Ill-posedness of the Tricomi problem for a multidimensional mixed hyperbolic-parabolic equation

Abstract

It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the environment. If the medium
is non-conducting, then we obtain multidimensional hyperbolic equations. If the medium has high conductivity, then we come to multidimensional parabolic equations.
Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) is reduced to multidimensional hyperbolic-parabolic equations.
It is also known that vibrations of elastic membranes in space according to Hamilton’s principle can be modeled by multidimensional hyperbolic equations.
The study of the process of heat propagation in a medium filled with mass leads to multidimensional parabolic equations.
Consequently, studying the mathematical modeling of the heat propagation process in vibrating elastic membranes, we also come to multidimensional hyperbolic-parabolic equations.
When studying these applications, it becomes necessary to obtain an explicit representation of
the solutions of the problems under study.
Boundary value problems for hyperbolic-parabolic equations in the plane are well-explored, but their multidimensional analogues have been studied very little. The Tricomi problem for the above equations has been previously investigated on a plane, but far as is known, this problem in space has not been analyzed. In this paper, we show that for the multidimensional model mixed hyperbolic-parabolic equation, the Tricomi problem is non-uniquely solvable. An explicit form of this solution is provided.