}\\
&\quad{ +C\int^t_{t/2}(1+t-\tau)^{-\frac{n}{4}-\frac{1}{2}} \|\partial^k_xf(u)\|_{L^1}d\tau}\\
&\quad { +C\int^t_0e^{-c(t-\tau)} \|\partial^{k}_xf(u)\|_{L^2}d\tau}\\
&\leq { C(1+t)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}(\|u_0\|_{\dot{H}^{-1}_1}}+ \|u_1\|_{\dot{H}^{-2}_1})
+  Ce^{-ct}(\|u_0\|_{H^{s+2}+\|u_1\|_{H^s})}\\
&\quad{ +C\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}} \|u\|^2_{L^2}\|u\|^{\theta-1}_{L^\infty}d\tau}\\
&\quad{ +C\int^t_{t/2}(1+t-\tau)^{-\frac{n}{4}-\frac{1}{2}}
\|\partial^k_x u\|^2_{L^2}\|u\|^{\theta-1}_{L^\infty}d\tau}
{ +C\int^t_0e^{-c(t-\tau)} \|\partial^k_xu\|_{L^{2}}\|u\|^{\theta}_{L^\infty}d\tau}
\\
&\leq { C(1+t)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}(\|u_0\|_{\dot{H}^{-1}_1}}+ \|u_1\|_{\dot{H}^{-2}_1})
+  Ce^{-ct}(\|u_0\|_{H^{s+2}+\|u_1\|_{H^s})}\\
&\quad{ +CR^{\theta+1}\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}} (1+\tau)^{-(\frac{n}{2}+1)}(1+\tau)^{-(\frac{n}{2}+\frac{1}{2})(\theta-1)}d\tau}\\
&\quad{ +CR^{\theta+1}\int^{t}_{t/2}(1+t-\tau)^{-\frac{n}{4}-\frac{1}{2}} (1+\tau)^{-\frac{n}{2}-k-1}(1+\tau)^{-(\frac{n}{2}+\frac{1}{2})(\theta-1)}d\tau}\\
&\quad { +CR^{\theta+1}\int^t_0e^{-c(t-\tau)} (1+\tau)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}(1+\tau)^{-(\frac{n}{2}+\frac{1}{2})\theta}d\tau}\\
&\leq  C(1+t)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}\{(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1}+\|u_0\|_{H^{s+2}}+\|u_1\|_{H^s})
+R^{\theta+1}\}.
\end{align*}
Thus
\begin{equation}
(1+t)^{\frac{n}{4}+\frac{k}{2}+\frac{1}{2}}\|\partial^k_x \Psi(u)\|_{L^{2}}
\leq CE_0+CR^{\theta+1}. \label{e5.5}
\end{equation}
It follows from \eqref{e5.4} that
\begin{equation}
\Psi(u)_t=G_t(t)*u_1+H_t(t)*u_0+\int^t_0G_t(t-\tau)*(I-a\Delta)^{-1}
\Delta f(u(\tau))d\tau. \label{e5.6}
\end{equation}
Using \eqref{e3.11}-\eqref{e3.12}, \eqref{e3.14} Lemma \ref{lem5.1} and \eqref{e5.1},
for $h \leq s$ we have
\begin{align*}
&{ \|\partial^h_x \Psi(u)_t\|_{L^2}}\\
&\leq { C\|\partial^h_xG_t(t)*u_1\|_{L^2}+   C\|\partial^h_xH_t(t)*u_0\|_{L^2} }\\
&\quad { +C\int^{t}_0\|\partial^h_xG_t(t-\tau)*(I-a\Delta)^{-1}\Delta f(u(\tau))\|_{L^2}d\tau}\\
&\leq { C(1+t)^{-\frac{n}{4}-\frac{h}{2}-1}(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1})+ Ce^{-ct}(\|u_0\|_{H^{s+2}}+\|u_1\|_{H^s})}\\
&\quad { +C\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{h}{2}-1} \|f(u)\|_{L^1}d\tau}\\
&\quad { +C\int^t_{t/2}(1+t-\tau)^{-\frac{n}{4}-1} \|\partial^h_xf(u)
 \|_{L^1}d\tau}
+C\int^t_0e^{-c(t-\tau)} \|\partial^{h}_xf(u)\|_{L^2}d\tau
\\
&\leq { C(1+t)^{-\frac{n}{4}-\frac{h}{2}-1}(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1})
+  Ce^{-ct}(\|u_0\|_{H^{s+2}}+\|u_1\|_{H^s})}\\
&\quad { +C\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{h}{2}-1} \|u\|^2_{L^2}\|u\|^{\theta-1}_{L^\infty}d\tau}\\
&\quad { +C\int^t_{t/2}(1+t-\tau)^{-\frac{n}{4}-1} \|\partial^h_x
 u\|^2_{L^2}\|u\|^{\theta-1}_{L^\infty}d\tau}
+C\int^t_0e^{-c(t-\tau)} \|\partial^h_xu\|_{L^{2}}\|u\|^{\theta}_{L^\infty}d\tau
\\
&\leq { C(1+t)^{-\frac{n}{4}-\frac{h}{2}-1}(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1})
+  Ce^{-ct}(\|u_0\|_{H^{s+2}}+\|u_1\|_{H^s})}\\
&\quad { +CR^{\theta+1}\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{h}{2}-1} (1+\tau)^{-(\frac{n}{2}+1)}(1+\tau)^{-(\frac{n}{2}+\frac{1}{2})(\theta-1)}d\tau}\\
&\quad { +CR^{\theta+1}\int^{t}_{t/2}(1+t-\tau)^{-\frac{n}{4}-1} (1+\tau)^{-\frac{n}{2}-h-1}(1+\tau)^{-(\frac{n}{2}+\frac{1}{2})(\theta-1)}d\tau}\\
&\quad { +CR^{\theta+1}\int^t_0e^{-c(t-\tau)} (1+\tau)^{-\frac{n}{4}-\frac{h}{2}-1}(1+\tau)^{-(\frac{n}{2}+\frac{1}{2})\theta}d\tau}\\
&\leq  C(1+t)^{-\frac{n}{4}-\frac{h}{2}-1}\{(\|u_0\|_{\dot{H}^{-1}_1}+ \|u_1\|_{\dot{H}^{-2}_1}+\|u_0\|_{H^{s+2}}+\|u_1\|_{H^s})
+R^{\theta+1}\}.
\end{align*}
Thus
\begin{equation}
(1+t)^{\frac{n}{4}+\frac{h}{2}+1}\|\partial^h_x \Psi(u)_t\|_{L^{2}}
\leq CE_0+CR^{\theta+1}. \label{e5.7}
\end{equation}
Combining   \eqref{e5.5}, \eqref{e5.7} and taking $E_0$  and $R$ suitably
 small yields
\begin{equation}
\|\Psi(u)\|_X\leq R. \label{e5.8}
\end{equation}
For $\tilde{u},\bar{u}\in X_R$, by using \eqref{e5.4}, we have
\begin{equation}
\Psi(\tilde{u})-\Psi(\bar{u})
=\int^t_0G(t-\tau)*(I-a\Delta)^{-1}\Delta[f(\tilde{u})-f(\bar{u})]d\tau. \label{e5.9}
\end{equation}
Using  \eqref{e5.9}, \eqref{e3.13} and Lemma \ref{lem5.2}, \eqref{e5.1},
for $ k\leq s+2 $ we obtain
\begin{align*}
&{ \|\partial^k_x\Psi(\tilde{u})-\Psi(\bar{u}))\|_{L^2}} \\
&\leq
{\int^t_0\|\partial^k_xG(t-\tau)*(I-a\Delta)^{-1}\Delta[f(\tilde{u})-f(\bar{u})]\|_{L^2}d\tau}\\
&\leq { C\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}\|(f(\tilde{u})-f(\bar{u}))\|_{L^1}d\tau}\\
&\quad { +C\int^t_{t/2}(1+t-\tau)^{-\frac{n}{4}-\frac{1}{2}}\|\partial^k_x(f(\tilde{u})-f(\bar{u}))\|_{L^1}d\tau}\\
&\quad { +C\int^t_0e^{-c(t-\tau)}\|\partial^k_x(f(\tilde{u})-f(\bar{u}))\|_{L^2}d\tau}\\
&\leq { C\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}(\|\tilde{u}\|_{L^2}+\|\bar{u}\|_{L^2}) \|\tilde{u}-\bar{u}\|_{L^2}}\\
&\quad { \times(\|\tilde{u}\|_{L^\infty}+\|\bar{u}\|_{L^\infty})^{\theta-1}d\tau}\\
&\quad { +C\int^t_{t/2}(1+t-\tau)^{-\frac{n}{4}-\frac{1}{2}}\{(\|\partial^k_x\tilde{u}\|_{L^2}+\|\partial^k_x\tilde{u}\|_{L^2})
\|\tilde{u}-\bar{u}\|_{L^2}}\\
&\quad { +(\|\tilde{u}\|_{L^2}+\|\bar{u}\|_{L^2})\|\partial^k_x(\tilde{u}-\bar{u})\|_{L^2}\}(\|\tilde{u}\|_{L^\infty}
+\|\bar{u}\|_{L^\infty})^{\theta-1}d\tau}\\
&\quad { +C\int^t_0e^{-c(t-\tau)} \{(\|\partial^k_x\tilde{u}\|_{L^2}+\|\partial^k_x\tilde{u}\|_{L^2})\|\tilde{u}-\bar{u}\|_{L^\infty}}\\
&\quad { +(\|\tilde{u}\|_{L^\infty}+\|\bar{u}\|_{L^\infty})\|\partial^k_x(\tilde{u}-\bar{u})\|_{L^2}\}(\|\tilde{u}\|_{L^\infty}
+\|\bar{u}\|_{L^\infty})^{\theta-1}d\tau}\\
&\leq { CR^\theta\|\tilde{u}-\bar{u}\|_X\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{k}{2}-\frac{1}{2}}(1+\tau)^{-(\frac{n}{2}+\frac{1}{2})\theta}d\tau}\\
&\quad { +CR^\theta\|\tilde{u}-\bar{u}\|_X\int^t_{t/2}(1+t-\tau)^{-\frac{n}{4}-\frac{1}{2}}(1+\tau)^{-(\frac{\theta}{2}(n+1)+\frac{k+1}{2})}d\tau}\\
&\quad { +CCR^\theta\|\tilde{u}-\bar{u}\|_X\int^t_0e^{-c(t-\tau)}(1+\tau)^{-(\frac{n}{4}+\frac{n}{2}\theta+\frac{k}{2}+\frac{1}{2})}d\tau}\\
&\leq  { CR^\theta(1+t)^{-\frac{n}{4}- \frac{k}{2}-\frac{1}{2}} \|\tilde{u}-\bar{u}\|_{X},}
\end{align*}
which implies
\begin{equation}
(1+t)^{\frac{n}{4}+ \frac{k}{2}+\frac{1}{2}}\|\partial^k_x (\Psi(\tilde{u})
-\Psi(\bar{u}))\|_{L^{2}}\leq  CR^\theta \|\tilde{u}-\bar{u}\|_{X}. \label{e5.10}
\end{equation}
Similarly for $h\leq s$, from \eqref{e5.6}, \eqref{e3.14} and \eqref{e5.1}, we have
\begin{align*}
{ \|\partial^h_x(\Psi(\tilde{u})-\Psi(\bar{u}))_t\|_{L^2}}
&\leq
{\int^t_0\|\partial^h_xG_t(t-\tau)*(I-a\Delta)^{-1}\Delta[f(\tilde{u})-f(\bar{u})]\|_{L^2}d\tau}\\
&\leq { C\int^{t/2}_0(1+t-\tau)^{-\frac{n}{4}-\frac{h}{2}-1}\|(f(\tilde{u})-f(\bar{u}))\|_{L^1}d\tau}\\
&\quad { +C\int^t_{t/2}(1+t-\tau)^{-\frac{n}{4}-1}\|\partial^h_x(f(\tilde{u})-f(\bar{u}))\|_{L^1}d\tau}\\
&\quad { +C\int^t_0e^{-c(t-\tau)}\|\partial^h_x(f(\tilde{u})-f(\bar{u}))\|_{L^2}d\tau}\\
&\leq  { CR^\theta(1+t)^{-\frac{n}{4}- \frac{h}{2}-1} \|\tilde{u}-\bar{u}\|_{X},}
\end{align*}
which implies
\begin{equation}
(1+t)^{\frac{n}{4}+ \frac{h}{2}+1}\|\partial^h_x (\Psi(\tilde{u})
-\Psi(\bar{u}))_t\|_{L^{2}}\leq  CR^\theta \|\tilde{u}-\bar{u}\|_{X}. \label{e5.11}
\end{equation}
Using  \eqref{e5.10}, \eqref{e5.11} and taking $R$ suitably small yields
\begin{equation}
\|\Psi(\tilde{u})-\Psi(\bar{u})\|_{X}\leq  \frac{1}{2} \|\tilde{u}-\bar{u}\|_{X}.
 \label{e5.12}
\end{equation}
From \eqref{e5.8} and \eqref{e5.12}, we know that $\Psi$ is strictly contracting
 mapping. Consequently, we conclude that there
exists a fixed point $u \in X_R$  of the mapping $\Psi$, which is a classical
solution to \eqref{e1.1}, \eqref{e1.2}.
 This completes the proof.
\end{proof}

\subsection*{Acknowledgements} 
The author would like to thank the anonymous referee for his/her comments
and suggestions. This work was supported in part by 
grants 11101144 from the  NNSF of China,
and  201031 from the Research Initiation Project for High-level Talents
of North China  University of Water Resources and Electric Power.


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\end{document}