\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\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.} \vspace{9mm}} \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} \maketitle \numberwithin{equation}{section} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}[thm]{Proposition} \newtheorem{lemm}[thm]{Lemma} \newtheorem{rmrk}[thm]{Remark} \allowdisplaybreaks \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}