IXED PROBLEM FOR THREE-DIMENSIONAL HYPERBOLIC EQUATIONS WITH DEGENERATION OF TYPE AND ORDER

It is known that in space during mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conductive, then we get degenerating multidimensional hyperbolic equations. Therefore, the analysis of electromagnetic fields in complex environments (for example, if the conductivity of the medium changes) is reduced to degenerating multidimensional hyperbolic equations. It is also known that oscillations of elastic membranes in space according to the Hamilton principle can be modeled by degenerating multidimensional hyperbolic equations. Therefore, by studying mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also come to degenerating multidimensional hyperbolic equations. When studying these applications, it becomes necessary to obtain a clear representation of the solutions to the investigated problems. The mixed problem for degenerating multidimensional hyperbolic equations in generalized spaces is well researched. This task is also studied in the works of S. A. Aldashev, where it is shown that its correctness significantly depends on the height of the cylindrical region under consideration. A.V. Bitsadze drew attention to the importance of studies of multidimensional hyperbolic equations with degeneration of type and order. Mixed problems for these equations have not previously been studied. In this work, the solvability of a mixed problem is shown and a clear form of a classical solution for three-dimensional hyperbolic equations with degeneration of type and order is obtained.