\documentclass[twoside]{article}
\usepackage{amsmath}
\pagestyle{myheadings}
\markboth{\hfil A three-point boundary-value problem \hfil EJDE--2002/62}
{EJDE--2002/62\hfil S. Mesloub \& S. A. Messaoudi \hfil}
\begin{document}

\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 62, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)} \vspace{\bigskipamount} \\
%
  A three-point boundary-value problem for a hyperbolic equation with a
  non-local condition
%
 \thanks{\emph{Mathematics Subject Classifications:}
35L20, 35L67, 34B15.  \hfil\break\indent
\emph{Key words:} Wave equation, Bessel operator, nonlocal condition.
\hfil\break \indent
\copyright 2002 Southwest Texas State University. \hfil\break \indent
Submitted April 16, 2002. Published July 3, 2002.} }
\date{}

\author{Said Mesloub \& Salim A. Messaoudi}
\maketitle

\begin{abstract}
  We use an energy method to solve a three-point boundary-value 
  problem for a hyperbolic equation with a Bessel operator and an 
  integral condition. The proof is based on an energy inequality 
  and  on the fact that the range of the operator generated is dense.
\end{abstract}

\newtheorem{theorem}{Theorem}[section] 
\newtheorem{proposition}[theorem]{Proposition} 
\newtheorem{corollary}[theorem]{Corollary} 
\newtheorem{remark}[theorem]{Remark} 
\newtheorem{definition}[theorem]{Definition} 
\numberwithin{equation}{section}

\allowdisplaybreaks

\section{Introduction}

In this paper, we investigate a boundary-value problem for a one-dimensional
hyperbolic equation with a weighted nonlocal boundary integral condition of
the form 
\[
\int_{l_{1}}^{l}\xi u(\xi ,t)d\xi =E(t),\quad 0<t<T, 
\]
where $l_{1}$ is a real number in $(0,l)$ and $E(\cdot )$ is a given
function.

Evolution problems dealing with nonlocal conditions were first studied a
long time ago by Samarskii [12] and Cannon [2]. The latter author considered
the problem 
\begin{equation}
\begin{gathered} u_{t}-u_{xx}=0,\quad x>0,\mbox{ }t>0, \\ u(x,0)=\varphi
(x),\quad x>0, \\ u(0,t)=g(t), \\ \int_{0}^{x(t)}u(\xi ,t)dx=f(t),
\end{gathered}  \label{e1}
\end{equation}
for $x(t)$ and $f(t)$ given functions. Introducing $g\equiv u(0,t)$ as the
unknown, it is proved in [2] that (1.1) is equivalent to a Volterra integral
equation of the second kind for the function $g$. The author proved the
existence and uniqueness of the solution with the aid of the integral
equation. Shi [11] considered weak solutions of the problem 
\begin{equation}
\begin{gathered} u_{t}-u_{xx}=f+g_{x},\quad (x,t)\in (0,1)\times (0,T), \\
u(x,0)=\varphi (x),\quad 0<x<1, \\ u_{x}(1,t)=0,\quad 0<t<T, \\
\int_{0}^{b}u(\xi ,t)dx=E(t),\quad 0<t<T \end{gathered}  \label{e5}
\end{equation}
and discussed the well-posedness of (1.2) in a weighted fractional Sobolev
space. Along a different line, (1.2) was also considered by Ionkin [5],
Makarov and Kulyev [8], and Yurchuk [13].

In this work, we are concerned with the mixed evolution problem 
\begin{equation}
\begin{gathered} \mathcal{L}u=u_{tt}-\frac{1}{x}\left( xu_{x}\right)
_{x}=F(x,t), \quad (x,t)\in Q, \\ \ell _{1}u=u(x,0)=\varphi _{1}(x),\quad
x\in (0,l), \\ \ell _{2}u=u_{t}(x,0)=\varphi _{2}(x),\quad x\in (0,l), \\
u_{x}(l,t)=E_{1}(t),\quad t\in (0,T), \\
\int_{l_{1}}^{l}xu(x,t)dx=E_{2}(t),\quad 0\leq l_{1}\leq l, \;t\in (0,T),
\end{gathered}  \label{e9}
\end{equation}
where $Q=(0,l)\times (0,T)$, with $0<l<\infty $, $0<T<\infty $, $F(x,t)$, $%
\varphi _{1}(x)$, $\varphi _{2}(x)$, $E_{1}(t)$, and $E_{2}(t)$ are known
functions satisfying, for compatibility, 
\begin{equation}
\begin{gathered} \varphi _{1}'(l)=E_{1}(0), \\ \int_{l_{1}}^{l}x\varphi
_{1}(x)dx=E_{2}(0),\\ \varphi _{2}'(l)=E_{1}'(0), \\
\int_{l_{1}}^{l}x\varphi _{2}(x)dx=E_{2}'(0). \end{gathered}  \label{e14}
\end{equation}

Problem (1.3), for $l_{1}=0$, has been studied by Mesloub and Bouziani [9].
We also refer the reader to Denche and Marhoune [3] for a similar result in
the parabolic case and to Yurchuk [13], Kartynik [6] and Bouziani [1] for
related results in both parabolic and hyperbolic cases, where the Bessel
operator was replaced by $\left( a(x,t)u_{x}\right) _{x}$. It should be
noted that the used method was developed first by Ladyzhenskaya [7]. Our
interest lies in proving the existence and uniqueness of a strong solution
of problem (1.3). In point of view of the used method, it is preferable to
transform inhomogeneous boundary conditions to homogeneous ones by
introducing a new unknown function $v$ defined as follows: 
\begin{equation}
v(x,t)=u(x,t)-\Phi (x,t),
\end{equation}
where 
\begin{equation}
\Phi (x,t)=x\big( x-\frac{4(x-l)^{2}}{l}\big) E_{1}(t)+\frac{12(x-l)^{2}}{%
l^{4}}E_{2}(t).
\end{equation}
Then problem (1.3) becomes 
\begin{equation}
\begin{gathered} \mathcal{L}v=F(x,t)-\mathcal{L}\Phi =f(x,t),\\ \ell
_{1}v=\varphi _{1}-\ell _{1}\Phi =\varphi (x),\\ \ell _{2}v=\varphi
_{2}-\ell _{2}\Phi =\psi (x)\\ v_{x}(l,t)=0, \\ \int_{l_{1}}^{l}xv(x,t)dx=0.
\end{gathered}  \label{e20}
\end{equation}
The solution to (1.3) is then given by $u(x,t)=v(x,t)+\Phi (x,t)$.

We now introduce appropriate function spaces. First let 
\[
\theta (x)=%
\begin{cases}
1+l_{1}^{2})x, &\mbox{if } 0<x\leq l_{1} \\
x+x^{3},&\mbox{if } l_{1}\leq x<l
\end{cases}
\]
and 
\[
\Im _{x}v=\int_{x}^{l}v(\xi ,t)d\xi ,\quad \Im
_{x}^{2}v=\int_{x}^{l}\int_{\xi }^{l}v(\eta ,t)d\eta d\xi . 
\]
Let $L^{2}(Q)$ be the space of square integrable functions with the norm 
\[
\left\| v\right\| _{L^{2}(Q)}^{2}=\int_{Q}v^{2}\,dx\,dt 
\]
and $L_{\theta }^{2}(Q)$ be the weighted $L^{2}$-space with the norm 
\[
\left\| v\right\| _{L_{\theta }^{2}(Q)}^{2}=\int_{Q}\theta (x)v^{2}\,dx\,dt. 
\]
We then define $W_{\theta ,2}^{1,0}(Q)$ to be the subspace of $L^{2}(Q)$
with the norm 
\[
\left\| v\right\| _{W_{\theta ,2}^{1,0}(Q)}^{2}=\left\| v\right\|
_{L_{\theta }^{2}(Q)}^{2}+\left\| v_{x}\right\| _{L_{\theta }^{2}(Q)}^{2} 
\]
and $W_{\theta ,2}^{1,1}(Q)$ to be the subspace of $W_{\theta ,2}^{1,0}(Q)$
whose elements satisfy $\sqrt{\theta (x)}v_{t}\in L^{2}(Q)$. In general, a
function in the space $W_{\theta ,2}^{q,p}(Q)$, with $q,p$ nonnegative
integers, possesses $x$-derivatives up to $q$th order in $L_{\theta }^{2}(Q)$
and $t$-derivatives up to $p$th order in $L_{\theta }^{2}(Q)$. We use also
weighted subspaces on the interval $(0,l)$ such as $W_{\theta
,2}^{1}((0,l))=H_{\theta }^{1}((0,l))$, whose definition is analogous to the
space on $Q$. For example, $H_{\theta }^{1}((0,l))$ is the subspace of $%
L^{2}(0,l)$ with the norm 
\[
\left\| \varphi \right\| _{H_{\theta }^{1}((0,l))}^{2}=\left\| \varphi
\right\| _{L_{\theta }^{2}((0,l))}^{2}+\left\| \varphi _{x}\right\|
_{L_{\theta }^{2}((0,l))}^{2}. 
\]

We associate with problem (1.7) the operator $L=(\mathcal{L},\ell _{1},\ell
_{2})$ whose domain of definition is $D(L)$, the set of functions $v\in
L^{2}(Q)$ for which $v_{t},v_{x},v_{tt},v_{xt},v_{xx}\in L^{2}(Q)$ and
satisfying conditions in (1.7). The operator $L$ maps $E$ into $F$; $E$ is
the Banach space of functions $v\in L^{2}(Q)$ satisfying conditions in
(1.7), with the norm 
\begin{equation}
\begin{aligned} \left\| v\right\| _{E}^{2} =&\max_{0\leq t\leq T}\left\|
v(.,\tau )\right\| _{W_{\theta ,2}^{1,1}((0,l))}^{2} \\ =&\max_{0\leq t\leq
T}\left\{ \left\| v(.,\tau )\right\| _{L_{\theta }^{2}((0,l))}^{2}+\left\|
v_{x}(.,\tau )\right\| _{L_{\theta }^{2}((0,l))}^{2} +\left\| v_{t}(.,\tau
)\right\| _{L_{\theta }^{2}((0,l))}^{2}\right\} \nonumber \end{aligned}
\label{e25}
\end{equation}
and $F$ is the Hilbert space $L_{\theta }^{2}(Q)\times H_{\theta
}^{1}((0,l))\times L_{\theta }^{2}((0,l))$, which consists of elements $%
\mathcal{F}=(f,\varphi ,\psi )$ with the norm 
\begin{equation}
\left\| \mathcal{F}\right\| _{F}^{2}=\left\| f\right\| _{L_{\theta
}^{2}(Q)}^{2}+\left\| \varphi \right\| _{H_{\theta }^{1}((0,l))}^{2}+\left\|
\psi \right\| _{L_{\theta }^{2}((0,l))}^{2}.
\end{equation}
Then, we establish an energy inequality: 
\begin{equation}
\left\| v\right\| _{E}\leq K\left\| Lv\right\| _{F},\quad \forall v\in D(L),
\end{equation}
and show that the operator $L$ has a closure $\overline{L}$.

\begin{definition}
\label{def1.1} \textrm{A solution of the operator equation 
\[
\overline{L}v=(f,\varphi ,\psi ),
\]
is called a strong solution of the problem (1.7). }
\end{definition}

Since the points of the graph of the operator $\overline{L}$ are limits of
sequences of points of the graph of $L$, we can extend the a priori estimate
(1.9) to be applied to strong solutions by taking limits, that is we have
the inequality 
\begin{equation}
\left\| v\right\| _{E}\leq K\left\| \overline{L}v\right\| _{F},\quad \forall
v\in D(\overline{L}).  \label{e28}
\end{equation}
From this inequality, We deduce the uniqueness of a strong solution, if it
exists, and that the range of the operator $\overline{L}$ coincides with the
closure of the range of $L$.

\begin{proposition}
\label{prop1.2} The operator $L$ admits a closure.
\end{proposition}

The proof of this proposition is similar to that in [9]; therefore we omit
it.

\section{A priori bound}

This section is devoted to the proof of the uniqueness and continuous
dependence of the solution on the given data.\newline

\begin{theorem}
\label{thm2.1} For any function $v\in D(L)$, we have the inequality 
\begin{equation}
\left\| v\right\| _{E}\leq c\left\| Lv\right\| _{F},\quad   \label{e29}
\end{equation}
where the positive constant $c$ is independent of the function $v$.
\end{theorem}

\paragraph{Proof}

We define 
\[
Mv=%
\begin{cases}
x(1+l_{1}^{2})v_{t}&\mbox{if } 0<x<l_{1} \\
(x+x^{3})v_{t}-x\Im _{x}^{2}(\xi v_{t})+x\Im _{x}(\xi ^{2}v_{t})
&\mbox{if } l_{1}<x<l\,.
\end{cases}
\]
Then we perform the scalar product in $L^{2}(Q^{\tau })$ of equation (1.7)
and $Mv$ to get 
\begin{equation}
\begin{split}
& \int_{Q^{\tau }}\theta (x)v_{t}v_{tt}\,dx\,dt-\int_{0}^{\tau
}\int_{0}^{l_{1}}(l_{1}^{2}+1)(xv_{x})_{x}v_{t}\,dx\,dt \\
& -\int_{0}^{\tau
}\int_{l_{1}}^{l}(x^{2}+1)(xv_{x})_{x}v_{t}\,dx\,dt-\int_{0}^{\tau
}\int_{l_{1}}^{l}xv_{tt}\Im _{x}^{2}(\xi v_{t})\,dx\,dt \\
& +\int_{0}^{\tau }\int_{l_{1}}^{l}(xv_{x})_{x}\Im _{x}^{2}(\xi
v_{t})\,dx\,dt+\int_{0}^{\tau }\int_{l_{1}}^{l}xv_{tt}\Im _{x}(\xi
^{2}v_{t})\,dx\,dt \\
& -\int_{0}^{\tau }\int_{l_{1}}^{l}(xv_{x})_{x}\Im _{x}(\xi
^{2}v_{t})\,dx\,dt \\
& =\int_{Q^{\tau }}\theta (x)v_{t}\mathcal{L}vdxdt-\int_{0}^{\tau
}\int_{l_{1}}^{l}x\mathcal{L}v\Im _{x}^{2}(\xi v_{t})\,dx\,dt+\int_{0}^{\tau
}\int_{l_{1}}^{l}x\mathcal{L}v\Im _{x}(\xi ^{2}v_{t})\,dx\,dt.
\end{split}
\label{e30}
\end{equation}
Integrating by parts each term of (2.2) and using conditions (1.7), we
obtain the following equations: 
\begin{equation}
\int_{Q^{\tau }}\theta (x)v_{t}v_{tt}\,dx\,dt=\frac{1}{2}\int_{0}^{l}\theta
(x)v_{t}^{2}(x,\tau )dx-\frac{1}{2}\int_{0}^{l}\theta (x)\psi ^{2}(x,\tau
)dx\,,  \label{e31}
\end{equation}
\begin{eqnarray}
\lefteqn{-\int_{0}^{\tau
}\int_{0}^{l_{1}}(l_{1}^{2}+1)(xv_{x})_{x}v_{t}\,dx\,dt}  \nonumber
\label{e32} \\
&=&\frac{1}{2}\int_{0}^{l_{1}}(l_{1}^{2}+1)xv_{x}^{2}(x,\tau )dx-\frac{1}{2}%
\int_{0}^{l_{1}}(l_{1}^{2}+1)x\varphi _{x}^{2}dx \\
&&-\int_{0}^{\tau }(l_{1}^{2}+1)l_{1}v_{t}(l_{1},t)v_{x}(l_{1},t)dt\,, 
\nonumber
\end{eqnarray}
\begin{eqnarray}
\lefteqn{-\int_{0}^{\tau }\int_{l_{1}}^{l}(x^{2}+1)(xv_{x})_{x}v_{t}\,dx\,dt}
\nonumber  \label{e33} \\
&=&\frac{1}{2}\int_{l_{1}}^{l}(x^{3}+x)v_{x}^{2}(x,\tau )dx-\frac{1}{2}%
\int_{l_{1}}^{l}(x^{3}+x)\varphi _{x}^{2}dx \\
&&+2\int_{0}^{\tau }\int_{l_{1}}^{l}x^{2}v_{x}v_{t}\,dx\,dt+\int_{0}^{\tau
}(l_{1}^{2}+1)l_{1}v_{t}(l_{1},t)v_{x}(l_{1},t)dt\,,  \nonumber
\end{eqnarray}
\begin{equation}
-\int_{0}^{\tau }\int_{l_{1}}^{l}xv_{tt}\Im _{x}^{2}(\xi v_{t})\,dx\,dt=%
\frac{1}{2}\int_{l_{1}}^{l}(\Im _{x}(\xi v_{t}(\xi ,\tau )))^{2}dx-\frac{1}{2%
}\int_{l_{1}}^{l}(\Im _{x}(\xi \psi ))^{2}dx\,,  \label{e34}
\end{equation}
\begin{eqnarray}
\lefteqn{\int_{0}^{\tau }\int_{l_{1}}^{l}(xv_{x})_{x}\Im _{x}^{2}(\xi
v_{t})\,dx\,dt}  \nonumber  \label{e35} \\
&=&-l_{1}\int_{0}^{\tau
}\int_{l_{1}}^{l}x^{2}v_{x}(l_{1},t)v_{t}\,dx\,dt+\int_{0}^{\tau
}\int_{l_{1}}^{l}xv_{x}\Im _{x}(\xi v_{t})\,dx\,dt\,,
\end{eqnarray}
\begin{eqnarray}
\lefteqn{\int_{0}^{\tau }\int_{l_{1}}^{l}xv_{tt}\Im _{x}(\xi
^{2}v_{t})\,dx\,dt}  \nonumber  \label{e36} \\
&=&-\frac{1}{2}\int_{l_{1}}^{l}(\Im _{x}(\xi v_{t}(\xi ,\tau )))^{2}dx+\frac{%
1}{2}\int_{l_{1}}^{l}(\Im _{x}(\xi \psi ))^{2}dx+\int_{0}^{\tau
}\int_{l_{1}}^{l}x^{3}v_{x}v_{t}\,dx\,dt  \nonumber \\
&&-\int_{0}^{\tau }\int_{l_{1}}^{l}xv_{x}\Im _{x}(\xi
v_{t})\,dx\,dt+\int_{0}^{\tau }\int_{l_{1}}^{l}x^{2}\Im _{x}(\xi v_{t})%
\mathcal{L}v\,dx\,dt\,,
\end{eqnarray}
\begin{eqnarray}
\lefteqn{-\int_{0}^{\tau }\int_{l_{1}}^{l}(xv_{x})_{x}\Im _{x}(\xi
^{2}v_{t})\,dx\,dt }\nonumber\\
&=&l_{1}\int_{0}^{\tau
}\int_{l_{1}}^{l}x^{2}v_{x}(l_{1},t)v_{t}\,dx\,dt-\int_{0}^{\tau
}\int_{l_{1}}^{l}x^{3}v_{x}v_{t}\,dx\,dt\,.  \label{e37}
\end{eqnarray}
Substituting (2.3)-(2.9) in (2.2) yields 
\begin{eqnarray}
\lefteqn{ \frac{1}{2}\int_{0}^{l}\theta (x)v_{t}^{2}(x,\tau )dx+\frac{1}{2}
\int_{0}^{l}\theta (x)v_{x}^{2}(x,\tau )dx  }  \nonumber \\  
&=&\frac{1}{2}\int_{0}^{l}\theta (x)\psi ^{2}dx+\frac{1}{2}
\int_{0}^{l}\theta (x)\varphi _{x}^{2}dx-2\int_{0}^{\tau
}\int_{l_{1}}^{l}x^{2}v_{x}v_{t}\,dx\,dt  \nonumber \\
&&+\int_{0}^{\tau }\int_{l_{1}}^{l}x\mathcal{L}v\Im _{x}(\xi
^{2}v_{t})\,dx\,dt-\int_{0}^{\tau }\int_{l_{1}}^{l}x^{2}\mathcal{L}v
\Im_{x}(\xi v_{t})\,dx\,dt \nonumber \\
 &&+\int_{Q^{\tau }}\theta (x)v_{t}\mathcal{L}v\,dx\,dt-\int_{0}^{\tau
 }\int_{l_{1}}^{l}x\mathcal{L}v\Im _{x}^{2}(\xi v_{t})\,dx\,dt.  \label{e38} 
\end{eqnarray}
Using Young's inequality and 
\[
\int_{l_{1}}^{l}(\Im _{x}^{2}v)^{2}dx\leq \frac{(l-l_{1})^{2}}{2}%
\int_{l_{1}}^{l}(\Im _{x}v)^{2}dx, 
\]
to estimate the last five terms on the right-hand side of (2.10), we obtain
the following inequalities: 
\begin{equation}
-2\int_{0}^{\tau }\int_{l_{1}}^{l}x^{2}v_{x}v_{t}\,dx\,dt\leq \int_{0}^{\tau
}\int_{l_{1}}^{l}xv_{x}^{2}\,dx\,dt+\int_{0}^{\tau
}\int_{l_{1}}^{l}x^{3}v_{t}^{2}\,dx\,dt,  \label{e40}
\end{equation}
\begin{eqnarray}
\lefteqn{\int_{0}^{\tau }\int_{l_{1}}^{l}x\mathcal{L}v\Im _{x}(\xi
^{2}v_{t})\,dx\,dt }\nonumber\\
&\leq& \frac{(l-l_{1})}{2}\int_{0}^{\tau }\int_{l_{1}}^{l}x(
\mathcal{L}v)^{2}\,dx\,dt+\frac{(l-l_{1})^{5}}{4}\int_{0}^{\tau
}\int_{l_{1}}^{l}xv_{t}^{2}\,dx\,dt  \label{e41}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{-\int_{0}^{\tau }\int_{l_{1}}^{l}x^{2}\mathcal{L}v\Im _{x}
(\xi v_{t})\,dx\,dt } \nonumber \\
\label{e42} 
&\leq& \frac{(l-l_{1})^{3}}{2}\int_{0}^{\tau }\int_{l_{1}}^{l}x(\mathcal{L}%
v)^{2}\,dx\,dt+\frac{(l-l_{1})^{3}}{4}\int_{0}^{\tau
}\int_{l_{1}}^{l}xv_{t}^{2}\,dx\,dt,
\end{eqnarray}
\begin{equation}
\int_{Q^{\tau }}\theta (x)v_{t}\mathcal{L}v\,dx\,dt\leq \frac{1}{2}%
\int_{Q^{\tau }}\theta (x)v_{t}^{2}\,dx\,dt+\frac{1}{2}\int_{Q^{\tau
}}\theta (x)(\mathcal{L}v)^{2}\,dx\,dt,  \label{e43}
\end{equation}
\begin{eqnarray}
\lefteqn{-\int_{0}^{\tau }\int_{l_{1}}^{l}x\mathcal{L}v\Im _{x}^{2}(\xi
v_{t})\,dx\,dt}  \nonumber \\
&\leq &\frac{(l-l_{1})^{3}}{4}\int_{0}^{\tau }\int_{l_{1}}^{l}(\Im _{x}(\xi
v_{t}))^{2}\,dx\,dt+\frac{l-l_{1}}{2}\int_{0}^{\tau }\int_{l_{1}}^{l}x(%
\mathcal{L}v)^{2}\,dx\,dt  \label{e44} \\
&\leq &\frac{(l-l_{1})^{5}}{8}\int_{0}^{\tau
}\int_{l_{1}}^{l}xv_{t}^{2}\,dx\,dt+\frac{l-l_{1}}{2}\int_{Q^{\tau }}\theta
(x)(\mathcal{L}v)^{2}\,dx\,dt.  \nonumber
\end{eqnarray}
We also have 
\begin{equation}
\frac{1}{2}\int_{0}^{l}\theta (x)v^{2}(x,\tau )dx\leq \frac{1}{2}%
\int_{0}^{l}\theta (x)\varphi ^{2}dx+\frac{1}{2}\int_{Q^{\tau }}\theta
(x)v^{2}\,dx\,dt+\frac{1}{2}\int_{Q^{\tau }}\theta (x)v_{t}^{2}\,dx\,dt.
\label{e45}
\end{equation}
Indeed, we have 
\[
\frac{\partial u^{2}}{\partial t}=2uu_{t}, 
\]
multiplying both sides by $\theta (x)$ then integrating with respect to $t$
from $0$ to $\tau $, and using Young's inequality, we obtain 
\[
\theta (x)v^{2}(x,\tau )-\theta (x)\varphi ^{2}(x)=2\int_{0}^{\tau }\theta
(x)vv_{t}dt\leq \int_{0}^{\tau }\theta (x)v^{2}dt+\int_{0}^{\tau }\theta
(x)v_{t}^{2}dt. 
\]
Multiplying by $(1/2)$ and integration of both sides of this last inequality
with respect to $x$ from $0$ to $l$ yields (2.16). Substituting
(2.11)-(2.15) in (2.10) and adding the resulting inequality with (2.16),
each side, gives

\begin{eqnarray}
\lefteqn{\frac{1}{2}\int_{0}^{l}\theta (x)v_{t}^{2}(x,\tau )dx+\frac{1}{2}%
\int_{0}^{l}\theta (x)v_{x}^{2}(x,\tau )dx+\frac{1}{2}\int_{0}^{l}\theta
(x)v^{2}(x,\tau )dx}  \nonumber \\
&\leq &\frac{1}{2}\int_{0}^{l}\theta (x)\psi ^{2}dx+\frac{1}{2}%
\int_{0}^{l}\theta (x)\varphi _{x}^{2}dx+\frac{1}{2}\int_{0}^{l}\theta
(x)\varphi ^{2}dx  \nonumber \\
&&\int_{0}^{\tau }\int_{l_{1}}^{l}xv_{x}^{2}\,dx\,dt+\big( \frac{%
3(l-l_{1})^{5}}{8}+\frac{(l-l_{1})^{3}}{4}\big) \int_{0}^{\tau
}\int_{l_{1}}^{l}xv_{t}^{2}\,dx\,dt  \nonumber \\
&&\int_{Q^{\tau }}\theta (x)v_{t}^{2}\,dx\,dt+\int_{0}^{\tau
}\int_{l_{1}}^{l}x^{3}v_{t}^{2}\,dx\,dt+\frac{1}{2}\int_{Q^{\tau }}\theta
(x)v^{2}\,dx\,dt  \label{e46} \\
&&+\big( \frac{1}{2}+\frac{(l-l_{1})}{2}\big) \int_{Q^{\tau }}\theta (x)(%
\mathcal{L}v)^{2}\,dx\,dt  \nonumber \\
&&+\big( \frac{(l-l_{1})^{3}}{2}+\frac{(l-l_{1})}{2}\big) \int_{Q^{\tau }}x(%
\mathcal{L}v)^{2}\,dx\,dt.  \nonumber
\end{eqnarray}
When we add the term $\int_{0}^{\tau
}\int_{l_{1}}^{l}x^{3}v_{x}^{2}\,dx\,dt+\int_{0}^{\tau
}\int_{0}^{l_{1}}(1+l_{1}^{2})xv_{x}^{2}\,dx\,dt$ to the right-hand side of
(2.17) and use the definition of $\theta (x)$, (2.17) takes the form 
\begin{eqnarray}
\lefteqn{\int_{0}^{l}\theta (x)v_{t}^{2}(x,\tau )dx+\int_{0}^{l}\theta
(x)v_{x}^{2}(x,\tau )dx+\int_{0}^{l}\theta (x)v^{2}(x,\tau )dx}  \nonumber \\
&\leq &K\Big( \int_{0}^{l}\theta (x)\psi ^{2}dx+\int_{0}^{l}\theta
(x)\varphi _{x}^{2}dx+\int_{0}^{l}\theta (x)\varphi ^{2}dx  \label{e47} \\
&&+\int_{Q^{\tau }}\theta (x)(\mathcal{L}v)^{2}\,dx\,dt+\int_{Q^{\tau
}}\theta (x)v^{2}\,dx\,dt  \nonumber \\
&&\int_{Q^{\tau }}\theta (x)v_{x}^{2}\,dx\,dt+\int_{Q^{\tau }}\theta
(x)v_{t}^{2}\,dx\,dt\Big) ,  \nonumber
\end{eqnarray}
where $K=\max \left\{ c_{1},c_{2}\right\} $, $c_{1}=\max \big\{ 3+\frac{%
3(l-l_{1})^{5}}{4}+\frac{3(l-l_{1})^{3}}{2},5\big\}$, and $%
c_{2}=1+2(l-l_{1})+(l-l_{1})^{3}$. By [4, Lemma 7.1], we obtain, from
inequality (2.18), 
\begin{eqnarray*}
\left\| v(x,\tau )\right\| _{W_{\theta ,2}^{1,1}((0,l))}^{2} &\leq
&Ke^{K\tau }\left\{ \left\| \varphi \right\| _{H_{\theta
}^{1}((0,l))}^{2}+\left\| \psi \right\| _{L_{\theta
}^{2}((0,l))}^{2}+\left\| \mathcal{L}v\right\| _{L_{\theta }^{2}(Q^{\tau
})}^{2}\right\} \\
&\leq &Ke^{KT}\left\{ \left\| \varphi \right\| _{H_{\theta
}^{1}((0,l))}^{2}+\left\| \psi \right\| _{L_{\theta
}^{2}((0,l))}^{2}+\left\| \mathcal{L}v\right\| _{L_{\theta
}^{2}(Q)}^{2}\right\} .
\end{eqnarray*}
By taking the supremum with respect to $\tau $, over $[0,T]$, the energy
inequality (2.1) follows with $c=\sqrt{K}e^{KT/2}$.

The a priori bound (1.10) leads to the following results.

\begin{corollary}
\label{coro2.2} If a strong solution of the problem (1.7) exists, it is
unique and depends continuously on the data $\mathcal{F}=(f,\varphi ,\psi
)\in F$.
\end{corollary}

\begin{corollary}
\label{coro2.3} The range $R(\overline{L})$ of the operator $\overline{L}$
is closed and coincides with the set $\overline{R(L)}$ and $\overline{L}^{-1}%
\mathcal{F}=\overline{L^{-1}}\mathcal{F}$ where $\overline{L^{-1}}$ is the
continuous extension of $L^{-1}$ from $R(L)$ to $\overline{R(L)}$.
\end{corollary}

\section{Existence of a solution}

The main result in this paper reads as follows.

\begin{theorem}
\label{thm3.1} For each $f\in L_{\theta }^{2}(Q)$, $\varphi \in H_{\theta
}^{1}((0,l))$, $\psi \in L_{\theta }^{2}((0,l))$, there exists a unique
strong solution $v=\overline{L}^{-1}\mathcal{F}=\overline{L^{-1}}\mathcal{F}$
of problem (1.7) satisfying the estimate 
\begin{equation}
\max_{0\leq t\leq T}\left\| v(.,\tau )\right\| _{W_{\theta
,2}^{1,1}((0,l))}^{2}\leq c^{2}\left( \left\| f\right\| _{L_{\theta
}^{2}(Q)}^{2}+\left\| \varphi \right\| _{H_{\theta }^{1}((0,l))}^{2}+\left\|
\psi \right\| _{L_{\theta }^{2}((0,l))}^{2}\right)   \label{e49}
\end{equation}
where $c$ is a positive constant independent of $v$.
\end{theorem}

\begin{remark}
\label{rm3.1} \textrm{\ According to corollary 2.3, to prove the existence
of the solution in the sense of Definition 1.1, for any $(f,\varphi ,\psi
)\in F$, it is sufficient to prove that $R(L)^{\perp }=\left\{ 0\right\} $.
For this purpose we need the following statement. }
\end{remark}

\begin{proposition}
\label{prop3.2} Let $D_{0}(L)=\left\{ v\in D(L):\ell _{1}v=\ell
_{2}v=0\right\} $. If for all $\omega $ in $L^{2}(Q)$ and all $v$ in $%
D_{0}(L)$, 
\begin{equation}
\int_{Q}\omega \mathcal{L}v\,dx\,dt=0,  \label{e50}
\end{equation}
then $\omega $ vanishes almost everywhere in $Q$.
\end{proposition}

\paragraph{Proof}

Assume that relation (3.2) holds for any function $v\in D_{0}(L)$. Using
this fact, (3.2) can be expressed in a special form. First define the
function $\beta $ by the formula 
\begin{equation}
\beta (x,t)=\int_{t}^{T}\omega (x,\tau )\,d\tau \,.  \label{e51}
\end{equation}
Let $v_{tt}$ be a solution of 
\begin{equation}
\beta =\begin{cases} xl_{1}v_{tt}, &\mbox{if }0\leq x<l_{1} \\
\frac{1}{2}(x^{2}+xl_{1})v_{tt}+x\Im _{x}(\xi v_{t}), & \mbox{if } l_{1}<x<l
\end{cases}  \label{e52}
\end{equation}
and let 
\begin{equation}
v=\begin{cases} 0, &\mbox{if } 0\leq t\leq s \\ \int_{s}^{t}(t-\tau )v_{\tau
\tau }d\tau ,&\mbox{if } s\leq t\leq T. \end{cases}  \label{e53}
\end{equation}
It follows that 
\begin{equation}
\omega =\begin{cases} -xl_{1}v_{ttt},& \mbox{if } 0\leq x<l_{1} \\
-\frac{1}{2}(x^{2}+xl_{1})v_{ttt}-x\Im _{x}(\xi v_{tt}),& \mbox{if }
l_{1}<x<l. \end{cases}  \label{e54}
\end{equation}

By [10, Lemma 4.2], the function $v$ defined by the relations (3.4) and
(3.5) has derivatives with respect to $t$ up to the third order belonging to
the space $L^{2}(Q_{s})$, where $Q_{s}=(0,l)\times (s,T)$. By replacing the
function $\omega $, given by its representation (3.6), in (3.2) we get 
\begin{eqnarray}
-\int_{s}^{T}\int_{0}^{l_{1}}l_{1}xv_{ttt}\big( v_{tt}-\frac{1}{x}%
(xv_{x})_{x}\big) \,dx\,dt &&  \nonumber  \label{e55} \\
-\frac{1}{2}\int_{s}^{T}\int_{l_{1}}^{l}(l_{1}x+x^{2})v_{ttt}\big( v_{tt}-%
\frac{1}{x}(xv_{x})_{x}\big) \,dx\,dt && \\
-\int_{s}^{T}\int_{l_{1}}^{l}x\Im _{x}(\xi v_{tt})\big( v_{tt}-\frac{1}{x}%
(xv_{x})_{x}\big) \,dx\,dt &=&0.  \nonumber
\end{eqnarray}
In light of conditions (1.7) and the special form of $v$ given by relations
(3.4), (3.5), we integrate by parts each term of (3.7) to obtain the
following equations: 
\begin{eqnarray}
\lefteqn{-\int_{s}^{T}\int_{0}^{l_{1}}l_{1}xv_{ttt}\big(v_{tt}-\frac{1}{x}%
(xv_{x})_{x}\big)\,dx\,dt}  \label{e56} \\
&=&\frac{1}{2}\int_{0}^{l_{1}}l_{1}xv_{tt}^{2}(x,s)dx+\frac{1}{2}%
\int_{0}^{l_{1}}l_{1}xv_{tx}^{2}(x,T)dx-%
\int_{s}^{T}l_{1}^{2}v_{tx}(l_{1},t)v_{tt}(l_{1},t)dt,  \nonumber
\end{eqnarray}
\begin{eqnarray}
\lefteqn{-\frac{1}{2}\int_{s}^{T}\int_{l_{1}}^{l}(l_{1}x+x^{2})v_{ttt}\big%
(v_{tt}-\frac{1}{x}(xv_{x})_{x}\big)\,dx\,dt}  \nonumber  \label{e57} \\
&=&\frac{1}{4}\int_{l_{1}}^{l}(l_{1}x+x^{2})v_{tt}^{2}(x,s)dx+\frac{1}{4}%
\int_{l_{1}}^{l}(l_{1}x+x^{2})v_{tx}^{2}(x,T)dx \\
&&+\int_{s}^{T}l_{1}^{2}v_{tx}(l_{1},t)v_{tt}(l_{1},t)dt+\frac{1}{2}%
\int_{s}^{T}\int_{l_{1}}^{l}xv_{tx}v_{tt}\,dx\,dt,  \nonumber
\end{eqnarray}
\begin{equation}
-\int_{s}^{T}\int_{l_{1}}^{l}x\Im _{x}(\xi v_{tt})\big( v_{tt}-\frac{1}{x}%
(xv_{x})_{x}\big) \,dx\,dt=\int_{s}^{T}\int_{l_{1}}^{l}x^{2}v_{x}v_{tt}\,dx%
\,dt.  \label{e58}
\end{equation}
Substituting (3.8)-(3.10) in (3.7) yields 
\begin{eqnarray}
\lefteqn{\frac{l_{1}}{2}\int_{0}^{l_{1}}xv_{tt}^{2}(x,s)dx+\frac{l_{1}}{2}%
\int_{0}^{l_{1}}xv_{tx}^{2}(x,T)dx}  \nonumber  \label{e59} \\
\lefteqn{+\frac{1}{4}\int_{l_{1}}^{l}(l_{1}x+x^{2})v_{tt}^{2}(x,s)dx+\frac{1%
}{4}\int_{l_{1}}^{l}(l_{1}x+x^{2})v_{tx}^{2}(x,T)dx} \\
&=&-\frac{1}{2}\int_{s}^{T}\int_{l_{1}}^{l}xv_{tt}v_{tx}\,dx\,dt-%
\int_{s}^{T}\int_{l_{1}}^{l}x^{2}v_{tt}v_{x}\,dx\,dt\,.  \nonumber
\end{eqnarray}
Using Young's and Poincare's inequalities, we estimate the right-hand side
of (3.11) as follows 
\begin{equation}
-\frac{1}{2}\int_{s}^{T}\int_{l_{1}}^{l}xv_{tt}v_{tx}\,dx\,dt\leq \frac{1}{4}%
\int_{s}^{T}\int_{l_{1}}^{l}xv_{tx}^{2}\,dx\,dt+\frac{1}{4}%
\int_{s}^{T}\int_{l_{1}}^{l}xv_{tt}^{2}\,dx\,dt,
\end{equation}
\begin{eqnarray}
-\int_{s}^{T}\int_{l_{1}}^{l}x^{2}v_{tt}v_{x}\,dx\,dt &\leq &\frac{1}{2}%
\int_{s}^{T}\int_{l_{1}}^{l}x^{2}v_{x}^{2}\,dx\,dt+\frac{1}{2}%
\int_{s}^{T}\int_{l_{1}}^{l}x^{2}v_{tt}^{2}\,dx\,dt  \nonumber  \label{e61}
\\
&\leq &\frac{d}{2}\int_{s}^{T}\int_{l_{1}}^{l}x^{2}v_{xt}^{2}\,dx\,dt+\frac{1%
}{2}\int_{s}^{T}\int_{l_{1}}^{l}x^{2}v_{tt}^{2}\,dx\,dt.
\end{eqnarray}
Combining (3.11)-(3.13), we arrive at 
\begin{eqnarray}
\lefteqn{\int_{0}^{l_{1}}xv_{tt}^{2}(x,s)dx+%
\int_{0}^{l_{1}}xv_{tx}^{2}(x,T)dx}  \nonumber  \label{e62} \\
\lefteqn{+\int_{l_{1}}^{l}(x+x^{2})v_{tt}^{2}(x,s)dx+%
\int_{l_{1}}^{l}(x+x^{2})v_{tx}^{2}(x,T)dx} \\
&\leq &\delta \Big( \int_{s}^{T}\int_{l_{1}}^{l}(x+x^{2})v_{tt}^{2}\,dx\,dt+%
\int_{s}^{T}\int_{l_{1}}^{l}(x+x^{2})v_{tx}^{2}\,dx\,dt\Big) ,  \nonumber
\end{eqnarray}
where $\delta =2\max \left\{ d,1\right\} /\min \left\{ l_{1},1\right\} $.
When we add to the right-hand side of (3.14) the quantity 
\[
\delta \int_{s}^{T}\int_{0}^{l_{1}}xv_{tt}^{2}\,dx\,dt+\delta
\int_{s}^{T}\int_{0}^{l_{1}}xv_{tx}^{2}\,dx\,dt,
\]
and define the function 
\[
\rho (x)=%
\begin{cases}
x &\mbox{if } 0<x<l_{1} \\
x+x^{2}& \mbox{if }l_{1}<x<l
\end{cases}
\]
we deduce, from (3.14), that 
\begin{eqnarray}
\lefteqn{\int_{0}^{l}\rho (x)v_{tt}^{2}(x,s)dx+\int_{0}^{l}\rho
(x)v_{tx}^{2}(x,T)dx}  \nonumber  \label{e63} \\
&\leq &\delta \big\{ \int_{Q_{s}}\rho (x)v_{tt}^{2}\,dx\,dt+\int_{Q_{s}}\rho
(x)v_{tx}^{2}\,dx\,dt\big\}.
\end{eqnarray}
This inequality is basic in our proof. To use it, we introduce the new
function 
\[
\eta (x,t)=\int_{t}^{T}v_{\tau \tau }d\tau .
\]
Then 
\[
v_{t}(x,t)=\eta (x,s)-\eta (x,t),\quad v_{t}(x,T)=\eta (x,s).
\]
Thus inequality (3.15) becomes 
\begin{eqnarray}
\lefteqn{\int_{0}^{l}\rho (x)v_{tt}^{2}(x,s)dx+(1-2\delta
(T-s))\int_{0}^{l}\rho (x)\eta _{x}^{2}(x,s)dx}  \nonumber  \label{e64} \\
&\leq &2\delta \Big\{ \int_{s}^{T}\int_{0}^{l}\rho
(x)v_{tt}^{2}\,dx\,dt+\int_{s}^{T}\int_{0}^{l}\rho (x)\eta
_{x}^{2}(x,t)\,dx\,dt\Big\} .
\end{eqnarray}
Hence, when $s_{0}>0$ satisfies $T-s_{0}=1/4\delta $, (3.16) implies 
\begin{eqnarray}
\lefteqn{\int_{0}^{l}\rho (x)v_{tt}^{2}(x,s)dx+\int_{0}^{l}\rho (x)\eta
_{x}^{2}(x,s)dx}  \nonumber  \label{e65} \\
&\leq &4\delta \Big\{ \int_{s}^{T}\int_{0}^{l}\rho
(x)v_{tt}^{2}\,dx\,dt+\int_{s}^{T}\int_{0}^{l}\rho (x)\eta
_{x}^{2}(x,t)\,dx\,dt\Big\}
\end{eqnarray}
for all $s\in [T-s_{0},T]$. If, in (3.17) we put 
\[
g(s)=\int_{s}^{T}\int_{0}^{l}\rho
(x)v_{tt}^{2}\,dx\,dt+\int_{s}^{T}\int_{0}^{l}\rho (x)\eta
_{x}^{2}(x,t)\,dx\,dt,
\]
then we have $\frac{-dg}{ds}\leq 4\delta g(s)$, from which it follows that 
\[
%\label{e66}
\frac{-d}{ds}\left( g(s)\exp (4\delta s)\right) \leq 0.
\]
Integrating this equation over $(s,T)$ and taking in account that $g(T)=0$,
we obtain 
\[
%67
g(s)\exp (4\delta s)\leq 0.
\]
This inequality guarantees that $g(s)=0$ for all $s\in [T-s_{0},T]$, which
implies that $v_{tt}=0$ on $Q_{s}$ where $s\in [T-s_{0},T]$. Hence it
follows, from (3.6), that $\omega \equiv 0$ almost everywhere on $Q_{T-s_{0}}
$. Proceeding this way step by step along the rectangle with side $s_{0}$,
we prove that $\omega \equiv 0$ almost everywhere on $Q$. This completes the
proof of the Proposition 3.3.

\paragraph{Proof of Theorem 3.1}

Suppose that for some $W=(\omega ,\omega _{1},\omega _{2})\in R(L)^{\perp }$%
, 
\begin{equation}
(\mathcal{L}v,\omega )_{L_{\theta }^{2}(Q)}+(\ell _{1}v,\omega
_{1})_{H_{\theta }^{1}((0,l))}+(\ell _{2}v,\omega _{2})_{L_{\theta
}^{2}((0,l))}=0.  \label{e68}
\end{equation}
Then we must prove that $W\equiv 0$. Putting $v\in D_{0}(L)$ into (3.18), we
have 
\[
(\mathcal{L}v,\omega )_{L_{\theta }^{2}(Q)}=0. 
\]
Hence Proposition 3.3 implies that $\omega \equiv 0$. Thus (3.18) takes the
form 
\begin{equation}
(\ell _{1}v,\omega _{1})_{H_{\theta }^{1}((0,l))}+(\ell _{2}v,\omega
_{2})_{L_{\theta }^{2}((0,l))}=0,\quad \forall v\in D(L).  \label{e69}
\end{equation}
Since the quantities $\ell _{1}v$ and $\ell _{2}v$ can vanish independently
and the ranges of the trace operators $\ell _{1}$ and $\ell _{2}$ are dense
in the spaces $H_{\theta }^{1}((0,l))$ and $L_{\theta }^{2}((0,l))$
respectively, the equation (3.19) implies that $\omega _{1}\equiv 0$, $%
\omega _{2}\equiv 0$. Hence $W\equiv 0$. The proof of Theorem 3.1 is
established.

\paragraph{Acknowledgment}

This work was completed while the first author was visiting the Mathematical
Sciences Department at KFUPM. Both authors would like to thank KFUPM for its
support.

\begin{thebibliography}{99}
\bibitem{b1}  A. Bouziani, \textit{Solution forte d'un probl\`{e}me mixte
avec condition int\'{e}grale pour une classe d'\'{e}quations hyperboliques,
Bull. CL. Sci., Acad. Roy. Belg.} \textbf{8} (1997), 53 - 70.

\bibitem{c1}  J. R. Cannon, \textit{The solution of heat equation subject to
the specification of energy, Quart. Appl. Math}. \textbf{21} (1963), 155 -
160.

\bibitem{d1}  M. Denche and A. L. Marhoune, \textit{A Three-point boundary
value problem with an integral condition for parabolic equations with the
Bessel operator, Appl. Math. Letters} \textbf{13} (2000), 85 - 89.

\bibitem{g1}  L. Garding, \textit{Cauchy's problem for hyperbolic equations, 
}University of Chicago, 1957.

\bibitem{i1}  N. I. Ionkin, \textit{Solution of a boundary value problem in
heat conduction with a nonclassical boundary condition, Differential
Equations} \textbf{13} (1977), 294 - 304.

\bibitem{k1}  A. V. Kartynnik, \textit{Three-point boundary value problem
with an integral space-variable condition for a second order parabolic
equation, Differential Equations} \textbf{26} (1990), 1160 - 1162.

\bibitem{l1}  O. A. Ladyzhenskaya, Boundary-value problems for partial
differential equations, Dokladi Academii nauk SSSR 97 n.3 (1954).

\bibitem{m1}  V. L. Makarov and D. T. Kulyev, \textit{Solution of a boundary
value problem for a quasi-parabolic equation with a nonclassical boundary
condition, Differential Equations} \textbf{21} (1985), 296 - 305.

\bibitem{m2}  S. Mesloub and A. Bouziani, \textit{On a class of singular
hyperbolic equation with a weighted integral condition,} \textit{Internat.
J. Math. \& Math. Sci.} \textbf{22} No. 3 (1999), 511 - 519.

\bibitem{m3}  S. Mesloub, \textit{On a nonlocal problem for a pluriparabolic
equation, Acta Sci. Math. (Szeged) }\textbf{67} (2001), 203 - 219.

\bibitem{s1}  P. Shi, \textit{Weak solution to an evolution problem with a
nonlocal constraint, SIAM J. Math. Anal.} \textbf{24} No. 1
(1993), 46 - 58.

\bibitem{s2}  A. A. Samarskii, \textit{Some problems in differential
equations theory, Differentsial'nye Uravnenya} \textbf{16} No. 
11 (1980), 1221 - 1228.

\bibitem{y1}  N. I. Yurchuk, \textit{Mixed problem with an integral
condition for certain parabolic equations, Differential equations} 
\textbf{22} No. 12 (1986), 1457 - 1463.
\end{thebibliography}

\noindent \textsc{Said Mesloub}\newline
Department of Mathematics, University of Tebessa, Tebessa 12002, Algeria
\newline
e-mail : mesloubs@yahoo.com \smallskip

\noindent \textsc{Salim A. Messaoudi }\newline
Department of Mathematical Sciences, KFUPM, Dhahran 31261\newline
Saudi Arabia\newline
e-mail : messaoud@kfupm.edu.sa

\end{document}