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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 48, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2007/48\hfil Existence and asymptotic expansion]
{Existence and asymptotic expansion of solutions
to a nonlinear wave equation with a memory condition at the
boundary}

\author[N. T. Long, L. X. Truong\hfil EJDE-2007/48\hfilneg]
{Nguyen Thanh Long, Le Xuan Truong}  % in alphabetical order

\address{Nguyen Thanh Long \hfill\break
 Department of Mathematics,
 Hochiminh City National University,
 227 Nguyen Van Cu, Q5, HoChiMinh City, Vietnam}
\email{longnt@hcmc.netnam.vn, longnt2@gmail.com}

\address{Le Xuan Truong \hfill\break
 Department of Mathematics, Faculty of General Science,
 University of Technical Education in HoChiMinh City, 01 Vo Van Ngan Str.,
 Thu Duc Dist., HoChiMinh City, Vietnam}
\email{lxuantruong@gmail.com}

\thanks{Submitted October 27, 2006. Published March 20, 2007.}
\subjclass[2000]{35L20, 35L70}
\keywords{Nonlinear wave equation; linear integral equation; \hfill\break\indent
 existence and uniqueness; asymptotic expansion}

\begin{abstract}
 We study the initial-boundary value problem for the nonlinear
 wave equation
\begin{gather*}
   u_{tt} - \frac{\partial }{\partial x} (\mu ({x,t})u_x )
   + K|u |^{p - 2} u + \lambda |u_t  |^{q - 2} u_t  = f(x,t),  \\
   u(0,t) = 0  \\
   - \mu (1,t)u_x (1,t) = Q(t),  \\
   u(x,0) = u_0 (x),\quad u_t (x,0) = u_1 (x), \\
\end{gather*}
 where $p\geq 2$, $q \geq 2$, $K, \lambda$ are given constants and
 $u_0, u_1, f,\mu$ are given functions. The unknown function
 $u(x,t)$ and the unknown boundary value $Q(t)$ satisfy the
 linear integral equation
\[
 Q(t)=K_1(t)u(1,t)+\lambda_1(t)u_t(1,t)-g(t)-\int_0^t {k(t-s)u(1,s)ds},
\]
 where $K_1, \lambda_1, g, k$ are given functions satisfying some
 properties stated in the next section. This paper consists of two
 main sections. First, we prove the existence and uniqueness for
 the solutions in a suitable function space. Then,  for the case
 $K_1(t)=K_1\geq 0$, we find the asymptotic expansion in
 $K,\lambda, K_1$ of the solutions, up to order $N+1$.
\end{abstract}

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

\section{Introduction}

In this paper, we consider the following problem: Find a pair
 of functions $(u,Q)$ satisfying
\begin{gather} \label{e1.1}
u_{tt}-\frac{\partial }{\partial x}(\mu (x,t)u_{x})
+F(u,u_t)=f(x,t), \quad 0<x<1,\; 0<t<T, \\
\label{e1.2} u(0,t)=0, \\
\label{e1.3} -\mu (1,t)u_{x}(1,t)=Q(t), \\
\label{e1.4} u(x,0)=u_0(x), \quad u_t(x,0)=u_1(x),
\end{gather}
where $F(u,u_t)=K|u|^{p-2}u+\lambda | u_t|^{q-2}u_t$, with
$p$, $q\geq 2$, $K$, $\lambda $ are given constants and $u_0$,
$u_1$, $f$, $\mu$ are given functions satisfying conditions
specified later; the unknown function $u(x,t)$ and the unknown
boundary value $Q(t)$ satisfy the  integral equation
\begin{equation} \label{e1.5}
Q(t)=K_1(t)u(1,t)+\lambda _1(t)u_t(1,t)-g(t)-\int_0^t k(t-s)u(1,s)ds,
\end{equation}
where $g$, $k$, $K_1$, $\lambda _1$ are given functions.
Santos [10] studied the asymptotic behavior of solution of problem
\eqref{e1.1}, \eqref{e1.2} and \eqref{e1.4} associated with a
boundary condition of memory type at $x=1$ as follows
\begin{equation} \label{e1.6}
u(1,t)+\,\int_0^t g(t-s)\mu (1,s)u_{x}(1,s)ds=0,\quad t>0.
\end{equation}
To make such a difficult condition simpler, Santos transformed
\eqref{e1.6} into \eqref{e1.3}, \eqref{e1.5} with
$K_1(t)=\frac{g'(0)}{g(0)}$, and $\lambda
_1(t)=\frac{1}{g(0)}$  positive constants.

In the case $\lambda _1(t)\equiv 0$, $K_1(t)=h\geq 0$,
 $\mu (x,t)\equiv 1$, the problem \eqref{e1.1}--\eqref{e1.5} is formed from the
problem \eqref{e1.1}--\eqref{e1.4} wherein, the unknown function
$u(x,t)$ and the unknown boundary value $Q(t)$ satisfy the
following Cauchy problem for ordinary differential equations
\begin{equation} \label{e1.7}
\begin{gathered}
Q''(t)+\omega ^2Q(t)=hu_{tt}(1,t),\quad 0<t<T,  \\
Q(0)=Q_0,\quad Q'(0)=Q_1,
\end{gathered}
\end{equation}
where $h\geq 0$, $\omega >0$, $Q_0$, $Q_1$ are given constants
[6].

An and Trieu [1]  studied a special case of problem
\eqref{e1.1}--\eqref{e1.4} and \eqref{e1.7} with
$u_0=u_1=Q_0=0$ and $F(u,u_t)=Ku+\lambda u_t$, with
$K\geq 0$, $\lambda \geq 0$ are given constants. In the later case
the problem \eqref{e1.1}--\eqref{e1.4} and \eqref{e1.7} is a
mathematical model describing the shock of a rigid body and a
linear viscoelastic bar resting on a rigid base [1].

From \eqref{e1.7} we represent $Q(t)$ in terms of $Q_0$,
$Q_1$, $\omega $, $h$, $u_{tt}(1,t)$ and then by integrating by
parts, we have
\begin{equation} \label{e1.8}
Q(t)=hu(1,t)-g(t)-\int_0^t k(t-s)u(1,s)ds,
\end{equation}
where
\begin{gather} \label{e1.9}
g(t)=-(Q_0-hu_0(1))\cos \omega t -\frac{1}{\omega
}(Q_1-hu_1(1))\sin \omega t, \\
 \label{e1.10}
k(t)=h\omega \sin \omega t.
\end{gather}
 Bergounioux, Long and Dinh [2] studied problem
\eqref{e1.1}, \eqref{e1.4} with the mixed boundary conditions
\eqref{e1.2}, \eqref{e1.3} standing for
\begin{gather} \label{e1.11}
u_{x}(0,t)=hu(0,t)+g(t)-\int_0^t k(t-s)u(0,s)ds, \\
\label{e1.12} u_{x}(1,t)+K_1u(1,t)+\lambda _1u_t(1,t)=0,
\end{gather}
where
\begin{gather} \label{e1.13}
g(t)=(Q_0-hu_0(0))\cos \omega t+\frac{1}{\omega
}(Q_1-hu_1(0))\sin \omega t, \\
 \label{e1.14} k(t)=h\omega \sin \omega t.
\end{gather}
where $h\geq 0$, $\omega >0$, $Q_0$, $Q_1$, $K$, $\lambda $,
$K_1$, $\lambda _1$ are given constants.

 Long, Dinh and Diem [7] obtained the unique existence, regularity and
asymptotic behavior of the problem \eqref{e1.1}, \eqref{e1.4} in
the case of $\mu (x,t)\equiv 1$,
$Q(t)=K_1u(1,t)+\lambda u_t(1,t)$,
$u_{x}(0,t)=P(t)$ where $P(t)$ satisfies
\eqref{e1.7} with $u_{tt}(1,t)$ is replaced
by $u_{tt}(0,t)$.

Long, Ut and Truc [9] gave the unique existence, stability,
regularity in time variable and asymptotic behavior for the
solution of problem \eqref{e1.1}--\eqref{e1.5} when
$F(u,u_t)=Ku+\lambda u_t$. In this case, the problem
\eqref{e1.1}--\eqref{e1.5} is the mathematical model describing a
shock problem involving a linear viscoelastic bar.

The present paper consists of two main parts. In Part 1 we prove a
theorem of global existence and uniqueness of weak solutions $(u, Q)$ of problem \eqref{e1.1} - \eqref{e1.5}. The proof is based on
a Galerkin type approximation associated to various energy
estimates-type bounds, weak-convergence and compactness arguments.
The main difficulties encountered here are the boundary condition
at $x=1$ and with the advent of the nonlinear term of
$F(u,u_t)$. In order to solve these particular difficulties,
stronger assumptions on the initial conditions $u_0$, $u_1$
and parameters $K$, $ \lambda $ will be modified. We remark that
the linearization method in the papers [3, 7] cannot be used in
[2, 5, 6]. In addition, in the case of
$ K_1(t)\equiv K_1\geq 0$,
we receive a theorem related to the asymptotic expansion of
the solutions with respect to $K$, $\lambda $, $K_1$ up to order
$N+1$. The results obtained here may be considered as the
generalizations of those in An and Trieu [1] and in Long, Dinh, Ut
and Truc [2, 3], [5-10].

\section{The existence and uniqueness theorem of solution}

Put $\Omega =(0,1)$, $Q_{T}=\Omega \times (0,T)$, $T>0$. We omit
the definitions of usual function spaces: $C^{m}(\overline{\Omega
}) $, $L^{p}(\Omega) $, $W^{m,p}(\Omega) $.  We denote
$W^{m,p}=W^{m,p}(\Omega) $, $ L^{p}=W^{0,p}(\Omega) $,
$H^{m}=W^{m,2}(\Omega) $, $1\leq p\leq \infty $, $m=0,1,\dots $
 The norm in $L^2$ is denoted by $\|\cdot \|$. We also denote by
  $\langle \cdot ,\cdot \rangle $
the scalar product in $ L^2$ or pair of dual scalar product of
continuous linear functional with an element of a function space.
We denote by $\|\cdot \|_{X}$ the norm of a
Banach space $X$ and by $X'$ the dual space of $X$. We denote by
$L^{p}(0,T;X)$, $1\leq p\leq \infty $ for the Banach space of the
real functions $u:(0,T)\rightarrow X$ measurable, such that
$$
\|u\|_{L^{p}(0,T;X)}=\Big(\int_0^{T}\|
u(t)\|_{X}^{p}dt\Big) ^{1/p}<\infty \quad \text{for } 1\leq p<\infty,
$$
and
$$
\|u\|_{L^{\infty }(0,T;X)}= \mathop{\rm ess\,sup}_{0<t<T}
\|u(t)\|_{X}\quad\text{for }p=\infty .
$$
Let $u(t)$, $u'(t)=u_t(t)$, $u''(t)=u_{tt}(t)$, $u_{x}(t)$, and
$u_{xx}(t)$ denote $u(x,t)$, $\frac{\partial u}{\partial t}(x,t)$,
$\frac{\partial ^2u}{\partial t^2}(x,t)$,
$\frac{\partial u}{\partial x}(x,t)$, and
$\frac{\partial ^2u}{\partial x^2}(x,t)$, respectively.
We put
\begin{gather} \label{e2.1}
V=\{v\in H^{1}(0,1):v(0)=0\}, \\
 \label{e2.2}
a(u,v)=\int_0^{1}\frac{\partial u}{\partial x}\frac{\partial v}{\partial x}
dx.
\end{gather}
The set $V$  is a closed subspace of $ H^{1}$ and on $V$, $\|v\|_{H^{1}}$
and $\|v\|_{V}=\sqrt{a(v,v)} =\|v_{x}\|$ are two
equivalent norms. Then we have the following result.

\begin{lemma} \label{lem2.1}
The imbedding $V\hookrightarrow C^{0}([0,1])$ is
compact and
\begin{equation} \label{e2.3}
\|v\|_{C^{0}([0,1])}\leq \|v\|_{V}, \text{ for all } \ v\in V.
\end{equation}
\end{lemma}
The proof is straightforward and we omit the details.
 We make the following assumptions:
\begin{itemize}
\item[(H1)] $K, \lambda \geq 0$,
\item[(H2)] $u_0\in V\cap H^2$, $u_1\in H^{1}$,
\item[(H3)] $ g, K_1, \lambda _1\in H^{1}(0,T)$,
$\lambda _1(t)\geq \lambda _0>0$, $K_1(t)\geq 0$,
\item[(H4)] $ k\in H^{1}(0,T)$,
\item[(H5)] $ \mu \in C^{1}(\overline{Q_{T}})$,
 $\mu _{tt}\in L^{1}(0,T;L^{\infty })$,
 $\mu (x,t)\geq \mu_0>0$,  for all $(x,t)\in \overline{Q_{T}}$,
\item[(H6)] $f, f_t\in L^2(Q_{T})$.
\end{itemize}
Then we have the following theorem.

\begin{theorem} \label{thm2.2}
Let (H1)--(H6)  hold. Then, for every $T>0$,
there exists a unique weak solution $(u, Q)$ of problem
\eqref{e1.1}--\eqref{e1.5} such that
\begin{equation} \label{e2.4}
\begin{gathered}
u\in L^{\infty }(0,T;V\cap H^2), \\
u_t\in L^{\infty }(0,T;V), \quad u_{tt}\in L^{\infty }(
0,T;L^2),\\
u(1,\cdot )\in H^2(0,T),\quad Q\in H^{1}( 0,T).
\end{gathered}
\end{equation}
\end{theorem}

\begin{remark} \label{rmk2.1} \rm
(i) Noting that with the regularity
obtained by \eqref{e2.4}, it follows that the component $u$ in the weak
solution $(u, Q)$ of problem \eqref{e1.1}--\eqref{e1.5}
satisfies
\begin{equation} \label{e2.5}
\begin{gathered}
u\in L^{\infty }(0,T;V\cap H^2)\cap C^{0}( 0,T;V)
\cap C^{1}(0,T;L^2),  \\
u_t\in L^{\infty }(0,T;V), u_{tt}\in L^{\infty }(
0,T;L^2),\quad u(1,\cdot )\in H^2(0,T).
\end{gathered}
\end{equation}

\noindent (ii)  From \eqref{e2.4} we can see that $u$, $u_{x}$, $u_t$,
$u_{xx}$, $u_{xt}$, $u_{tt}\in L^{\infty }(0,T;L^2)
\subset L^2(Q_{T})$. Also if $(u_0,u_1)\in
(V\cap H^2)\times H^{1}$, then the component $u$ in the weak
solution $(u, Q)$ of problem \eqref{e1.1}--\eqref{e1.5} belongs
to $H^2(Q_{T})\cap L^{\infty }(0,T;V\cap H^2)
\cap C^{0}(0,T;V)\cap C^{1}(0,T;L^2)$. So the
solution is almost classical which is rather natural since the
initial data $u_0$ and $u_1$ do not belong
necessarily to $V\cap C^2(\overline{\Omega })$ and
$C^{1}(\overline{\Omega })$, respectively.
\end{remark}

\begin {proof}[Proof of the Theorem 2.2]
The proof consists of Steps four steps.

\noindent{\bf Step 1.}  The Galerkin approximation. Let $\{w_{j}\}$ be
a denumerable base of $ V\cap H^2$. We find the approximate
solution of problem \eqref{e1.1}- \eqref{e1.5} in the form
\begin{equation} \label{e2.6}
u_{m}(t)=\sum_{j=1}^{m} c_{mj}(t)w_{j},
\end{equation}
where the coefficient functions $c_{mj}$ satisfy the system of
ordinary differential equations as follows
\begin{gather} \label{e2.7}
\begin{aligned}
&\langle u_{m}''(t),w_{j}\rangle  +\langle \mu
(t)u_{mx}(t),w_{jx}\rangle  +Q_{m}(t)w_{j}(1)
 +\langle F(u_{m}(t),u_{m}'(t)),w_{j}\rangle\\
&=\langle f\,(t),w_{j}\rangle, 1\leq j\leq m,
\end{aligned}  \\  \label{e2.8}
Q_{m}(t)=K_1(t)u_{m}(1,t)+\lambda
_1(t)u_{m}'(1,t)-\int_0^t k(t-s)u_{m}(1,s)ds-g(t), \\
\label{e2.9} \begin{gathered}
u_{m}(0)=u_{0m}=\sum_{j=1}^{m}\alpha _{mj}w_{j}\rightarrow u_0
\quad \text{ strongly in } V\cap H^2,  \\
u_{m}'(0)=u_{1m}=\sum_{j=1}^{m}\beta _{mj}w_{j}\rightarrow
u_1\quad \text{strongly in }H^{1}.
\end{gathered}
\end{gather}
From the assumptions of Theorem 2.2, system \eqref{e2.7}--\eqref{e2.9} has
solution $( u_{m}, Q_{m})$ on an interval $[0,T_{m}]$. The
following estimates allow one to take $T_{m}=T$ for all $m$.

\noindent{\bf Step 2.} A priori estimates:
\emph{A priori estimates I.} Substituting \eqref{e2.8} into \eqref{e2.7}, then
multiplying the $j^{th}$ equation of \eqref{e2.7} by $c_{mj}'(t)$,
summing up with respect to $j$ and afterwards integrating with
respect to the time variable from $0$ to $t$, we get after some
rearrangements
\begin{equation} \label{e2.10}
\begin{aligned}
S_{m}(t)&=S_{m}(0)+\int_0^t ds\int_0^{1}\mu
'(x,s)u_{mx}^2(x,s)dx+\int_0^t K_1'(s)u_{m}^2(1,s)ds\\
&\quad +2 \int_0^t g(s)u_{m}'(1,s)ds
+2\int_0^t u_{m}'(1,s)(\int_0^{s}k(s-\tau )u_{m}(1,\tau
)d\tau) ds\\
&\quad +2\int_0^t \langle f(s),u_{m}'(s)\rangle ds,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e2.11}
\begin{aligned}
S_{m}(t)&=\|u_{m}'(t)\|^2+\|\sqrt{\mu(
t)}u_{mx}(t)\|^2+K_1(t)u_{m}^2(1,t)+\frac{2K}{p}
\|u_{m}(t)\|_{L^{p}}^{p} \\
&\quad +2\lambda \int_0^t \|u_{m}'(s)\|
_{L^{q}}^{q}ds+2\int_0^t \lambda _1(s)|
u_{m}'(1,s)|^2ds .
\end{aligned}
\end{equation}
Using the inequality
\begin{equation} \label{e2.12}
2ab\leq \beta a^2+\frac{1}{\beta }b^2,\quad \forall a,b\in
\mathbb{R}, \forall \beta >0,
\end{equation}
and the following inequalities
\begin{gather} \label{e2.13}
S_{m}(t)\geq \|u_{m}'(t)\|^2+\mu
_0\|u_{mx}(t)\|^2+2\lambda_0\int_0^t |
u_{m}'(1,s)|^2ds, \\
 \label{e2.14}
|u_{m}(1,t)|\leq \|u_{m}(t)\|_{C^{0}(
\overline{\Omega })}\leq \|u_{mx}(t)\|\leq \sqrt{\frac{
S_{m}(t)}{\mu _0}},
\end{gather}
we shall estimate respectively the following terms on the right-hand
side of \eqref{e2.10} as follows
\begin{gather} \label{e2.15}
\int_0^t ds\int_0^{1}\mu '(x,s)u_{mx}^2(x,s)dx\leq
\frac{1}{\mu _0}\|\mu '\|
_{C^{0}(\overline{Q_{T}})
}\int_0^t S_{m}(s)ds, \\
 \label{e2.16}
\int_0^t K_1'(s)u_{m}^2(1,s)ds\leq \frac{1}{\mu _0}
\int_0^t |K_1'(s)|S_{m}(s)ds, \\
 \label{e2.17}
2\int_0^t g(s)u_{m}'(1,s)ds\leq \frac{1}{\beta }\|
g\|_{L^2(0,T)}^2+\frac{\beta }{2\lambda _0} S_{m}(t), \\
 \label{e2.18} \begin{aligned}
&2\int_0^t u_{m}'(1,s)\Big(\int_0^{s}k(s-\tau )u_{m}(1,\tau )d\tau\Big) ds\\
&\leq \frac{\beta }{2\lambda _0}S_{m}(t)+\frac{1}{\beta \mu _0}
T\|k\|_{L^2(0,T)}^2\int_0^t S_{m}(s)ds,  \end{aligned}\\
 \label{e2.19}
2\int_0^t \langle f\,(s),u_{m}'(s)\rangle ds\leq
\|f\|_{L^2(Q_{T})}^2+\int_0^t S_{m}(s)ds.
\end{gather}
In addition, from the assumptions
(H1), (H2), (H5) and the embedding
$H^{1}(0,1)\hookrightarrow L^{p}(0,1)$, $ p>1$, there exists a
positive constant $C_1$ such that for all $m$,
\begin{equation} \label{e2.20}
S_{m}(0)=\|u_{1m}\|^2+\|\sqrt{\mu (
0)}u_{0mx}\|^2+K_1(0)u_{0m}^2(1)+\frac{2K
}{p}\|u_{0m}\|_{L^{p}}^{p}\leq C_1
\end{equation}
Combining \eqref{e2.10}, \eqref{e2.11}, \eqref{e2.15}--\eqref{e2.20}
 and choosing $\beta=\frac{ \lambda _0}{2}$, we obtain
\begin{equation} \label{e2.21}
S_{m}(t)\leq M_{T}^{(1)}+\int_0^t N_{T}^{(1)}(s)S_{m}(s)ds,
\end{equation}
where
\begin{equation} \label{e2.22}
\begin{gathered}
M_{T}^{(1)}=2C_1+\frac{4}{\lambda _0}\|g\|
_{L^2(0,T)}^2+2\|f\|_{L^2(Q_{T})}^2, \\
N_{T}^{(1)}(s)=2[1+\frac{2}{\lambda _0\mu _0}T\|
k\|_{L^2(0,T)}^2+\frac{1}{\mu _0}\|
\mu'\|_{C^{0}(\overline{Q_{T}})}+\frac{1}{\mu _0}
|K_1'(s)|] ,  \\
N_{T}^{(1)}\in L^{1}(0,T).
\end{gathered}
\end{equation}
By Gronwall's lemma, we deduce from \eqref{e2.21}, \eqref{e2.22}, that
\begin{equation} \label{e2.23}
S_{m}(t)\leq M_{T}^{(1)}\exp (\int_0^t N_{T}^{(1)}(s)ds)
\leq
C_{T},\quad \text{for all }t\in [ 0,T].
\end{equation}

\noindent\emph{A priori estimates }II.
 Now differentiating \eqref{e2.7} with
respect to $t$ , we have
\begin{equation} \label{e2.24}
\begin{aligned}
&\langle u_{m}'''(t),w_{j}\rangle +\langle \mu
(t)u_{mx}'(t)+\mu
'(t)u_{mx}(t),w_{jx}\rangle +Q_{m}'(t)w_{j}(1) \\
&+K(p-1)\langle | u_{m}| ^{p-2}u_{m}',w_{j}\rangle
+\lambda (q-1)\langle |u_{m}'|
^{q-2}u_{m}'',w_{j}\rangle \\
&=\langle f\,'(t),w_{j} \rangle ,
\end{aligned}
\end{equation}
for all $1\leq j\leq m$.
Multiplying the  $j^{th}$ equation of (2.24) by $c_{mj}''(t)$,
summing up with respect to $j$  and then integrating with respect
to the time variable from $0$  to $t$, we have after some
rearrangements
\begin{equation} \label{e2.25}
\begin{aligned}
X_{m}(t)
&=X_{m}(0)+2\langle \mu
'(0)u_{0mx},u_{1mx}\rangle -2\langle \mu '(t)u_{mx}(t),u_{mx}'(t)\rangle \\
&\quad +2\int_0^t \langle \mu ''(s)u_{mx}(s),u_{mx}'(s)\rangle
ds+3\int_0^t ds\int_0^{1}\mu '(x,s)|u_{mx}'(x,s)|^2dx \\
&\quad -2\int_0^t \big(K_1'(s)-k(0)\big)
u_{m}(1,s)u_{m}''(1,s)ds \\
&\quad -2\int_0^t \big(K_1(s)+\lambda _1'(s)\big)
u_{m}'(1,s)u_{m}''(1,s)ds \\
&\quad +2\int_0^t u_{m}''(1,s)\big(g'(s)+\int_0^{s}k'(s-\tau
)u_{m}(1,\tau )d\tau\big) ds \\
&\quad -2(p-1)K\int_0^t \langle |u_{m}(s)|
^{p-2}u_{m}'(s),u_{m}''(s)\rangle
ds+2\int_0^t \langle f^{ /}(s),u_{m}''(s)\rangle ds,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e2.26}
\begin{aligned}
X_{m}(t)&=\|u_{m}''(t)\|^2+\| \sqrt{\mu (t)}u_{mx}'(t)\|
^2+2\int_0^t \lambda_1(s)|u_{m}''(1,s)|^2ds \\
&\quad +\frac{8}{q^2}(q-1)\lambda \int_0^t \|\frac{
\partial }{\partial s}\big(|u_{m}'(s)|^{\frac{q-2}{
2}}u_{m}'(s)\big)\|^2ds.
\end{aligned}
\end{equation}
From the assumptions (H1), (H2) ,
(H5), (H6) and the imbedding $H^{1}(0,1)\hookrightarrow L^{p}(0,1)$, $p>1$,
there exists positive constant $\widetilde{D}_1$ depending on $\mu $,
$u_0$, $u_1$, $K$, $\lambda $, $ p$, $q$, $f$ such that
\begin{equation} \label{e2.27}
\begin{aligned}
&X_{m}(0)+2\langle \mu '(0)u_{0mx},u_{1mx}\rangle\\
&=\|u_{m}''(0)\|^2+\|\sqrt{\mu (0)} u_{1mx}\|^2+2\langle \mu
'(0)u_{0mx},u_{1mx}\rangle \\
&\leq \|\mu (0)u_{0mxx}+\mu_{x}(0)u_{0mx}-K|u_{0m}|^{p-2}u_{0m}-\lambda
|u_{1m}|^{q-2}u_{1m}+f(0)\|^2 \\
&\quad +\|\sqrt{\mu (0)}u_{1mx}\|^2
 +2\|\mu '(0)\|_{L^{\infty }(\Omega)}\|u_{0mx}\|\|u_{1mx}\|
 \leq \widetilde{D}_1,
\end{aligned}
\end{equation}
for all $m$. Using the inequality \eqref{e2.12} where $\beta $ is
replaced by $ \beta _1$ and the following inequalities
\begin{gather} \label{e2.28}
X_{m}(t)\geq \|u_{m}''(t)\|^2+\mu_0\|
u_{mx}'(t)\|^2+2\lambda _0\overset{t}{\underset{0}{\int }}
|u_{m}''(1,s)|^2ds, \\
 \label{e2.29}
|u_{m}(1,t)|\leq \|u_{m}(t)\|_{C^{0}(
\overline{\Omega })}\leq \|u_{mx}(t)\|\leq \sqrt{\frac{
S_{m}(t)}{\mu _0}}\leq \sqrt{\frac{C_{T}}{\mu _0}}, \\
 \label{e2.30}
|u_{m}'(1,t)|\leq \|u_{m}'(t)\|
_{C^{0}(\overline{\Omega })}\leq \|u_{mx}'(t)\|
\leq \sqrt{\frac{X_{m}(t)}{\mu _0}},
\end{gather}
we estimate, without difficulty the following terms in the right-hand side
of \eqref{e2.25} as follows
\begin{equation} \label{e2.31}
-2\langle \mu '(t)u_{mx}(t),u_{mx}'(t)\rangle \leq
\beta _1X_{m}(t)+\frac{1}{\beta _1\mu _0}C_{T}\|\mu'\|
_{C^{0}(\overline{Q_{T}})}^2,
\end{equation}
\begin{equation} \label{e2.32}
\begin{aligned}
&2\int_0^t \langle \mu''(s)u_{mx}(s),u_{mx}'(s)\rangle ds\\
&\leq 2\int_0^t \|\mu ''(s)\|_{L^{\infty
}}\|u_{mx}(s)\|\|u_{mx}'(s)\|ds \\
&\leq \beta _1\frac{1}{\mu _0}\int_0^t \|\mu
''(s)\|_{L^{\infty }}\|u_{mx}(s)\|
^2ds+\beta _1\mu _0\int_0^t \|\mu ''(s)\|
_{L^{\infty }}\|u_{mx}'(s)\|^2ds \\
&\leq \beta _1\int_0^t \|\mu ''(s)\|
_{L^{\infty }}X_{m}(s)ds+\frac{C_{T}}{\beta _1\mu
_0}\|\mu ''\|_{L^{1}(0,T;L^{\infty })},
\end{aligned}
\end{equation}
\begin{equation} \label{e2.33}
3\int_0^t ds\int_0^{1}\mu '(x,s)| u_{mx}'(x,s)|^2dx\leq
\frac{3}{\mu _0}\|\mu
'\|_{C^{0}(\overline{Q_{T}})}\int_0^t X_{m}(s)ds,
\end{equation}
\begin{equation} \label{e2.34}
-2\int_0^t (K_1'(s)-k(0))
u_{m}(1,s)u_{m}''(1,s)ds\leq \frac{\beta _1}{2\lambda
_0}X_{m}(t)+ \frac{C_{T}}{\beta _1\mu _0}\|
K_1'-k(0)\|_{L^2(0,T)}^2,
\end{equation}
\begin{equation} \label{e2.35}
\begin{aligned}
&-2\int_0^t (K_1(s)+\lambda _1'(s))
u_{m}'(1,s)u_{m}''(1,s)ds \\
&\leq \frac{2}{\beta _1\mu _0}\int_0^t (| K_1(s)| ^2+|
\lambda _1'(s)|^2)X_{m}(s)ds+\frac{\beta _1}{
2\lambda _0}X_{m}(t),
\end{aligned}
\end{equation}
\begin{equation} \label{e2.36}
\begin{aligned}
&2\int_0^t u_{m}''(1,s)(g'(s)+\int_0^{s}k'(s-\tau
)u_{m}(1,\tau )d\tau) ds \\
&\leq \frac{\beta _1}{ 2\lambda
_0}X_{m}(t)+\frac{2}{\beta _1}[\|g'\|_{L^2(0,T)}^2
+\frac{C_{T}}{\mu _0} T\|k'\|_{L^{1}(0,T)}^2] ,
\end{aligned}
\end{equation}
\begin{equation} \label{e2.37}
-2(p-1)K\int_0^t \langle | u_{m}(s)|
^{p-2}u_{m}'(s),u_{m}''(s)\rangle ds
\leq 2\frac{p-1}{\sqrt{\mu
_0}}K(\frac{C_{T}}{\mu _0})^{\frac{p-2}{2}
}\int_0^t X_{m}(s)ds,
\end{equation}
\begin{equation} \label{e2.38}
2\int_0^t \langle f^{ /}(s),u_{m}''(s)\rangle ds\leq
\beta _1\int_0^t X_{m}(s)ds+\frac{1}{\beta _1}\|
f\,'\|
_{L^2(Q_{T})}^2.
\end{equation}
In terms of \eqref{e2.25}, \eqref{e2.27}, \eqref{e2.31}--\eqref{e2.38}
 and by the choice of $\beta _1>0$ such that
$$
\beta _1(1+\frac{3}{2\lambda_0})\leq \frac{1}{2},
$$
we obtain
\begin{equation} \label{e2.39}
X_{m}(t)\leq \widetilde{M}_{T}^{(2)}+\int_0^t N_{T}^{(2)}(s)X_{m}(s)ds,
\end{equation}
where
\begin{equation} \label{e2.40}
\begin{gathered}
\begin{aligned}
\widetilde{M}_{T}^{(2)}&=2\widetilde{D}_1+\frac{2C_{T}}{\beta
_1\mu _0} [\|\mu '\|
_{C^{0}(\overline{Q_{T}})}^2+\|\mu ''\|
_{L^{1}(0,T;L^{\infty })}+\|K_1'-k(0)\|
_{L^2(0,T)}^2] \\
&\quad +\frac{2}{\beta _1}[2\|g'\|_{L^2( 0,T)}^2
+\frac{2C_{T}}{\mu_0}T\|k'\|_{L^{1}(0,T)}^2+\|
f\,'\|_{L^2(Q_{T})}^2],
\end{aligned} \\
\begin{aligned}
N_{T}^{(2)}(s)
&=2\beta _1+4\frac{p-1}{\sqrt{\mu_0}}K(\frac{C_{T}}{ \mu _0})
^{\frac{p-2}{2}}+\frac{6}{\mu _0}\|\mu '\|
_{C^{0}(\overline{Q_{T}})}+2\beta _1\|\mu
''(s)\|_{L^{\infty }} \\
&\quad +\frac{4}{\beta _1\mu _0}(|
K_1(s)|^2+|\lambda _1'(s)|^2),
\end{aligned} \\
N_{T}^{(2)}\in L^{1}(0,T).
\end{gathered}
\end{equation}
 From \eqref{e2.39}--\eqref{e2.40} and applying Gronwall's inequality, we obtain
that
\begin{equation} \label{e2.41}
\begin{array}{c}
X_{m}(t)\leq M_{T}^{(2)}\exp \big(\int_0^t N_{T}^{(2)}(s)ds\big)
\leq C_{T}\quad \text{for all }t\in [0,T] .
\end{array}
\end{equation}
On the other hand, we deduce from \eqref{e2.8}, \eqref{e2.11},
\eqref{e2.23}, \eqref{e2.26} and
\eqref{e2.41}, that
\begin{equation} \label{e2.42}
\begin{aligned}
\|Q_{m}'\|_{L^2(0,T)}^2
&\leq \frac{5D_{T} }{2\lambda _0}\|\lambda _1\|
_{\infty }^2+\frac{ 5T^2C_{T}}{\mu _0}\,\|
k'\|_{L^2(0,T)}^2+5\|g'\|
_{L^2(0,T)}^2 \\
&\quad +\frac{5D_{T}}{\mu _0}(\|K_1+\lambda
_1'\|_{L^2(0,T)}^2+\|K_1'-\,k(0)\|_{L^2(0,T)}^2),
\end{aligned}
\end{equation}
where $\|\lambda _1\|_{\infty }=\|\lambda
_1\|_{L^{\infty }(0,T)}$.
From the assumptions (H3) and (H4), we
deduce from \eqref{e2.42}, that
\begin{equation} \label{e2.43}
\|Q_{m}\|_{H^{1}(0,T)}\leq C_{T}\quad \text{for all }m,
\end{equation}
where $C_{T}$ is a positive constant depending only on $T$.

\noindent{\bf Step 3.} Limiting process. From \eqref{e2.11}, \eqref{e2.23},
\eqref{e2.26}, \eqref{e2.41} and \eqref{e2.43}, we deduce the existence of a
subsequence of $\{(u_{m}$, $ Q_{m})\} $ still also so denoted,
such that
\begin{equation} \label{e2.44}
\begin{gathered}
u_{m}\rightarrow u \quad \text{in } L^{\infty }(0,T;V)\quad \text{weak*,}\\
u_{m}'\rightarrow u' \quad \text{in }  L^{\infty }(0,T;V) \quad
\text{weak*,}\\
u_{m}''\rightarrow u'' \quad \text{in }  L^{\infty }(0,T;L^2) \quad
 \text{ weak*,} \\
u_{m}(1,\cdot )\rightarrow u(1,\cdot ) \quad \text{in }  H^2(0,T) \quad
 \text{ weakly,} \\
Q_{m}\rightarrow \widetilde{Q} \quad \text{in }  H^{1}(0,T) \quad \text{weakly.}
\end{gathered}
\end{equation}
By the compactness lemma in Lions [4: p.57] and the imbedding
$H^2(0,T)\hookrightarrow C^{1}([0,T])$, we can deduce from
\eqref{e2.44}$_{1,2,3,4,5}$ the existence of a subsequence still denoted
by $ \{(u_{m},Q_{m})\}$ such that
\begin{equation} \label{e2.45}
\begin{gathered}
u_{m}\rightarrow u \quad \text{strongly in }  L^2(Q_{T}),  \\
u_{m}'\rightarrow u' \quad \text{strongly in } L^2(Q_{T}),  \\
u_{m}(1,\cdot )\rightarrow u(1,\cdot ) \quad \text{strongly in }  C^{1}([0,T]),  \\
Q_{m}\rightarrow \widetilde{Q}\quad  \text{strongly in } C^{0}([0,T]).
\end{gathered}
\end{equation}
From \eqref{e2.8}\ and \eqref{e2.45}$_3$ we have that
\begin{equation} \label{e2.46}
Q_{m}(t)\rightarrow K_1(t)u(1,t)+\lambda
_1(t)u'(1,t)-g(t)-\int_0^t k(t-s)\,u(1,s)ds\equiv Q(t)
\end{equation}
strongly  in $C^{0}([0,T])$.

Combining \eqref{e2.45}$_4$ and \eqref{e2.46}, we conclude that
\begin{equation} \label{e2.47}
\begin{array}{c}
Q(t)=\widetilde{Q}(t).
\end{array}
\end{equation}
By means of the  inequality
\begin{equation} \label{e2.48}
\big||x|^{\delta -2}x-|y|^{\delta -2}y\big|
\leq (\delta -1)R^{\delta -2}| x-y|quad \forall x,y\in [-R;R],
\end{equation}
for all $R>0$, $\delta \geq 2$, it follows from \eqref{e2.39}, that
\begin{equation} \label{e2.49}
||u_{m}|^{p-2}u_{m}-|u|^{p-2}u |\leq (p-1)R^{p-2}|
u_{m}-u|\quad \text{with }R=\sqrt{\frac{C_{T}}{\mu _0}}.
\end{equation}
Hence, it follows from \eqref{e2.45}$_1$ and \eqref{e2.49}, that
\begin{equation} \label{e2.50}
|u_{m}|^{p-2}u_{m}\rightarrow |u|
^{p-2}u\quad \text{strongly in }L^2(Q_{T}).
\end{equation}
By the same way, we deduce from \eqref{e2.48}, with
$R=\sqrt{\frac{C_{T}}{\mu _0}}$ and \eqref{e2.44}$_3$, \eqref{e2.45}$_2$,
that
\begin{equation} \label{e2.51}
|u_{m}'|^{q-2}u_{m}'\rightarrow |
u'|^{q-2}u'\quad \text{strongly in }L^2(Q_{T}).
\end{equation}
Passing to the limit in \eqref{e2.7}--\eqref{e2.9} by \eqref{e2.44}$_{1,5}$,
\eqref{e2.46}, \eqref{e2.47}, \eqref{e2.50} and \eqref{e2.51}
 we have $(u, Q)$ satisfying
\begin{gather} \label{e2.52}
\begin{aligned}
&\langle u''(t),v\rangle +\langle \mu (t)u_{x}(t),v_{x}\rangle
 +Q(t)v(1) +\langle K|u|^{p-2}u+\lambda |u'|^{q-2}u',v\rangle \\
&=\langle f(t),v\rangle ,\quad \forall v\in V,
\end{aligned} \\
\label{e2.53}
u(0)=u_0, \quad u'(0)=u_1, \\
 \label{e2.54}
Q(t)=K_1(t)u(1,t)+\lambda_1(t)u_t(1,t)-g(t)
-\int_0^t k(t-s)u(1,s)ds,
\end{gather}
On the other hand,  from \eqref{e2.44}$_{5}$, \eqref{e2.52} and
assumptions (H5)-(H6) we have
\begin{equation} \label{e2.55}
u_{xx}=\frac{1}{\mu (x,t)}(u''-\mu _{x}u_{x}+ K| u|
^{p-2}u+\lambda |u'|
^{q-2}u'-f )\in L^{\infty }(0,T;L^2).
\end{equation}
Thus $u\in L^{\infty }(0,T;V\cap H^2)$ and the existence of the
theorem is proved completely.

\noindent{\bf Step 4. } Uniqueness of the solution.
Let $(u_1, Q_1)$, $(u_2, Q_2)$ be two weak solutions of
problem \eqref{e1.1}--\eqref{e1.5}, such that
\begin{equation} \label{e2.56}
\begin{gathered}
u_{i}\in L^{\infty }(0,T;V\cap H^2),\quad
u_{i}'\in L^{\infty }(0,T;H^{1}),\quad
u_{i}''\in L^{\infty }(0,T;L^2),  \\
u_{i}(1,\cdot )\in H^2(0,T),\quad
Q_{i}\in H^{1}(0,T),\quad i=1, 2.
\end{gathered}
\end{equation}
Then $(u,Q)$ with  $u=u_1-u_2$ and $Q=Q_1-Q_2$ satisfy the
variational problem
\begin{equation} \label{e2.57}
\begin{gathered}
\langle u''(t),v\rangle +\langle \mu
(t) u_{x}(t),v_{x}\rangle +Q(t)v(1)+K\langle |
u_1|^{p-2}u_1-|u_2|^{p-2}u_2\,\,,v \rangle  \\
 + \lambda \langle |u_1'|^{q-2}u_1'-| u_2'| ^{q-2}u_2',
 v\rangle =0 \quad \forall  v\in V, \\
u(0)=\quad u'(0)=0,
\end{gathered}
\end{equation}
and
\begin{equation} \label{e2.58}
\,Q(t)=K_1(t)u(1,t)+\lambda _1(t)u'(1,t)-\int_0^t k(t-s)\,u(1,s)ds.
\end{equation}
We take $v=u'$ in \eqref{e2.57}$_1$, and integrating with
respect to $t$, we obtain
\begin{equation} \label{e2.59}
\begin{aligned}
\sigma (t)&\leq \int_0^t \|\sqrt{|\mu'(s)|}u_{x}(s)\|
^2ds+\int_0^t K_1'(s)u^2(1,s)ds \\
&\quad +2\int_0^t u'(1,s)ds\int_0^{s}k(s-\tau )\,u(1,\tau )d\tau \\
&\quad -2K\int_0^t \langle |u_1|^{p-2}u_1-|u_2| ^{p-2}u_2,
u'\rangle ds,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e2.60}
\sigma (t)=\|u'(t)\|^2+\|\sqrt{\mu
(t)} u_{x}(t)\|
^2+K_1(t)u^2(1,t)+2\int_0^t \lambda
_1(s)|u'(1,s)|^2ds.
\end{equation}
Noting that
\begin{gather} \label{e2.61}
\sigma (t)\geq \|u'(t)\|^2+\mu _0\|
u_{x}(t)\|^2+2\lambda _0\int_0^t |
u'(1,s)|^2ds, \\
 \label{e2.62}
|u(1,t)|\leq \|u(t)\|_{C^{0}(
\overline{\Omega })}\leq \|u_{x}(t)\|\leq \sqrt{\frac{
\sigma (t)}{\mu _0}}.
\end{gather}
We again use inequalities \eqref{e2.12} and \eqref{e2.48} with
$\delta =p$, $R=\max_{i=1,2} \|u_{i}\|_{L^{\infty }(0,T;V)}$,
then, it follows from \eqref{e2.59}--\eqref{e2.62}, that
\begin{equation} \label{e2.63}
\begin{aligned}
\sigma (t)&\leq \frac{1}{\mu _0}\int_0^t (\|\mu
'\|_{C^{0}(\overline{Q_{T}})}+|K_1'(s)|) \sigma (s)ds
+\frac{\beta }{2\lambda _0} \sigma (t) \\
&\quad +\frac{T}{\beta \mu _0}\|k\|
_{L^2(0,T)}^2\int_0^t \sigma (\tau )d\tau +\frac{1}{\sqrt{\mu _0}}
(p-1)KR^{p-2}\int_0^t \sigma (s)ds.
\end{aligned}
\end{equation}
Choosing $\beta >0$, such that $\beta \frac{1}{2\lambda _0}\leq 1/2$,
we obtain from \eqref{e2.63}, that
\begin{equation} \label{e2.64}
\sigma (t)\leq \int_0^t q_1(s)\sigma (s)ds,
\end{equation}
where
\begin{equation} \label{e2.65}
\begin{gathered}
q_1(s)=\frac{2}{\mu _0}(\|\mu '\|
_{C^{0}(\overline{Q_{T}})}+| K_1'(s)|
) +\frac{2T}{\beta \mu _0}\|k\|_{L^2(0,T)}^2+
\frac{2}{\sqrt{\mu _0}}(p-1)KR^{p-2},  \\
q_1\in L^2(0,T).
\end{gathered}
\end{equation}
By Gronwall's lemma, we deduce that $\sigma \equiv 0$ and Theorem
2.2 is completely proved.
\end{proof}


\begin{remark} \label{rmk2.2} \rm
 In the case $p$, $q>2$, $K<0$, and
$\lambda <0$, the question of existence for the solutions of
problem \eqref{e1.1}--\eqref{e1.5} is still open. However we have
also obtained the answer of problem \eqref{e1.1}--\eqref{e1.5} when
$ p=q=2$ and $K$, $\lambda \in \mathbb{R}$ published in [9].
\end{remark}

\section{Asymptotic expansion of the solution}

In this part, we consider two given functions $u_0$, $u_1$ as
$ \widetilde{u}_0$, $\widetilde{u}_1$, respectively. Then we
assume that $ K_1(t)=K_1$ is a nonnegative constant and
$(\widetilde{u}_0$, $ \widetilde{u}_1$, $f$, $\mu$, $g$,
$k$, $\lambda _1)$ satisfy the
assumptions (H2)-(H6). Let $(K, \lambda , K_1) \in
\mathbb{R}_{+}^{3}$.
 By Theorem 2.2, the problem \eqref{e1.1}--\eqref{e1.5}
has a unique weak solution $(u, Q)$ depending on $(K, \lambda, K_1)$:
$$
u=u(K,\lambda ,K_1),\quad Q=Q(K,\lambda ,K_1).
$$
We consider the following perturbed problem, where $K$, $\lambda $,
$K_1$ are small parameters such that, $0\leq K\leq K_{\ast }$,
$0\leq \lambda \leq \lambda _{\ast }$, $0\leq K_1\leq K_{1\ast }$:
\begin{equation} \label{PKLK1} %(P_{K, \lambda, K_1})
\begin{gathered}
Au\equiv u_{tt}-\frac{\partial }{\partial x}(\mu (x,t)u_{x})
=-KF(u)-\lambda G(u_t)+f(x,t),\quad 0<x<1,0<t<T, \\
u(0,t)=0,\\
Bu\equiv -\mu (1,t)u_{x}(1,t)=Q(t),\\
u(x,0)=\widetilde{u}_0(x),\quad u_t(x,0)=\widetilde{u}_1(x),\\
Q(t)=K_1u(1,t)+\lambda _1(t)u_t(1,t)-g(t)-\int_0^t k(t-s)u(1,s)ds,
\end{gathered}
\end{equation}
 where $F(u)=|u|^{p-2}u$, $G(u_t)=|u_t|^{q-2}u_t$,
$p>N\geq 2$, $q>N\geq 2$. We shall study the
asymptotic expansion of the solution of problem
$(P_{K,\lambda ,K_1})$ with respect to $($ $K$, $\lambda $, $K_1)$.
We use the following notation. For a multi-index
$\gamma =(\gamma _1,\gamma _2,\gamma _3)\in
\mathbb{Z}_{+}^{3}$ and $\overrightarrow{K}=(K, \lambda, K_1)\in
\mathbb{R}_{+}^{3}$, we put
\begin{gather*}
|\gamma |=\gamma _1+\gamma _2+\gamma _3,\quad
\gamma !=\gamma _1!\gamma _2!\gamma _3!,\\
\|\overrightarrow{K}\|=\sqrt{K^2+\lambda ^2+K_1^2} ,\quad
\overrightarrow{K}^{\gamma }=K^{\gamma_1}\lambda ^{\gamma
_2}K_1^{\gamma _3},  \\
\alpha ,\beta \in \mathbb{Z}
_{+}^{3},\quad \beta \leq \alpha \Longleftrightarrow \beta
_{i}\leq
\alpha _{i}\quad \forall i=1,2,3.
\end{gather*}
 First, we shall need the following Lemma.

\begin{lemma} \label{thm3.1}
Let $m$, $N \in \mathbb{N}$ and $v_{\alpha }\in \mathbb{R}$,
$\alpha \in \mathbb{Z}_{+}^{3}$, $1\leq |\alpha |\leq N$. Then
\begin{equation} \label{e3.1}
(\sum_{1\leq |\alpha |\leq N}v_{\alpha }
\overrightarrow{K}^{ \alpha })^{m}=\sum_{m\leq |\alpha
|\leq mN}T^{(m)}[v]_{\alpha }\overrightarrow{K}^{ \alpha },
\end{equation}
 where the coefficients $T^{(m)}[v]_{\alpha }$, $m\leq |\alpha
|\leq mN$  depending on $v=(v_{\alpha })$, $\alpha \in
\mathbb{Z}_{+}^{3}$, $1\leq |\alpha |\leq N$ are defined by the
recurrence formulas
\begin{equation} \label{e3.2}
\begin{gathered}
T^{(1)}[v]_{\alpha }=v_{\alpha },\quad 1\leq |\alpha |\leq N,\\
T^{(m)}[v]_{\alpha }=\sum_{\beta \in A_{\alpha }^{(m)}}
v_{\alpha -\beta }T^{(m-1)}[v]_{\beta },\quad m\leq |\alpha
|\leq mN, m\geq 2, \\
A_{\alpha }^{(m)}=\{\beta \in \mathbb{Z} _{+}^{3}:\beta \leq
\alpha , 1\leq |\alpha -\beta
|\leq N, m-1\leq |\beta |\leq (m-1)N\}.
\end{gathered}
\end{equation}
\end{lemma}

The proof of the above lemma can be found in [11].
Let $(u_0$, $Q_0)\equiv (u_{0,0,0}$, $Q_{0,0,0})$ be a unique
weak solution of the following problem  (as in Theorem 2.2)
corresponding to $ (K, \lambda , K_1) =(0, 0, 0)$;
i.e.,
\begin{gather*} %(P_{0,0,0} )
Au_0=P_{0,0,0}\equiv f(x,t),\quad 0<x<1, 0<t<T, \\
u_0(0,t)=0,\quad Bu_0=Q_0(t),\\
u_0(x,0)=\widetilde{u}_0(x),\quad u_0'(x,0)=\widetilde{u}_1(x), \\
Q_0(t)=-g(t)+\lambda
_1(t)u_0'(1,t)-\int_0^t k(t-s)u_0(1,s)ds,  \\
u_0\in C^{0}(0,T;V)\cap C^{1}(0,T;L^2)\cap
L^{\infty }(0,T;V\cap H^2),  \\
u_0'\in L^{\infty }(0,T;H^{1}),\quad
u_0''\in L^{\infty }(0,T;L^2), \\
u_0(1,\cdot )\in H^2(0,T),\quad Q_0\in
H^{1}(0,T).
\end{gather*}
 Let us consider the sequence of weak solutions
$(u_{\gamma } , Q_{\gamma })$, $\gamma \in
\mathbb{Z}_{+}^{3}$, $1\leq |\gamma |\leq N$, defined by the following
problems ($\widetilde{P}_{\gamma }$):
\begin{equation} \label{tPg}
\begin{gathered}
Au_{\gamma }=P_{\gamma },\quad 0<x<1,\; 0<t<T, \\
u_{\gamma }(0,t)=0,\quad Bu_{\gamma }=Q_{\gamma }(t), \\
u_{\gamma }(x,0)=u_{\gamma }'(x,0)=0,\\
Q_{\gamma }(t)=\widehat{Q}_{\gamma }(t)+\lambda _1(t)u_{\gamma
}'(1,t)-\int_0^t k(t-s)u_{\gamma }(1,s)ds,  \\
u_{\gamma }\in C^{0}(0,T;V)\cap C^{1}( 0,T;L^2)
\cap L^{\infty }(0,T;V\cap H^2),  \\
u_{\gamma }'\in L^{\infty }(0,T;H^{1}),\quad
u_{\gamma }''\in L^{\infty }(0,T;L^2), \\
u_{\gamma }(1,\cdot )\in H^2(0,T),\quad
Q_{\gamma }\in H^{1}(0,T),
\end{gathered}
\end{equation}
where $P_{\gamma }$, $\widehat{Q}_{\gamma }$, $|\gamma |\leq N$ are defined
by the recurrence formula
\begin{equation} \label{e3.3}
\begin{gathered}
\widehat{Q}_{\gamma }(t)=0,\quad 1\leq |\gamma |\leq N,\quad \gamma _3=0, \\
\widehat{Q}_{\gamma }(t)=u_{\gamma _1,\gamma _2,\gamma
_3-1}(1,t), \quad  1\leq |\gamma |\leq N,\quad
\gamma _3\geq 1,
\end{gathered}
\end{equation}
and
\begin{equation} \label{e3.4}
\begin{gathered}
P_{1,0,0}=-F(u_0),\quad
P_{0,1,0}=-G(u_0'), \quad P_{0,0,1}=0,\\
P_{0,0,\gamma _3}=0,\quad 2\leq \gamma _3\leq N,\\
P_{0,\gamma _2,\gamma _3}=-\sum_{m=1}^{|\gamma |-1}
\frac{1}{m!}G^{(m)}(u_0')T^{(m)}[u']_{0,\gamma
_2-1,\gamma _3},\quad 2\leq \gamma _2+\gamma _3\leq N,
\gamma _2\geq 1,  \\
P_{\gamma _1,0,\gamma _3}=-\sum_{m=1}^{|\gamma |-1}
\frac{1}{m!}F^{(m)}(u_0)T^{(m)}[u]_{\gamma_1-1,0,\gamma
_3},\quad 2\leq \gamma _1+\gamma _3\leq N, \gamma _1\geq 1
,\quad  \\
P_{\gamma }=-\sum_{m=1}^{|\gamma |-1}\frac{1}{m!}[
F^{(m)}(u_0)T^{(m)}[u]_{\gamma _1-1,\gamma _2,\gamma
_3}+G^{(m)}(u_0')T^{(m)}[u']_{\gamma _1,\gamma
_2-1,\gamma _3}] ,  \\
2\leq |\gamma |\leq N, \gamma _1\geq 1,  \gamma_2\geq 1,
\end{gathered}
\end{equation}
here we have used the notation $u=(u_{\gamma })$,
$\gamma\in \mathbb{Z}_{+}^{3}$, $|\gamma |\leq N$.
Let $(u,Q)=(u_{K,\lambda ,K_1}, Q_{K,\lambda,K_1})$ be a unique
weak solution of problem \eqref{PKLK1}.
Then $(v, R)$, with
$$
v=u-\,\sum_{|\gamma |\leq N}u_{\gamma }
\overrightarrow{K}^{\gamma }\equiv u-h,\quad
R=Q-\sum_{|\gamma |\leq N}Q_{\gamma }\overrightarrow{K}^{ \gamma },
$$
satisfies the problem
\begin{equation} \label{e3.5}
\begin{gathered}
\begin{aligned}
Av&\equiv v_{tt}-\frac{\partial }{\partial x}(\mu (x,t)v_{x})\\
&=-K \big[F(v+h)-F(h)\big] -\lambda \big[G(v_t+h_t)-G(h_t)\big]\\
&\quad +\widetilde{E}_{N}(\overrightarrow{K}),\quad 0<x<1,\; 0<t<T,
 \end{aligned} \\
v(0,t)=0,\quad Bv\equiv -\mu (1,t)v_{x}(1,t)=R(t), \\
R(t)=K_1v(1,t)+\lambda _1(t)v_t(1,t)+\widetilde{G}_{N}(\overrightarrow{
K})-\int_0^t k(t-s)v(1,s)ds, \\
v(x,0)=v_t(x,0)=0,\\
v\in C^{0}(0,T;V)\cap C^{1}(0,T;L^2)\cap L^{\infty }(0,T;V\cap H^2),\\
v'\in L^{\infty }(0,T;H^{1}),\quad v''\in L^{\infty }(0,T;L^2), \\
v(1,\cdot )\in H^2(0,T),\quad R\in H^{1}( 0,T),
\end{gathered}
\end{equation}
where
\begin{gather} \label{e3.6}
\widetilde{E}_{N}(\overrightarrow{K})=f(x,t)-KF(h)-\lambda
G(h_t)-\sum_{|\gamma |\leq N}P_{\gamma }
\overrightarrow{K}^{ \gamma }, \\
 \label{e3.7}
\widetilde{G}_{N}(\overrightarrow{K})=\sum_{|\gamma |
=N+1, \gamma _3\geq 1}u_{\gamma _1,\gamma _2,\gamma _3-1}(1,t)
\overrightarrow{K}^{ \gamma }.
\end{gather}
Then, we have the following lemma.

\begin{lemma} \label{lem3.2}
Let (H2)--(H6) hold. Then
\begin{gather} \label{e3.8}
\|\widetilde{E}_{N}(\overrightarrow{K})\|_{L^{\infty
}(0,T;V)}\leq \widetilde{C}_{1N}\|\overrightarrow{K}\|^{N+1}, \\
 \label{e3.9}
\|\widetilde{G}_{N}(\overrightarrow{K})\|
_{H^2(0,T)}\leq \widetilde{C}_{2N}\|\overrightarrow{K}\|^{N+1},
\end{gather}
for all  $\overrightarrow{K}=(K$, $\lambda $, $K_1)\in
\mathbb{R}_{+}^{3}$, $\|\overrightarrow{K}\|
\leq \|\overrightarrow{K_{\ast }}\|$ with
$\overrightarrow{K_{\ast } }=(K_{\ast }, \lambda _{\ast },
K_{1\ast })$, where $\widetilde{C} _{1N}$,
$\widetilde{C}_{2N}$ are positive constants depending only on the
constants $\|\overrightarrow{K_{\ast }}\|$,
$\|u_{\gamma }\|_{L^{\infty }(0,T;V)}$,
$\|u_{\gamma }'\|_{L^{\infty }(0,T;V)}$,
$(|\gamma |\leq N)$,
$\|u_{\gamma _1,\gamma _2,\gamma_3-1}(1,\cdot )\|_{H^2(0,T)}$,
 $(|\gamma | =N+1$, $\gamma _3\geq 1)$.
\end {lemma}

\begin{proof} In the case of $N=1$, the proof of Lemma 3.2 is easy, hence we
omit the details, which we only prove with $N\geq 2$. Put
\begin{equation} \label{e3.10}
h=u_0+h_1, h_1=\sum_{1\leq |\gamma |\leq N}u_{\gamma }\overrightarrow{K}^{ \gamma }.
\end{equation}
By using Taylor's expansion of the function
$F(h)=F(u_0+h_1)$ around the point $u_0$ up to order $N-1$,
we obtain
\begin{equation} \label{e3.11}
F(h)=F(u_0)+\,\sum_{m=1}^{N-1}\frac{1}{m!}F^{(m)}(u_0)h_1^{m}+\frac{1}{
N!}F^{(N)}(u_0+\theta _1h_1)h_1^{N},
\end{equation}
where $0<\theta _1<1$. By Lemma 3.1, we obtain from \eqref{e3.11}, after some
rearrangements in order to of $\overrightarrow{K}^{ \gamma }$, that
\begin{equation} \label{e3.12}
\begin{aligned}
KF(h)& = KF(u_0)\\
&\quad +\sum_{2\leq |\gamma |\leq N,\quad \gamma _1\geq 1}\sum_{m=1}^{|\gamma |-1}\frac{1}{m!
}F^{(m)}(u_0)T^{(m)}[u]_{\gamma _1-1,\gamma _2,\gamma _3}
\overrightarrow{K}^{ \gamma }+R^{(1)}(F,\overrightarrow{K}),
\end{aligned}
\end{equation}
where
\begin{equation} \label{e3.13}
\begin{aligned}
&R^{(1)}(F,\overrightarrow{K})\\
&=K \sum_{m=1}^{N-1}\frac{1}{m!} F^{(m)}(u_0)\sum_{N\leq |\gamma
|\leq mN}T^{(m)}[u]_{\gamma }\overrightarrow{K}^{ \gamma }+\frac{1}{N!}
F^{(N)}(u_0+\theta _1h_1)Kh_1^{N},
\end{aligned}
\end{equation}
Similarly, we use Taylor's expansion of the function
$G(h_t)=G(u_0'+h_1')$ around the point $u_0'$ up to
order $N-1$, we obtain
\begin{equation} \label{e3.14}
\begin{aligned}
\lambda G(h_t)
&=\lambda G(u_0') +\,\sum_{2\leq |\gamma |\leq N,\quad \gamma
_2\geq 1}\sum_{m=1}^{|\gamma |
-1}\frac{1}{m!}G^{(m)}(u_0')T^{(m)}[u']_{\gamma
_1,\gamma _2-1,\gamma _3}\overrightarrow{K}^{ \gamma }\\
&\quad +R^{(2)}(G,\overrightarrow{K}),
\end{aligned}
\end{equation}
where
\begin{equation} \label{e3.15}
\begin{aligned}
&R^{(2)}(G,\overrightarrow{K})\\
&=\lambda \,\sum_{m=1}^{N-1}\frac{1}{m!} G^{(m)}(u_0')\sum_{N\leq
|\gamma |\leq mN}T^{(m)}[u']_{\gamma }\overrightarrow{K}^{ \gamma }
+\lambda \frac{1}{N!}G^{(N)}(u_0'+\theta _2h_1')(h_1')^{N},
\end{aligned}
\end{equation}
and $0<\theta _2<1$.
Combining \eqref{e3.4}, \eqref{e3.6}, \eqref{e3.12}--\eqref{e3.15},
we then obtain
\begin{equation} \label{e3.16}
\begin{aligned}
\widetilde{E}_{N}(\overrightarrow{K})
& = f(x,t)-KF(u_0)-\lambda G(u_0')  \\
&\quad - \sum_{2\leq |\gamma | \leq N,\; \gamma _1\geq 1}
  \sum_{m=1}^{|\gamma |-1}\frac{1}{m!}F^{(m)}(u_0)T^{(m)}[u]_{\gamma _1-1,
  \gamma_2,\gamma _3}\overrightarrow{K}^{ \gamma } \\
&\quad -\sum_{2\leq |\gamma |\leq N,\;
\gamma _2\geq 1}\,\sum_{m=1}^{|\gamma |
-1}\frac{1}{m!}G^{(m)}(u_0')T^{(m)}[u']_{\gamma
_1,\gamma _2-1,\gamma _3}\overrightarrow{K}^{ \gamma } \\
&\quad -\sum_{|\gamma |\leq N}P_{\gamma }\overrightarrow{K}
^{ \gamma }-R^{(1)}(F,\overrightarrow{K})-R^{(2)}(G,\overrightarrow{K})
 \\
&=-R^{(1)}(F,\overrightarrow{K})-R^{(2)}(G,\overrightarrow{K}).
\end{aligned}
\end{equation}
We shall estimate respectively the following terms on the
right-hand side of \eqref{e3.16}.

\noindent\emph{Estimate for $R^{(1)}(F,\overrightarrow{K})$.}
 By the boundedness of the functions $u_{\gamma }$,
 $ \gamma \in \mathbb{Z} _{+}^{3}$, $|\gamma |\leq N$ in the function
space $ L^{\infty }(0,T;H^{1})$, we obtain from (3.13), that
\begin{equation} \label{e3.17}
\begin{aligned}
&\|R^{(1)}(F,\overrightarrow{K})\|_{L^{\infty}(0,T;L^2)} \\
&\leq |K| \sum_{m=1}^{N-1}\sum_{N\leq |\gamma |\leq mN}\frac{1 }{m!}\|
F^{(m)}(u_0)\|_{L^{\infty }(0,T;V)}\|
T^{(m)}[u]_{\gamma }\|_{L^{\infty }(0,T;L^2)}|
\overrightarrow{K}^{ \gamma }|\\
&\quad +\frac{1}{N!}K\|F^{(N)}(u_0+\theta _1h_1)\|_{L^{\infty }
 (0,T;V)}\|h_1\|_{L^{\infty }(0,T;V)}^{N}.
\end{aligned}
\end{equation}
Using the inequality
\begin{equation} \label{e3.18}
|\overrightarrow{K}^{ \gamma }|\leq \|
\overrightarrow{K}\|^{|\gamma |}\text{, \ for all
}\gamma \in \mathbb{Z}_{+}^{3}, |\gamma |\leq N,
\end{equation}
it follows from \eqref{e3.17} and \eqref{e3.18} that
\begin{equation} \label{e3.19}
\|R^{(1)}(F,\overrightarrow{K})\|
_{L^{\infty }(0,T;L^2)}\leq \widetilde{C}_{1N}^{(1)}\|
\overrightarrow{K} \|^{N+1},\quad \|
\overrightarrow{K}\|\leq
\|\overrightarrow{K_{\ast }}\|,
\end{equation}
where
\begin{equation} \label{e3.20}
\begin{aligned}
\widetilde{C}_{1N}^{(1)}
&=\sum_{m=1}^{N-1}\sum_{N\leq |\gamma
|\leq mN}C_{p-1}^{m}\|u_0\|_{L^{\infty
}(0,T;V)}^{p-m-1}\|T^{(m)}[\widehat{u}]_{\gamma
}\|_{L^{\infty }(0,T;L^2)}\|
\overrightarrow{K_{\ast }}\|^{|\gamma |-N} \\
&\quad +C_{p-1}^{N}\|\overrightarrow{K_{\ast }}\|
^{-N}( \sum_{|\gamma |\leq N}\|u_{\gamma }\|
_{L^{\infty }(0,T;V)}\|\overrightarrow{K_{\ast }}
\|^{|\gamma |})^{p-1},
\end{aligned}
\end{equation}
$\overrightarrow{K_{\ast }}=(K_{\ast },\lambda _{\ast },K_{1\ast
})$, and $C_{p-1}^{m}=\frac{(p-1)(p-2)\dots (p-m)}{m!}$.

\noindent\emph{Estimate for $R^{(2)}(G,\overrightarrow{K})$.}
 From \eqref{e3.15} We obtain  in a similar manner corresponding to
the above part, that
\begin{equation} \label{e3.21}
\|R^{(2)}(G,\overrightarrow{K})\|
_{L^{\infty }(0,T;L^2)}\leq \widetilde{C}_{1N}^{(2)}\|
\overrightarrow{K} \|^{N+1},\quad \|
\overrightarrow{K}\|\leq
\|\overrightarrow{K_{\ast }}\|,
\end{equation}
where
\begin{equation} \label{e3.22}
\begin{aligned}
\widetilde{C}_{1N}^{(2)}
&=\sum_{m=1}^{N-1}\sum_{N\leq |\gamma
|\leq mN}C_{q-1}^{m}\|u_0'\|_{L^{\infty
}(0,T;V)}^{q-m-1}\|T^{(m)}[\widehat{u}']_{\gamma
}\|_{L^{\infty }(0,T;L^2)}\|
\overrightarrow{K_{\ast }}\|
^{|\gamma |-N} \\
&\quad +C_{q-1}^{N}\|\overrightarrow{K_{\ast }}\|
^{-N}( \sum_{|\gamma |\leq N}\|u_{\gamma }'\|
_{L^{\infty }(0,T;V)}\|\overrightarrow{K_{\ast }}
\|^{|\gamma |})^{q-1}.
\end{aligned}
\end{equation}
Therefore, it follows from \eqref{e3.16}, \eqref{e3.19}--\eqref{e3.22} that
\begin{equation} \label{e3.23}
\begin{gathered}
\|\widetilde{E}_{N}(\overrightarrow{K})\|_{L^{\infty }(0,T;L^2)}
\leq (\widetilde{C}_{1N}^{(1)}+\widetilde{C} _{1N}^{(2)})
\|\overrightarrow{K}\|^{N+1}\equiv
\widetilde{C}_{1N}\|\overrightarrow{K}\|^{N+1},\\
\|\overrightarrow{K}\|\leq \|\overrightarrow{K_{\ast }}\|.
\end{gathered}
\end{equation}
Hence, the first part of Lemma 3.2 is proved.

With $\widetilde{G}_{N}(\overrightarrow{K})$, then, we obtain from
\eqref{e3.7} in a similar manner to the above part, that
\begin{equation} \label{e3.24}
\|\widetilde{G}_{N}(\overrightarrow{K})\|
_{H^2(0,T)}\leq \widetilde{C}_{2N}\|\overrightarrow{K}\|
^{N+1},
\end{equation}
where
\begin{equation} \label{e3.25}
\emph{\ }\widetilde{C}_{2N}=\sum_{|\gamma |=N+1,  \gamma
_3\geq 1}\|u_{\gamma _1,\gamma _2,\gamma
_3-1}(1,\cdot )\|_{H^2(0,T)}.
\end{equation}
The proof of Lemma 3.2 is complete.
\end{proof}


\begin{theorem} \label{thm3.3}
 Let (H2)--(H6) hold. Then, for every
$\overrightarrow{K}\in \mathbb{R}_{+}^{3}$, with $0\leq K\leq K_{\ast }$,
$0\leq \lambda \leq \lambda_{\ast }$, $0\leq K_1\leq K_{1\ast }$, problem
\eqref{PKLK1} has a unique weak solution $(u, Q)=(u_{K,\lambda ,K_1}$,
 $Q_{K,\lambda ,K_1})$ satisfying the asymptotic estimations up to
order $N+1$ as follows
\begin{equation} \label{e3.26}
\begin{aligned}
&\|u'-\sum_{|\gamma |\leq N}u_{\gamma }'
\overrightarrow{K}^{ \gamma }\|_{L^{\infty
}(0,T;L^2)}+\|u-\sum_{|\gamma |\leq N}u_{\gamma }
\overrightarrow{K}^{ \gamma }\|_{L^{\infty }(0,T;V)} \\
&+\|u'(1,\cdot )-\sum_{|\gamma |\leq N}u_{\gamma
}'(1,\cdot )\overrightarrow{K}^{ \gamma }\|
_{L^2(0,T)}\\
&\leq \widetilde{D}_{N}^{\ast }\|\overrightarrow{K}\|^{N+1},
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.27}
\|Q-\sum_{|\gamma |\leq N}Q_{\gamma }
\overrightarrow{K}^{ \gamma }\|_{L^2(0,T)}\leq \widetilde{D}
_{N}^{\ast \ast }\|\overrightarrow{K}\|^{N+1},
\end{equation}
for all $\overrightarrow{K}\in \mathbb{R} _{+}^{3}$,
$\|\overrightarrow{K}\|\leq \|\overrightarrow{K_{\ast }}\|$,
$\widetilde{D}_{N}^{\ast}$ and $\widetilde{D}_{N}^{\ast \ast }$
are positive constants independent of $\overrightarrow{K}$,
the functions $(u_{\gamma }$,
$ Q_{\gamma })$ are the weak solutions of problems
\eqref{tPg}, $\gamma \in \mathbb{Z}_{+}^{3}$, $|\gamma |\leq N$.
\end {theorem}

\begin{remark} \label{rmk3.1} \rm
 In [9], as in this special case for
problem \eqref{e1.1}--\eqref{e1.5}, Long, Ut and Truc have obtained
a result about the asymptotic expansion of the solutions with
respect to two parameters $(K, \lambda )$ up to order $ N+1$.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm3.3}]
 First, we note that, if the data $\overrightarrow{K}$ satisfy
\begin{equation} \label{e3.28}
0\leq K\leq K_{\ast },\quad 0\leq \lambda \leq \lambda _{\ast },\quad
0\leq K_1\leq K_{1\ast },
\end{equation}
where $K_{\ast }$, $\lambda _{\ast }$, $K_{1\ast }$ are fixed positive
constants. Therefore, the a priori estimates of the sequences $\{u_{m}\}$
and $\{Q_{m}\}$ in the proof of theorem 2.2 satisfy
\begin{gather} \label{e3.29}
\|u_{m}'(t)\|^2+\mu _0\|u_{mx}(t)\|^2
+2\lambda _0\int_0^t  |u_{m}'(1,s)|^2 ds
\leq M_{T}, \forall t\in [0,T], \\
\label{e3.30}
\|u_{m}''(t)\|^2+\mu _0\|u_{mx}'(t)\|^2
+2\lambda _0\int_0^t |u_{m}''(1,s)|^2ds
\leq M_{T}, \forall t\in [0,T], \\
\label{e3.31}
\|Q_{m}\|_{H^{1}(0,T)}\leq M_{T},
\end{gather}
where $M_{T}$ is a constant depending only on $T$,
$\widetilde{u}_0$, $ \widetilde{u}_1$, $\lambda _0$,
$\mu_0$, $f$, $g$, $k$, $\mu $, $ \lambda _1$, $K_{\ast }$,
$\lambda _{\ast }$, $K_{1\ast }$ (independent of
$\overrightarrow{K}$). Hence, the limit $(u, Q)$ in suitable
function spaces of the sequence $\{(u_{m}$, $Q_{m})\}$ defined by
\eqref{e2.7}-- \eqref{e2.9} is a weak solution of the problem
\eqref{e1.1}--\eqref{e1.5} satisfying the a priori estimates
\eqref{e3.29}--\eqref{e3.31}.

Multiplying the two sides of \eqref{e3.5}$_1$ with $v'$, and
integrating in $t$, we find without difficulty from Lemma 3.2 that
\begin{equation} \label{e3.32}
\begin{aligned}
\sigma (t)&\leq 2T(\frac{2}{\lambda _0}\widetilde{C}_{2N}^2+
\widetilde{C}_{1N}^2)\| \overrightarrow{K}\|^{2N+2}\\
&\quad +2[1+K+\frac{1}{\mu _0}\|\mu '\|_{C^{0}(\overline{Q_{T}})
}+\frac{2}{\lambda _0\mu _0}T\|k\|_{L^2(0,T)}^2]
\int_0^t \sigma (s)ds \\
&\quad +2K\int_0^t \|F(v+h)-F(h)\|^2ds,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e3.33}
\sigma (t)=\|v'(t)\|^2+\|\sqrt{\mu
(t)} v_{x}(t)\|^2+K_1v^2(1,t)+2\int_0^t \lambda
_1(s)|v'(1,s)|^2ds.
\end{equation}
By using the same arguments as in the above part we can show that
the component $u$ of the weak solution $(u, Q)$ of problem
$(P_{K,\lambda ,K_1})$ satisfies
\begin{equation} \label{e3.34}
\|u'(t)\|^2+\mu _0\|u_{x}(t)\|^2+2\lambda
_0\int_0^t |u'(1,s)|^2ds\leq
M_{T}, \forall t\in [ 0,T],
\end{equation}
where $M_{T}$ is a constant independent of $K$, $\lambda $, $K_1$.
On the other hand,
\begin{equation} \label{e3.35}
\|h\|_{L^{\infty }(0,T;V)}\leq \sum_{|\gamma|\leq N}
\|u_{\gamma }\|_{L^{\infty}(0,T;V)}\|\overrightarrow{K_{\ast }}\|
^{|\gamma |}\equiv R_1.
\end{equation}
We again use inequality \eqref{e2.48} with $\delta =p$,
$R= \max \{R_1,\sqrt{\frac{M_{T}}{\mu _0}}\}$, then, it follows from
\eqref{e3.33}--\eqref{e3.35}, that
\begin{equation} \label{e3.36}
\int_0^t \|F(v+h)-F(h)\|^2ds\leq \frac{1}{\mu _0}
(p-1)^2R^{2p-4} \int_0^t \sigma (s)ds.
\end{equation}
Combining \eqref{e3.32} and \eqref{e3.36}, we then obtain
\begin{equation} \label{e3.37}
\sigma (t)\leq 2T(\frac{2}{\lambda _0}\widetilde{C}_{2N}^2+
\widetilde{C}_{1N}^2)\|
\overrightarrow{K}\|
^{2N+2}\text{ }+\sigma _{1T}\int_0^t \sigma (s)ds,
\end{equation}
for all $t\in [ 0,T]$, where
\begin{equation} \label{e3.38}
\sigma _{1T}=2\big[1+K_{\ast }+\frac{1}{\mu _0}\|\mu
'\|_{C^{0}(\overline{Q_{T}})}+\frac{2}{\lambda
_0\mu _0}\,T\|k\|_{L^2(0,T)}^2+\frac{1}{\mu _0}
(p-1)^2R^{2p-4}K_{\ast }\big] ,
\end{equation}
By Gronwall's lemma, we obtain from \eqref{e3.37} that
\begin{equation} \label{e3.39}
\sigma (t)\leq 2T(\frac{2}{\lambda _0}\widetilde{C}_{2N}^2+
\widetilde{C}_{1N}^2)\|\overrightarrow{K}\|
^{2N+2}\exp (T\sigma _{1T})\equiv \widetilde{D}
_{T}^{(1)} \|\overrightarrow{K}\|^{2N+2},
\end{equation}
for all $t\in [ 0,T]$ and all
$ \overrightarrow{K}\in \mathbb{R}_{+}^{3}$,
$\| \overrightarrow{K}\|\leq \|\overrightarrow{K_{\ast }}\|$.
It follows that
\begin{equation} \label{e3.40}
\|v'(t)\|^2+\mu _0\|
v_{x}(t)\|^2+2\lambda
_0\int_0^t |v'(1,s)|^2ds\leq
\sigma (t)\leq \widetilde{D}_{T}^{(1)} \|\overrightarrow{K}
\|^{2N+2}.
\end{equation}
Hence
\begin{equation} \label{e3.41}
\|v'\|_{L^{\infty }(0,T;L^2)}+\|
v\|_{L^{\infty }(0,T;V)}+\|v'(1,\cdot)\|
_{L^2(0,T)}\leq \widetilde{D}_{N}^{\ast }\|\overrightarrow{K}
\|^{N+1},
\end{equation}
or
\begin{equation} \label{e3.42}
\begin{aligned}
&\|u'-\sum_{|\gamma |\leq N}u_{\gamma }'
\overrightarrow{K}^{ \gamma }\|_{L^{\infty
}(0,T;L^2)}+\|u-\sum_{|\gamma |\leq N}u_{\gamma }
\overrightarrow{K}^{ \gamma }\|_{L^{\infty }(0,T;V)} \\
&+\|u'(1,\cdot )-\sum_{|\gamma |\leq N}u_{\gamma }'(1,\cdot )
\overrightarrow{K}^{ \gamma }\|_{L^2(0,T)}\\
&\leq \widetilde{D}_{N}^{\ast }\|\overrightarrow{K}\|^{N+1},
\end{aligned}
\end{equation}
for all $ \overrightarrow{K}\in  \mathbb{R}_{+}^{3}$,
$\| \overrightarrow{K}\|\leq \|\overrightarrow{K_{\ast }}\|$, where
$\widetilde{D}_{N}^{\ast }$
is a constant independent of $\overrightarrow{K}$.
On the other hand, it follows from \eqref{e3.9}, \eqref{e3.41}, that
\begin{equation} \label{e3.43}
\begin{aligned}
\|R\|_{L^2(0,T)}
&\leq K_1\| v\|_{L^{\infty }(0,T;V)}+\|\lambda
_1\|_{\infty }\|v'(1,\cdot )\|_{L^2(0,T)}
+\| \widetilde{G}_{N}(\overrightarrow{K})\|_{L^2(0,T)} \\
&\quad +\sqrt{\frac{1}{\mu _0}T}\|k\|
_{L^2(0,T)}\Big(\int_0^{T}\sigma (s)ds\Big)^{1/2}\\
&\leq \widetilde{D}_{N}^{\ast \ast }\|\overrightarrow{K}\|^{N+1},
\end{aligned}
\end{equation}
hence,
\begin{equation} \label{e3.44}
\|Q-\sum_{|\gamma |\leq N}Q_{\gamma }
\overrightarrow{K}^{ \gamma }\|_{L^2(0,T)}\leq \widetilde{D}
_{N}^{\ast \ast }\|\overrightarrow{K}\|^{N+1},
\end{equation}
where $\widetilde{D}_{N}^{\ast \ast }$ is a constant independent
of $ \overrightarrow{K}$. The proof of  Theorem 3.3 is complete.
\end{proof}

\begin{remark} \label{rmk3.2} \rm
 For the case  $(K,\lambda ,K_1)\in \mathbb{R}^2\times \mathbb{R}_{+}$,
but $p=q=2$, we have received  a theorem of the asymptotic expansion
for the weak solution $(u, Q)$ of problem \eqref{e1.1}--\eqref{e1.5} with
respect to three mentioned parameters; however, the detailes of proof
have been omitted.
\end{remark}

\subsection*{Acknowledgements}
 The authors wish to express their sincere thanks to the referee
for his/her helpful comments, also to Professor Julio G. Dix
and to Professor Dung Le  for their helpful suggestions.

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