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\vspace*{0.3cm}
\begin{center}
NONCLASSICAL PROBLEMS FOR THE SECOND
ORDER HYPERBOLIC EQUATIONS

\vspace*{0.2cm}
{\it G. Avalishvili, D. Gordeziani} 

\vspace*{0.2cm}
{\it I. Vekua Institute of Applied Mathematics} \\
{\it I. Javakhishvili Tbilisi State University}
\end{center}

\vspace*{0.1cm}\par
Nonclassical initial and boundary conditions often arise in mathematical
modelling of various phenomena of physics, biology, ecology and other fields.
Conditions of such type contain certain relations either between the values
of an unknown function on the boundary of the domain and its internal
values or between the values
of an unknown function at the initial and some later moments of time.
Consequently the nonclassical problems of the first type are called
spatial nonlocal problems and the second ones $-$ nonlocal problems
in time.

First, a certain class of spatial nonlocal problems was
investigated by A.V. Bitsadze and A.A. Samarskii [1]. Further, in [2,3]
the problems of such a type were called Bitsadze-Samarskii
nonlocal problems and iteration methods of
resolution for the such type problems in the case of rather
general elliptic and parabolic equations were suggested. Later, in [4-17] interesting
generalizations of Bitsadze-Samarskii conditions were
suggested. It should be pointed out, that the majority
of workes devoted to the nonlocal problems studies elliptic
and parabolic cases.

Let $\Omega\subset{\Bbb R}^n$ be a bounded domain with a piecewise smooth
boundary $\Gamma=\partial\Omega$. $L^2(\Omega)$ denotes the space of real 
functions square integrable in $\Omega$ in the Lebesgue sense.
$L^{\infty}(\Omega)$ is a space of measurable and almost everywhere
bounded functions. $W^{2,k}(\Omega)=H^k(\Omega)$ denotes the Sobolev 
space of order $k$, i.e. the subspace
of $L^2(\Omega)$ of functions whose derivatives up to the order $k$
in the sense of distributions are in $L^2(\Omega)$.
$W^{2,k}_0(\Omega)=H^k_0(\Omega)$ is the closure of the set $D(\Omega)$ of 
real infinitely differentiable functions with compact support in $\Omega$ 
in the space $H^k(\Omega)$. $C^0(0,T;X)$ is a space of
continuous functions of $t$ with values in the Banach space $X$.

Suppose, that $a_{ij}\;(i,j=\overline{1,n})$ are prescribed real functions 
on $\Omega$, which satisfy the following conditions
\begin{equation}%1
\begin{array}{c}
a,\;\;a_{ij}\in L^{\infty}(\Omega),\\
a(x)\ge 0,\;\;a_{ij}(x)=a_{ji}(x),\\
\sum\limits_{i,j=1}^{n}a_{ij}(x)\xi_i\xi_j\ge\alpha(\xi^2_1+...+\xi^2_n),\;
\alpha>0,\;\forall \xi=(\xi_1,...,\xi_n)\in{\Bbb R}^n, 
\end{array}
\end{equation}
almost everywhere on $\Omega$. Let $A:H^1_0(\Omega)\to L^2(\Omega)$ be the
linear operator
$$
Au=-\sum\limits_{i,j=1}^{n}\frac{\partial}{\partial x_i}\left(a_{ij}(x)
\frac{\partial u}{\partial x_j}\right)+a(x)u.
$$ 
From conditions (1) it immediately follows, that the bilinear
form
$$
a(\varphi,\psi)=(A\varphi,\psi)=\sum_{i,j=1}^{n}\int\limits_{\Omega}
a_{ij}(x)\frac{\partial \psi}{\partial x_i} 
\frac{\partial \varphi}{\partial x_j}dx+\int\limits_{\Omega}a(x)\varphi(x)
\psi(x)dx
$$
is symmetric and coercive
$$
a(\varphi,\psi)=a(\psi,\varphi)\;\;\;\;\;\forall\varphi,\psi\in H^1_0(\Omega),
$$
$$
a(\varphi,\varphi)\ge\alpha\|\varphi\|^2_{H^1_0(\Omega)},\;\;\;\;\;\alpha >0,
$$
and eigen functions of the operator $A$ are dense in $L^2(\Omega)$.

Let us consider the nonlocal problem in time for the second order
hyperbolic equation
\begin{equation}%2
\displaystyle
\frac{\partial^2 u}{\partial t^2}+Au=f(x,t),\;\;\;\;(x,t)\in\Omega\times(0,T),
\end{equation}
\begin{equation}%3
\begin{array}{c}
\displaystyle
u(x,0)=\sum\limits_{i=1}^{k}\alpha_i u(x,T_i)+\sum\limits_{T_i<T_j}
\int\limits_{T_i}^{T_j}\rho_{ij}(\tau)u(x,\tau)d\tau+u_0(x),\\
\displaystyle
u_t(x,0)=\sum\limits_{i=1}^{k}\beta_i u_t(x,T_i)+\sum\limits_{T_i<T_j}
\int\limits_{T_i}^{T_j}\omega_{ij}(\tau)u_t(x,\tau)d\tau+u_1(x),\\
\end{array}
\;\;x\in \Omega,
\end{equation}%4
\begin{equation}
\displaystyle
u(x,t)=0,\;\;\;\;\;\;\;\;(x,t)\in\partial\Omega\times[0,T],
\end{equation}
where $\rho_{ij},\;\omega_{ij}$ are measurable, bounded functions,
$\alpha_i,\;\beta_i$ are real constant
numbers, $T_i\in(0,T],$ $u_0\in H^1_0(\Omega),$ $u_1\in L^2(\Omega)$. 
We have to find the function $u\in C^0(0,T;H^1_0(\Omega))$ with the 
generalized derivative $u^{\prime}\in C^0(0,T;L^2(\Omega))$, which 
satisfies the following equation
\begin{equation}%5
\frac{d}{dt}(u^{\prime}(\cdot),v)+a(u(\cdot),v)=(f(\cdot),v),\;\;\;\forall v\in 
H_0^1(\Omega),
\end{equation}
in the sense of distributions on $(0,T)$ and conditions (3) in the
sense of spaces $H^1_0(\Omega)$ and $L^2(\Omega)$ respectively.

Under these conditions the following theorem is true.

{\bf Theorem.} {\it If there exists a positive real number $q>0$, such that
$$
\begin{array}{l}
\displaystyle\left| \left( 1-\sum\limits_{i=1}^k \alpha_i cos(\lambda_n T_i)-
\sum\limits_{T_i<T_j}\int\limits_{T_i}^{T_j} \rho_{ij}(\tau)cos(\lambda_n \tau)
d\tau \right)
\displaystyle\left( 1-\sum_{i=1}^{k} \beta_i cos(\lambda_n T_i)-\right.
\right.\\
\displaystyle
\left.-\sum\limits_{T_i<T_j}\int\limits_{T_i}^{T_j} \omega_{ij}(\tau)cos(\lambda_n \tau)
d\tau \right)+
\displaystyle\left(\sum_{i=1}^k \alpha_i sin(\lambda_n T_i)+ 
\sum\limits_{T_i<T_j}\int\limits_{T_i}^{T_j} \rho_{ij}(\tau)sin(\lambda_n \tau)
d\tau \right)
\end{array}
$$
\begin{equation}%6
\displaystyle\left.\left(\sum_{i=1}^k \beta_i sin(\lambda_n T_i)+ 
\sum\limits_{T_i<T_j}\int\limits_{T_i}^{T_j} \omega_{ij}(\tau)sin(\lambda_n \tau)
d\tau \right)\right|>q,\;\;\;\;n=1,2,...
\end{equation}
where $\{\lambda_n\}_{n=1}^{\infty}$ are eigen values of the operator $A$,
$u_0\in H^1_0(\Omega),$ $u_1\in L^2(\Omega)$, $f\in L^2(0,T;L^2(\Omega)),$
then nonlocal problem {\rm(2)-(4)} has a unique solution.}

Let's formulate one useful sufficient condition for the fulfilment of
inequality (6).

{\bf Corollary.} If in the nonlocal conditions (4) integral addends are 
omitted and coefficients $\alpha_i,\;\beta_i\;(i=\overline{1,m})$ satisfy 
the following inequality
$$
\displaystyle\sum_{i=1}^k(|\alpha_i|+|\beta_i|)<1,
$$
then the nonlocal problem has a unique solution.

Now we consider some particular cases of nonlocal problem (2)-(4) and 
show an essential difference between classical and above mentioned problems.

Let $\displaystyle\Omega=(0,l),\;A\equiv-\frac{\partial^2}{\partial x^2},\;f\equiv 0$
and conditions (3) take the form
\begin{equation}%7
\begin{array}{c}
\displaystyle
u(x,0)=\sum\limits_{i=1}^{k}\alpha_i u(x,T_i)+u_0(x),\\
\displaystyle
u_t(x,0)=\sum\limits_{i=1}^{k}\alpha_i u_t(x,T_i)+u_1(x),
\end{array}
x\in \Omega.
\end{equation}
Hence, we get nonlocal problem in time for a string  oscillation 
equation, which has a unique solution if $\displaystyle \sum_{i=1}^k
|\alpha_i|<1$. In the degenerate case $\displaystyle \sum_{i=1}^k
|\alpha_i|=1$ if among 
the points $ \{T_i\}_{i=1}^{k} $ at least one is such, that quotient 
$ T_i/l$ is irrational, then the
nonlocal problem does not have more than one solution. It must be
pointed out, that for $k=1,\;|\alpha_1|=1$ the condition of irrationality of
$T_1/l$ is a
necessary and sufficient condition for the uniqueness of the solution
of the nonlocal problem.

Similarly we may consider two-dimensional and multidimensional
problem (2), (7) with homogeneous boundary conditions, when 
$\displaystyle\Omega=(0,l_1)\times(0,l_2)\times...\times(0,l_s)$, 
$\displaystyle A\equiv-\sum_{i=1}^{s}\frac{\partial^2}{\partial x_i^2}$, 
$f\equiv 0$. Taking into
account inequality (6), in the case 
$\displaystyle \sum_{i=1}^k |\alpha_i|=1$ we obtain, that if for some 
$T_i$ equation
$$
\displaystyle
\left( \frac{n_1}{l_1} \right)^2+
\left( \frac{n_2}{l_2} \right)^2+...+
\left( \frac{n_s}{l_s} \right)^2=\left( \frac{p}{T_i} \right)^2
$$
is unsolvable in integer numbers, i.e. for any integer 
$n_1,$ $n_2,...,n_s,p\in {\Bbb Z}$ quotients $n_1/l_1,$ $n_2/l_2,...,n_s/l_s,$ $p/T_i$ are not the 
so-called generalized Pythagorean numbers, then the nonlocal problem does
not have more than one solution.

So, for the nonlocal problems in time the uniqueness and existence
of the solution depend on algebraic properties of the quotients 
$T_1/l,$ $T_2/l,...,T_s/l$ (in the one-dimensional case), that doesn't take place
for classical problems.

\vspace*{0.2cm}
\footnotesize
\begin{center}
{\bf R e f e r e n c e s}

\vspace*{0.1cm}
\end{center}
\par
[1] Bitsadze A.V., Samarskii  A.A., On Some Simplest
Generalizations of Linear Elliptic Problems. Dokl. Akad. Nauk SSSR,
185, 4, 1969, 739-740, (Russian).

[2] Gordeziani D.G., On One Method of Resolution of
Bitsadze-Samarskii Boundary Value Problem. Abstracts of 
reports of Inst. Appl. Math. Tbilis. State Univ., 2, 1970, 38-40, (Russian).  

[3] Gordeziani D.G., Djioev T.Z., On Solvability of
One Boundary Value Problem for Non-Linear Elliptic Equation. 
Bull. Acad. Sci. Georgian SSR, 68, 2, 1972,  289-292, (Russian).

[4] Fridman A., Monotonic Decay of Solutions of
Parabolic Equations with Nonlocal Boundary Conditions. Quart. Appl. Math., 
44, 1986, 401-407.

[5] Gordeziani D.G., On Methods of Resolution for One
Class of Nonlocal Boundary Value Problems. Tbilisi University Press,
Tbilisi, 1981, (Russian).

[6] Gordeziani D., Gordeziani N., Avalishvili G., 
On One Class of Nonlocal Problems for Partial Differential Equations.
Reports of Enlarged Sessions of I.Vekua Institute of Applied 
Mathematics, 10, 3, 1995, 20-22.

[7] Gordeziani D., Gordeziani N., Avalishvili G.,
On the Investigation and Resolution of Nonlocal Boundary and Initial-Boundary
Value Problems.
Reports of Enlarged Sessions of I.Vekua Institute of Applied Mathemtics, 
12,  3, 1997.

[8] Gordeziani D., Gordeziani N., Avalishvili G.,
Nonlocal Boundary Value Problems in Mathematical Physics. Abstracts of
Intern. Symposium Dedicatied to the $90^{th}$ Birthday Anniversary of
Academician I. Vekua DEMPh, 1997, 127-128.

[9] Gordeziani D., Gordeziani N., Avalishvili G.,
Nonlocal Boundary Value Problems for Some Partial Differential Equations.
Bull. Georgian Acad. Sci., 157, 3, 1998, 365-368.

[10] Gordeziani D., Avalishvili G., Investigation of the Nonlocal 
Initial-Boundary Value Problems for the String Oscillation and Telegraph 
Equations. AMI, 2, 1997, 65-79.

[11] Gordeziani D., Grigalashvili Z., Nonlocal Problems in Time for Some
Equations of Mathematical Physics. 
Reports of Seminars of I.Vekua Institute of Applied Mathematics,
22, 1993, 108-114.
 
[12] Gushchin A.K., Mikha{\^i}lov V.P.,
Continuity of Solution of a Class of Nonlocal Problems for an Elliptic
Equation. Math. Sb., 2, 1995, 37-58.

[13] Il'in V.A., Moiseev E.I., Two Dimensional
Nonlocal Boundary Value Problems for Poisson's Operator in Differential and
Difference Variants.  Mat. Mod., 2, 1990, 139-159, (Russian).

[14] Kapanadze D.V., On the Bitsadze-Samarskii Nonlocal
Boundary Value Problem. Diff. Equat., 23, 1987, 543-545, (Russian).

[15] Mansourati Z.G., Campbell L.L., Non-Classical
Diffusion Equations Related to Birth-Death Processes With Two Boundaries.
Quart. Appl. Math., 54, 1996, 423-443.

[16] Paa C.V., Reaction Diffusion Equations With
Nonlocal Boundary and Nonlocal Initial Conditions. J. Math. Anal. Appl., 
195, 1995, 702-718.

[17] Skubachevskii A.L., Nonlocal 
Elliptic Problems and Multidimensional Diffusion Processes.
J. Math. Phys., 3, 1995, 327-360, (Russian).
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