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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 138, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/138\hfil Inverse problem of a hyperbolic equation]
{Inverse problem of a hyperbolic equation with an
integral overdetermination condition}

\author[T.-E. Oussaeif, A. Bouziani \hfil EJDE-2016/138\hfilneg]
{Taki-Eddine Oussaeif, Abdelfatah Bouziani}

\address{Taki-Eddine Oussaeif \newline
Department of Mathematics and Informatics,
The Larbi Ben M'hidi University,
Oum El Bouaghi, Alg\'erie}
\email{taki\_maths@live.fr}

\address{Bouziani Abdelfatah \newline
D\'epartement de Mathematiques et Informatique, Universit\'e
Larbi Ben M'hidi-Oum El Bouaghi, Alg\'erie}
\email{af\_bouziani@hotmail.com}

\thanks{Submitted October 27, 2015. Published June 8, 2016.}
\subjclass[2010]{35R30, 49K20}
\keywords{Inverse problem; hyperbolic equation; integral condition}

\begin{abstract}
 In this article we study the inverse problem of a hyperbolic equation
 with an integral overdetermination condition.
 The existence, uniqueness and  continuous dependence of the solution
 of the solution upon the data are established.
\end{abstract}

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

\section{Introduction}

In this article we study the unique solvability of the inverse problem of
determining a pair of functions $\{u,f\}$ satisfying the equation
\begin{equation} \label{e1.1}
u_{tt}-\Delta u+\beta u_{t}=f(t) g(x,t), \quad  x\in \Omega,\; t\in [ 0,T] ,
\end{equation}
with the initial conditions
\begin{gather}
u(x,0)=\varphi (x) ,\quad  x\in \Omega , \label{e1.2}\\
u_{t}(x,0)=\psi (x) ,\quad x\in \Omega ,  \label{e1.3}
\end{gather}
the boundary condition
\begin{equation} \label{e1.4}
u(x,t)=0,\quad  (x,t)\in \partial \Omega \times [ 0,T] ,
\end{equation}
and the nonlocal condition
\begin{equation} \label{e1.5}
\int_{\Omega }v(x) u(x,t) dx=\theta (t) , \quad t\in [ 0,T] .
\end{equation}
Here $\Omega $ is a bounded domain in $\mathbb{R}^{n}$ with smooth
 boundary $\partial \Omega .$ The functions $g$, $\varphi$, $\psi $, 
$\theta $ are known functions and $\beta $ is a positive
constant.

Inverse problems for hyperbolic PDEs arise naturally in geophysics, oil
prospecting, in the design of optical devices, and in many other areas where
the interior of an object is to be imaged using the response of the object
to acoustic waves (satisfying hyperbolic PDEs). Additional information about
the solution to the inverse problem is given in the form of integral
observation condition \eqref{e1.5}.

The parameter identification in a partial differential equation from the
data of integral overdetermination condition plays an important role in
engineering and physics \cite{c3,c4,f1,i2,i3,k4}.
From the physical point
of view, these conditions may be interpreted as measurements of the
temperature $u(x,t)$ by a device averaging over the domain of spatial
variables.

Note that inverse problems with integral overdetermination are closely
related to nonlocal problems \cite{c1,k2}. Studies have shown that
classical methods often do not work when we deal with nonlocal problems 
\cite{b1,i1}. To date, several methods have been proposed for overcoming
the difficulties arising from nonlocal conditions. The choice of method
depends on the kind of nonlocal conditions.

We note that the inverse problem for a parabolic equation with integral
condition \eqref{e1.5} and its unique solubility have been studied by many
authors (see for example \cite{c2,k1,k5,p1}).

Also, there are some papers devoted to the study of existence and uniqueness
of solutions of inverse problems for various parabolic equations with
unknown source functions. Inverse problems of determining the right-hand
side of a parabolic equation under a final overdetermination condition were
studied in papers \cite{k2,k3,p1,p2}.


In the present work, new studies are presented for the inverse problem for a
hyperbolic equation. The existence and uniqueness of the classical solution
of the problem \eqref{e1.1}-\eqref{e1.5} is reduced to a fixed point problem.

\section{Functional space}

Let us introduce certain notation used below. We set
\begin{equation} \label{e2.1}
g^{\ast }(t) =\int_{\Omega }v(x) g(x,t)
dx,\quad Q_{T}=\Omega \times [ 0,T] .
\end{equation}
The spaces $W_2^1(\Omega ) $,
$C((0,T),L_2(\Omega )) $ and $W_2^{2,1}(Q_{T}) $ with
corresponding norms are understood as follows: 
The Banach space $W_2^1(\Omega )$ consists of all functions from
$L_2(\Omega )$ having all weak derivatives of the first order that are
square integrable over with norm
\begin{equation*}
\| u\| _{W_2^1(\Omega ) }=(\|u\| _{L_2(\Omega )}^2
+\| u_{x}\|_{L_2(\Omega )}^2) ^{1/2},
\end{equation*}
We denote by $C((0,T) , L_2(\Omega )) $ the
space comprises of all continuous functions on $(0,T)$ with values in 
$L_2(\Omega )$. The corresponding norm is given by
\begin{equation*}
\| u\| _{C((0,T) , L_2(\Omega
)) }=\max_{(0,T) }\| u\| _{L_2(\Omega
)}<\infty .
\end{equation*}
Let us also introduce the Sobolev space $W_2^{2,1}(Q_{T}) $ of
functions $u(x,t)$ with finite norm
\begin{equation*}
\| u\| _{W_2^{2,1}(Q_{T}) }=\Big(\|
u\| _{L_2(Q_{T})}^2+\| u_{x}\|
_{L_2(Q_{T})}^2+\| D_{t}u\| _{L_2(Q_{T})}^2\Big)
^{1/2}
\end{equation*}
where
\begin{equation*}
\| u\| \equiv \| u\| _{L_2(\Omega )},
\end{equation*}
We note that the weighted arithmetic-geometric mean inequality 
(Cauchy's $\varepsilon $-inequality) is:
\begin{equation*}
2| ab| \leqslant \varepsilon a^2+\frac{1}{\varepsilon }
b^2,\text{ \ for }\varepsilon >0.
\end{equation*}
Also, we use the notation
\begin{equation*}
\| \nabla u\| =\Big(\int_{\Omega
}\sum_{i=1}^{n}u_{x_{i}}^2dx\Big) ^{1/2}\quad \text{and}\quad
\|\Delta u\| =\Big(\int_{\Omega
}\sum_{i,j=1}^{n}u_{x_{i},x_{j}}^2dx\Big) ^{1/2}.
\end{equation*}

\section{Existence and uniqueness of the solution to the inverse
problem}

We seek a solution of the original inverse problem in the form 
$\{u,f\}=\{z,f\}+\{y,0\}$ where $y$ is the solution of the direct problem
\begin{gather} \label{e3.1}
y_{tt}-\Delta y+\beta y_{t}=0, \quad (x,t) \in Q_{T}, \\
 \label{e3.2}
y(x,0)=\varphi (x) ,\quad  x\in \Omega , \\
 \label{e3.3}
y_{t}(x,0)=\psi (x) ,\quad x\in \Omega , \\
 \label{e3.4}
y(x,t)=0,\quad  (x,t)\in \partial \Omega \times [ 0,T] ,
\end{gather}
while the pair $\{z,f\}$ is the solution of the inverse problem
\begin{gather} \label{e3.5}
z_{tt}-\Delta z+\beta z_{t}=f(t) g(x,t),\quad (x,t) \in Q, \\
 \label{e3.6}
z(x,0)=0,\quad x\in \Omega , \\
 \label{e3.7}
z_{t}(x,0)=0,\quad x\in \Omega , \\
 \label{e3.8}
z(x,t)=0,\quad (x,t) \in \partial \Omega \times [ 0,T], \\
\label{e3.9}
\int_{\Omega }v(x) z(x,t) \,dx=E(t) ,\quad t\in [ 0,T] ,
\end{gather}
where
\begin{equation*}
E(t) =\theta (t) -\int_{\Omega }v(x) y(x,t) \,dx.
\end{equation*}
We shall assume that the functions appearing in the data for the problem are
measurable and satisfy the following conditions:
\begin{itemize}
\item[(H1)] $g\in C((0,T),L_2(\Omega ))$,
$v\in W_2^1(\Omega )$, $E\in W_2^2(0,T)$,
$\| g(x,t) \| \leqslant m$, $|g^{\ast }(t) | \geqslant p>0$, for
$p\in\mathbb{R}$, $(x,t) \in Q_{T}$,
$\varphi (x),\psi (x) \in W_2^1(\Omega )$
\end{itemize}
The correspondence between $f$ and $z$ may be viewed as one possible way of
specifying the linear operator
\begin{equation} \label{e3.10}
A:L_2(0,T) \to L_2(0,T,L_2(\Omega)) .
\end{equation}
with the values
\begin{equation} \label{e3.11}
(Af)(t)=\frac{1}{g^{\ast }}\Big\{ \int_{\Omega }\nabla
z\nabla vdx\Big\} .
\end{equation}
In this view, it is reasonable to refer to the linear equation of the second
kind for the function $f$ over the space $L_2(0,T) $:
\begin{equation} \label{e3.12}
f=Af+W,
\end{equation}
where
\begin{equation*}
W=\frac{E''+\beta E}{g^{\ast }}.
\end{equation*}

\begin{remark} \label{rmk1} \rm
As $\{u,f\}=\{z,f\}+\{y,0\}$ where $y$ is the solution of the direct problem
\eqref{e3.1}--\eqref{e3.4}.
Obviously, $y$ exists and is unique, so instead of proving the solvability
of the original problem \eqref{e1.1}--\eqref{e1.5}, we prove the existence 
and uniqueness of the solution of the inverse problem \eqref{e3.5}--\eqref{e3.9}.
\end{remark}

\begin{theorem} \label{thm1}
Suppose the input data of the inverse problem \eqref{e3.5}-\eqref{e3.9}
satisfies (H1).
Then the following assertions are valid: 
(i) if the inverse problem \eqref{e3.5}--\eqref{e3.9} is solvable,
then so is equation \eqref{e3.12}. 
(ii) if equation \eqref{e3.12} possesses a solution and the compatibility 
condition
\begin{equation} \label{e3.13}
E(0) =0,\quad E'(0) =0,
\end{equation}
holds, then there exists a solution of the inverse problem 
\eqref{e3.5}-\eqref{e3.9}.
\end{theorem}

\begin{proof}
(i) Assume that the inverse problem \eqref{e3.5}--\eqref{e3.9} is solvable.
We denote its solution by $\{z,f\}$. Multiplying both sides of \eqref{e3.5}
 by the function $v$ scalarly in $L_2(\Omega ) $, we obtain the relation
\begin{equation} \label{e3.14}
\frac{d}{dt}\int_{\Omega }z_{t}vdx+\int_{\Omega }\nabla
z\nabla vdx+\beta \int_{\Omega }z_{t}vdx=f(t) g^{\ast}(x,t).
\end{equation}
With \eqref{e3.9} and \eqref{e3.11}, it follows from \eqref{e3.14} that
 $f=Af+\frac{E''+\beta E}{g^{\ast }}$. This means that $f$ solves equation
\eqref{e3.12}.

(ii) By the assumption, equation \eqref{e3.12} has a solution in the space 
$L_2(0,T) $, say $f$.

When inserting this function in \eqref{e3.5}, the resulting relations 
\eqref{e3.5}--\eqref{e3.8} can be treated as a direct problem having a
unique solution $z\in W_{2,0}^{2,1}(Q_{T})$. Let us show that the function $z$
satisfies also the integral overdetermination condition \eqref{e3.9}.

Equation \eqref{e3.14} yields
\begin{equation} \label{e3.15}
\frac{d}{dt}\int_{\Omega }z_{t}vdx+\int_{\Omega }\nabla
z\nabla vdx+\beta \int_{\Omega }z_{t}vdx=f(t) g^{\ast
}(x,t).
\end{equation}
On the other hand, being a solution of \eqref{e3.12}, the function $z$
is subject to relation
\begin{equation} \label{e3.16}
E''+\beta E'+\int_{\Omega }\nabla
z\nabla v\,dx=f(t) g^{\ast }(x,t).
\end{equation}
Subtracting  \eqref{e3.15} from  \eqref{e3.16}, we obtain
\begin{equation*}
\frac{d}{dt}\int_{\Omega }z_{t}vdx
+\beta \int_{\Omega }z_{t}vdx=E''+\beta E'.
\end{equation*}
Integrating the preceding differential equation and taking into account the
compatibility condition \eqref{e3.13}, we find that the function $z$ satisfies
the overdetermination condition \eqref{e3.9} and the pair of functions $\{z,f\}$
is a solution of the inverse problem \eqref{e3.5}--\eqref{e3.9}.
This completes the proof.
\end{proof}

Now, we state some properties of the operator $A$.

\begin{lemma} \label{lem1}
Let  condition {\rm (H1)} hold. Then there exist a positive $\varepsilon $ for
which $A$ is a contracting operator in $L_2(0,T) $.
\end{lemma}

\begin{proof}
Obviously, \eqref{e3.11} yields the estimate
\begin{equation} \label{e3.17}
\| Af\| _{L_2(0,t) }\leq \frac{k}{p}\Big(
\int_0^{t}\| \nabla z(.,\tau ) \|
_{L_2(\Omega ) }^2d\tau \Big) ^{1/2},
\end{equation}
where
$k=\| \nabla v\| _{L_2(\Omega ) }$.
Multiplying both sides of \eqref{e3.5} by $z_{t}$ scalarly in 
$L_2(\Omega) $ and integrating the resulting expressions by parts, we obtain the
identity
\begin{equation*}
\frac{1}{2}\frac{d}{dt}\| z_{t}(\cdot,t) \|
_{L_2(\Omega ) }^2+\frac{1}{2}\frac{d}{dt}\|
\nabla z(\cdot,t) \| _{L_2(\Omega )
}^2+\beta \| z_{t}(\cdot,t) \| _{L_2(\Omega ) }^2
=f(t) \int_{\Omega }g(x,t)z_{t}dx.
\end{equation*}
So, by using the Cauchy's $\varepsilon $-inequality, we obtain the relation
\begin{equation} \label{e3.18}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\| z_{t}(\cdot,t) \|
_{L_2(\Omega ) }^2+\frac{1}{2}\frac{d}{dt}\|
\nabla z(\cdot,t) \| _{L_2(\Omega )
}^2+\beta \| z_{t}(\cdot,t) \| _{L_2(
\Omega ) }^2   \\
&\leq \frac{m^2}{2\varepsilon }| f(t) |
^2+\frac{\varepsilon }{2}\| z_{t}(\cdot,t) \|
_{L_2(\Omega ) }^2,
\end{aligned}
\end{equation}
Choosing $0<\varepsilon <2\beta $, and integrating \eqref{e3.18}
over $(0,t) $, and using \eqref{e3.6} and \eqref{e3.7}), we obtain
\begin{equation} \label{e3.19}
\begin{aligned}
&\frac{1}{2}\| z_{t}(\cdot,t) \| _{L_2(
\Omega ) }^2+\frac{1}{2}\| \nabla z(\cdot,t)\| _{L_2(\Omega ) }^2
+(\beta -\frac{\varepsilon }{2}) \int_0^{t}\| z_{t}(.,\tau )
\| _{L_2(\Omega ) }^2   \\
&\leq \frac{m^2}{2\varepsilon }\int_0^{t}| f(\tau) | ^2.
\end{aligned}
\end{equation}
Omitting some terms on the left-hand side \eqref{e3.19} and integrating
 over $(0,t)$, and using \eqref{e3.6}, leads to
\begin{equation} \label{e3.20}
\int_0^{t}\| \nabla z(.,\tau ) \|
_{L_2(\Omega ) }^2d\tau \leq \frac{m^2}{\varepsilon }
\int_0^{t}\| f(\tau ) \| _{L_2(0,T) }^2d\tau .
\end{equation}
So, according to \eqref{e3.17}  and \eqref{e3.20}, we obtain the
 estimate
\begin{equation} \label{e3.21}
\| Af\| _{L_2(0,t) }\leq \delta \Big(\int_0^{t}\| f(\tau ) \| _{L_2(
0,T) }^2d\tau \Big) ^{1/2},\quad 0\leqslant t\leqslant T,
\end{equation}
where
\begin{equation*}
\delta =\frac{km}{p\sqrt{\varepsilon }}.
\end{equation*}
So, we obtain
\begin{equation} \label{e3.22}
\| Af\| _{L_2(0,T) }\leq \delta \|f\| _{L_2(0,T, L_2(0,T) ) }.
\end{equation}
It follows from the foregoing that there exists a positive $\varepsilon $
such that
\begin{equation} \label{e3.23}
\delta <1.
\end{equation}
Inequality \eqref{e3.23} shows that the linear operator $A$ is a contracting
mapping on $L_2(0,T, L_2(0,T) ) $ and
completes the proof.
\end{proof}

Regarding the unique solvability of the inverse problem, the
following result will be useful.

\begin{theorem} \label{thm2}
Let  condition {\rm (H1)} and the compatibility condition \eqref{e3.13}
hold. Then the following assertions are valid: 
(i) a solution $\{z,f\}$ of the inverse problem \eqref{e3.5}-\eqref{e3.9}
exists and is unique, and 
(ii) with any initial iteration $f_0\in L_2(0,T,L_2(0,T)) $ the successive
approximations
\begin{equation} \label{e3.24}
f_{n+1}=\widetilde{A}f_{n}\,.
\end{equation}
converge to $f$ in the $L_2(0,T, L_2(0,T)) $-norm
(for $\widetilde{A}$ see below).
\end{theorem}

\begin{proof}
(ii) We  use the nonlinear operator
$\widetilde{A}:L_2(0,T) \to L_2(0,T,L_2(0,T) )$
acting in accordance to the rule
\begin{equation} \label{e3.25}
\widetilde{A}f=Af+\frac{E''+\beta E}{g^{\ast }},
\end{equation}
where the operator $A$ and the function $g^{\ast }$ arise from \eqref{e3.11}.
From \eqref{e3.24} it follows that  \eqref{e3.12} can be written as
\begin{equation} \label{e3.26}
f=\widetilde{A}f.
\end{equation}
Hence it is sufficient to show that operator $\widetilde{A}$ has a fixed
point in the space $L_2(0,T,L_2(0,T) ) $. By the relations
\begin{equation*}
\widetilde{A}f_1-\widetilde{A}f_2=Af_1-Af_2=A(f_1-f_2),
\end{equation*}
 from estimate \eqref{e3.22} we deduce that
\begin{equation} \label{e3.27}
\begin{aligned}
\| \widetilde{A}f_1-\widetilde{A}f_2\| _{L_2(0,T) }
&=\| A(f_1-f_2) \| _{L_2(0,T) } \\
&\leq \delta \| (f_1-f_2) \|_{L_2(0,T, L_2(0,T) ) }.
\end{aligned}
\end{equation}
From \eqref{e3.23}  and \eqref{e3.26}, we find that
$\widetilde{A}$ is a contracting mapping on $L_2(0,T,L_2(0,T) )$.

Therefore $\widetilde{A}$ has a unique fixed point $f$ in 
$L_2(0,T, L_2(0,T) ) $ and the successive approximations
\eqref{e3.24} converge to $f$ in the $L_2(0,T,L_2(0,T) ) $-norm
irrespective of the initial iteration
$f_0\in L_2(0,T, L_2(0,T) ) $.

(i) This shows that, equation \eqref{e3.26} and, in turn, equation
\eqref{e3.12} have a unique solution $f$ in $L_2(0,T,L_2(0,T) ) $.

According to Theorem \ref{thm1}, this confirms the existence of solution to the
inverse problem \eqref{e3.5}--\eqref{e3.9}.

It remains to prove the uniqueness of this solution.
Assume to the contrary that there were two distinct solutions
$\{z_1,f_1\}$  and $\{z_2,f_2\}$ of the inverse problem under
consideration.

We claim that in that case $f_1\neq f_2$ almost everywhere on 
$(0,T)$. If $f_1=f_2$, then applying the uniqueness theorem to the
corresponding direct problem \eqref{e3.1}--\eqref{e3.4} we
would have $z_1=z_2$ almost everywhere in $Q_{T}$.

Since both pairs satisfy identity \eqref{e3.14}, the functions 
$f_1 $ and $f_2$ give two distinct solutions to equation
\eqref{e3.26}. But this contradicts the uniqueness of the solution to
equation \eqref{e3.26} just established and proves the theorem.
\end{proof}

\begin{corollary} \label{coro1}
Under the conditions of Theorem \ref{thm2}, the solution $f$ to equation 
\eqref{e3.12} depends continuously upon the data $W$.
\end{corollary}

\begin{proof}
Let $W$ and $V$ be two sets of data, which satisfy the assumptions of
Theorem \ref{thm2}.
Let $f$ and $g$ be solutions of the equation \eqref{e3.12}
corresponding to the data $W$ and $V$, respectively. 
According to \eqref{e3.12},
we have
\begin{gather*}
f = Af+W, \\
g = Ag+V.
\end{gather*}
First, let us estimate the difference $f-g$. It is easy to see by using 
\eqref{e3.22}, that
\begin{align*}
\| f-g\| _{L_2(0,T, L_2(0,T)) }
&=\| (Af+W) -(Ag+V) \| _{L_2(0,T) } \\
&= \| A(f-g) +(W-V) \| _{L_2(0,T) } \\
&\leqslant \delta \| f-g\| _{L_2(0,T,L_2(0,T) ) }+\| (W-V) \|_{L_2(0,T) };
\end{align*}
so, we obtain
\[
\| f-g\| _{L_2(0,T, L_2(0,T)) }\leqslant \frac{1}{(1-\delta ) }\| (
W-V) \| _{L_2(0,T) }.
\]
The proof  is complete.
\end{proof}


\begin{thebibliography}{99}

\bibitem{b1} Bouziani, A.;
\emph{Solution forte d'un problem mixte avec
condition non locales pour uneclasse d'equations hyperboliques}; Bull. de la
Classe des Sciences, Academie Royale de Belgique. 1997. V. 8, pp. 53-70.

\bibitem{c1} Cannon, J. R.; Lin, Y.;
\emph{An inverse problem of finding a parameter in a semi-linear heat equation};
J. Math. Anal. Appl. 1990. V. 145. pp. 470-484.

\bibitem{c2} Cannon, J. R.; Lin, Y.;
\emph{Determination of a parameter $p(t)$ in some quasilinear parabolic
differential equations}, Inverse Problems 4:1 (1988), 35-45.

\bibitem{c3} Cannon, J. R.; Lin, Y.;  Wang, S.;
\emph{Determination of a control parameter in a parabolic partial differential 
equation}. J. Aust. Math. Soc. Ser. B, Appl. Math 33, 149-163 (1991).

\bibitem{c4} Cannon, J. R.; Lin, Y.;  Wang, S.;
\emph{Determination of source parameter in a parabolic equations}. 
Meccanica 27, 85-94 (1992).

\bibitem{f1} Fatullayev, A; Gasilov, N.; Yusubov, I.;
\emph{
Simultaneous determination of unknown coefficients in a parabolic equation}.
 Appl. Anal. 87, 1167-1177 (2008).

\bibitem{i1} Ionkin, N. I.;
\emph{Solution of one boundary value problem of
heat conduction theory with a nonclassical boundary condition}; Differ.
Uravn. 1977. V. 13, No. 2. pp. 294-304.

\bibitem{i2} Ismailov, M.; Kanca, F.;
\emph{An inverse coefficient problem for
a parabolic equation in the case of nonlocal boundary and overdetermination
conditions}. Math. Methods Appl. Sci. 34, 692-702 (2011).

\bibitem{i3} Ivanchov, M.; Pabyrivska, N.;
\emph{Simultaneous determination of two coefficients of a parabolic equation
in the case of nonlocal and integral conditions}. Ukr. Math. J. 53, 674-684 (2001).

\bibitem{k1} Kamynin, V. L.;
\emph{On convergence of the solutions of inverse
problems for parabolic equations with weakly converging coefficients},
Elliptic and Parabolic Problems (Pont- a-Mousson, 1994), Pitman Res. Notes
Math. Ser., vol. 325, 1995, pp. 130-151.

\bibitem{k2} Kamynin, V. L.;
\emph{Unique solvability of the inverse problem of
determination of the leading coefficient in a parabolic equation};
 Differ. Equ. 2011. V. 47, No. 1. pp. 91-101.

\bibitem{k3} Kamynin, V. L.; Vasin, I. A.;
\emph{Asymptotic behaviour of the solutions of inverse problems for parabolic
 equations with irregular coefficients}, Sobornik Math., 188(1997), 371--387.

\bibitem{k4} Kanca, F.; Ismailov, M.;
\emph{Inverse problem of finding the
time-dependent coefficient of heat equation from integral overdetermination
condition data}. Inverse Probl. Sci. Eng. 20, 463-476 (2012).

\bibitem{k5} Kostin, A. B.;
\emph{Inverse problem for the heat equation with
integral overdetermination}, Inverse problems for mathematical modelling of
physical processes. Collection of scientific reports, Moscow Institute of
Engineering Physics, Moscow 1991, pp. 45-49. (in Russian).


\bibitem{p1} Prilepko, A. I.; Orlovski, D. G.;
\emph{Determination of the parameter of an evolution equation and inverse 
problems of mathematical physics. II}, Differentsial'nye Uravneniya 21 (1985), 
694-700; English transl. in Differential equations 21 (1985).

\bibitem{p2} Prilepko, A. I.;  Orlovsky, D. G.;  Vasin, I. A.;
\emph{Methods for Solving Inverse Problems in Mathematical Physics}, 
Marcel Dekker Inc., 1999.

\end{thebibliography}

\end{document}