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{\em Electronic Journal of Differential Equations},
Vol. 2002(2002), No. 85, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\thanks{\copyright 2002 Southwest Texas State University.}
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\begin{document}
\title[\hfilneg EJDE--2002/85\hfil A nonlocal problem for hyperbolic equations]
{A nonlocal problem for fourth order hyperbolic equations with
multiple characteristics}
\author[Bidzina Midodashvili\hfil EJDE--2002/85\hfilneg]
{Bidzina Midodashvili }
\address{Bidzina Midodashvili \newline
Department of Theoretical Mechanics,\newline
Georgian Technical University,\newline
Tbilisi, Georgia}
\email{bidmid@hotmail.com}
\date{}
\thanks{Submitted July 16, 2002. Published October 4, 2002.}
\subjclass[2000]{35L35}
\keywords{Goursat problem, Riemann function}
\begin{abstract}
In this paper, we study fourth order differential equations
with multiple characteristics and dominated low terms.
We prove the existence and uniqueness of a Riemann function for this
equation, and then provide an integral representation of
the general solution of the Goursat problem. We also provide
sufficient conditions for the solvability of a nonlocal problem.
\end{abstract}
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\section{Introduction}
Partial differential equations of higher order with dominated
low terms are encountered when studying mathematical
models for certain natural and physical processes. As an example
of such type of equations, is the equation of moisture transfer [2]
$$
\frac {\partial w}{\partial t} = \frac {\partial}{\partial x}( D
\frac {\partial w}{\partial x} + A \frac {\partial^2 w}{\partial x
\partial t}),
$$
where $w$ is the concentration of moisture per unit, $D$ is the
coefficient of diffusivity, and $A>0$ is the varying coefficient
of Hallaire. Under the proper schematization of the process of
absorbing the soil moisture by the roots of plants, the pressure
$u(x,t)$ in the area of root absorption satisfies the equation of
form [4]
$$
(\frac {\partial}{\partial x} + \frac {1}{x})(u_{xt} + \lambda
u_{x}) = \mu u_{t}.
$$
Obviously, the equation
$$
\frac {\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial
x^2} - \frac{\partial^4 u}{\partial x^2 \partial t^2} = 0,
$$
which describes the longitudinal waves in a thin elastic stem
taking into account the effects of transversal inertia, is of the
same type [5].
In the present work, a class equations with fourth order partial
derivatives and dominated lower order terms is considered.
In the space $\mathbb{R}^3$ of the independent variables $x_1$, $x_2$
and $x_3$ let
$$
\Pi := \{(x_1,x_2,x_3)\in R^{3} : a_{i}