THE TRICOMI PROBLEM FOR SOME CLASSES OF MULTIDIMENSIONAL MIXED HYPERBOLIC-PARABOLIC EQUATIONS

Abstract

It is known that in the mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of themedium. If themedium is non-conducting,we obtain degenerate multidimensional hyperbolic equations. If the medium has a high conductivity, then we come to degenerate multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) is reduced to degenerate multidimensional hyperbolic - parabolic equations. It is also known that the oscillations of elastic membranes in space can be modeled according to the Hamilton principle by degenerate multidimensional hyperbolic equations. The study of the process of heat propagation in a medium filled with mass leads to degenerate multidimensional parabolic equations. Therefore, by studying the mathematical modeling of the heat propagation process in oscillating elastic membranes,we also arrive at degenerate multidimensional hyperbolic - parabolic equations. When studying these applications, it becomes necessary to obtain an explicit representation of the solutions to the problems under study. Boundary value problems for hyperbolic - parabolic equations on the plane are well studied, and their multidimensional analogues are little studied. The Tricomi problem for these equations was previously investigated. As far as we know, this problem has not been studied in space. In this paper, we show that for some classes of multidimensional mixed hyperbolicparabolic equations, the Tricomi problem is ambiguously solvable.