Well-posedness of the dirichlet problem in a multidimensional domain for a hyperbolic-parabolic equation

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 medium. If the medium is nonconducting, then we obtain multidimensional hyperbolic equations. If the medium has a large conductivity, then we get a multidimensional parabolic equation. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to a multidimensional hyperbolic-parabolic equation.It is also known that the vibrations of elastic membranes in space by the Hamiltonian principle can be modelled by multidimensional hyperbolic equations. Studying of the process of heat propagation in a medium filled with mass leads to multidimensional parabolic equations.Consequently, by investigating mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also arrive at multidimensional hyperbolic-parabolic equations. When studying these applications, it becomes necessary to obtain an explicit representation of solutions to the investigated problems.In this paper we give a multidimensional domain where the Dirichlet problem for a hyperbolic-parabolic equation is uniquely solvable and an explicit form of its classical solution is obtained.