CORRECTNESS OF A MIXED PROBLEM FOR DEGENERATE THREE-DIMENSIONAL HYPERBOLIC-PARABOLIC EQUATIONS

Abstract

In mathematical modeling of electromagnetic fields in space, the nature of electromagnetic process is determined by the properties of the medium. If the medium is non-conducting, we obtain degenerate three-dimensional hyperbolic equations. If the medium has a high conductivity, then we come to degeneratethree-dimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the mediums conductivity changes) is reduced to degenerate three-dimensional hyperbolic-parabolic equations. The mixed problem for multidimensional hyperbolic equations is well studied and has been previously considered in the works of various authors. In the articles of Professor S.A. Aldashev, the unique solvability of the mixed problem for degenerate multidimensional hyperbolic equations is proved. It is known that mixed problems for multidimensional hyperbolic-parabolic equations have not been studied much. The paper finds a new class of degenerate three-dimensional hyperbolic-parabolic equations for which the mixed problem has a unique solution and gives an explicit representation of its classical solution.