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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.

Multidimensional hyperbolic-elliptic equations describe important physical, astronomical, and geometric processes. It is known that the oscillations of elastic membranes in space according to Hamiltons prism can be modeled by multidimensional degenerate hyperbolic equations. Assuming that the membrane is in equilibrium in half the bend, from Hamiltons principle we also obtain degenerate elliptic equations. Consequently, vibrations of elastic membranes in space can be modeled as multidimensional degenerate hyperbolic-elliptic equations. When studying these applications, it is necessary to obtain an explicit representation of the investigated boundary value problems. The author has previously studied the Dirichlet problem for multidimensional hyperbolic-elliptic equations, where the unique solvability of this problem is shown, which essentially depends on the height of the cylindrical domain under consideration. However, the Dirichlet problem in a cylindrical domain for multidimensional degenerate hyperbolic-elliptic equations has not been studied previously. In this paper, the Dirichlet problem is studied for a class of degenerate multidimensional hyperbolic-elliptic equations. Moreover, the existence and uniqueness of the solution depends on the height of the considered cylindrical domain and on the degeneration of the equation. A uniqueness criterion for a regular solution is also obtained.

In the cylindrical domain of Euclidean space for one class of multidimensional hyperbolic parabolic equations the spectral Dirichlet problem with homogeneous boundary conditions is considered. The solution is sought in the form of an expansion in multidimensional spherical functions. The existence and uniqueness theorems of the solution are proved. Conditions for the unique solvability of the problem are obtained, which essentially depend on the height of the cylinder.

The paper shows the unique solvability of the classical Dirichlet problem in cylindrical domain for threedimensional elliptic equations with degeneration of type and order.

It is shown that the Dirichlet problem for a degenerating many-dimensional equation of mixed type in a cylindrical domain is uniquely solvable. A criterion for the uniqueness of a regular solution is also established.

We establish the unique solvability of a local boundary value problem in a cylindrical domain for multi-dimensional hyperbolic equations with the wave operator. This problem is a generalization of the Dirichlet and Poincare problems. We prove a criterion for the uniqueness of a regular solution. Bibliography: 17 titles.

We prove that the Dirichlet and Poincar´e problems for degenerate multidimensional hyperbolic equations with Gellerstedt operator in a cylindrical domain are uniquely solvable and establish a criterion of uniqueness for the regular solutions of these problems.

Some examples that show that the homogeneous Tricomi problem for a multidimensional hyperbolic-parabolic equation has the infinite set of nontrivial solutions are constructed.

We determine a many-dimensional domain in which the Dirichlet and Poincare problems for the wave equation are uniquely solvable.

It is shown that the Dirichlet problem for one class of multidimensional hyperbolic-elliptic equations in a cylindrical domain is uniquely solvable. We also find a criterion for the uniqueness of a regular solution.

We consider the Dirichlet spectral problem with the homogeneous boundary conditions in a cylindrical domain of Euclidean space for multidimensional hyperbolic equation with wave operator. We construct the solution as an expansion in multidimensional spherical functions; prove the existence and uniqueness theorems. The obtained conditions of the problem unique solvability essentially depend on the “height” of the cylinder.

For elliptic equations, boundary-value problems on the plane were shown to be well posed in [1] by using methods from the theory of analytic functions of a complex variable. When the number of independent variables is greater than two, difficulties of a fundamental nature arise. The highly attractive and convenient method of singular integral equations can hardly be applied, because the theory of multidimensional singular integral equations is still incomplete [2]. In the present paper, using a method from [3], we establish the unique solvability of the classical solution of the Dirichlet problem in a cylindrical domain for degenerating multidimensional elliptic equations.

It is shown that the Dirichlet problem in a cylindrical domain for a class of multidimensional hyperbolic-parabolic equations is uniquely solvable. The criterion of uniqueness of a regular solution is obtained.

It is shown that the Dirichlet problem in a multidimensional domain for the Lavrent’ev–Bitsadze equation is uniquely solvable. A criterion of the uniqueness of the solution is obtained.

We determine eigenvalues and eigenfunctions of the Gellerstedt problem for the multidimensional Lavrent’ev–Bitsadze equation.

We obtain a criterion for the unique solvability of the spectral problem in a cylindrical domain for a multidimensional Lavrent’ev-Bitsadze equation.

It is known that waves (acoustic waves, radio waves, elastic waves, and electric waves) in cylindrical tubes are described by the wave equation. In the theory of hyperbolic-type partial differential equations, boundary-value problems with data on the whole boundary serve as examples of ill-posedness of the posed problems. In this work, it is shown that the Poincar´e problem in a cylindrical domain for the higher-dimensional wave equation is uniquely solvable. A uniqueness criterion for a regular solution is also obtained.

We show that there exists a countable set of eigenfunctions of the Tricomi spectral problem for multidimensional mixed hyperbolic–parabolic equations.

It is shown that the solution of the Gellerstedt space problem is not unique for one class of multidimensional hyperbolic-elliptic equations.

Inverse problems for hyperbolic equations are found in geophysical prospecting and seismology, and their multidimensional analogues are especially important for applied work. However, whereas results have been established for the some narrow classes of hyperbolic equations, no results exist for more general classes. This paper proves the solvability of the inverse problem for a general class of multidimensional hyperbolic equations. Our approach consists of properly choosing the shape of the overidentifying condition that is needed to determine the right-hand side of the hyperbolic PDE and then applying the Fourier series method. We are then able to establish the results of the existence of solution for the cases when the unknown right-hand side is time- independent or space independent.

We obtain a criterion for the uniqueness of a regular solution of the Darboux-Protter problem for multidimensional hyperbolic equations with Chaplygin operator. We also prove a theorem on the uniqueness of solutions of the dual problem.

We obtain a criterion for the uniqueness of a regular solution of the Darboux–Protter problem for degenerate multidimensional hyperbolic equations.

We prove the well-posedness of the Cauchy, Goursat, and Darboux problems for multidimensional in-tegro-differential equations of the hyperbolic type encountered in biology.

For the generalized many-dimensional Lavrent’ev-Bitsadze equation, we prove the unique solvability of the Dirichlet and Tricomi problems. We also establish the existence and uniqueness of a solution of the Dirichlet problem in the hyperbolic part of a mixed domain.

We prove the unique solvability of the Dirichlet problems for the many-dimensional wave equation and Lavrent’ev-Bitsadze equation.

It is proved that multi-dimensional Darboux problems for the wave equation are correct in the domain\(D_3 \subset E_{m + 1} \) bounded by the surfaces ¦x ¦=t + ɛ and ¦x ¦=1- t and the planet = 0, 0 ≤ ɛ < 1. The behavior of the solutions as ɛ→0 is studied.

In the theory of hyperbolic PDEs, the boundary-value problems with conditions on the entire boundary of the domain serve typically as the examples of the ill-posedness. The paper shows the unique solvability of the Dirichlet problem in the cylindric domain for the multidimensional wave equation. We also establish the criterion for the unique solvability of the equation.

For the linear hyperbolic equations $$\sum\limits_{i,j = 1}^{m + 1} {a_{ij} \left( {x,x_{m + 1} } \right)u_{x_i x_j } + \sum\limits_{i = 1}^{m + 1} {a_i \left( {x,x_{m + 1} } \right)u_{x_i } + c\left( {x,x_{m + 1} } \right)u = 0,x = \left( {x_1 ,...,x_m } \right)} ,} m \geqslant 2,$$ the correctness of multidimensional analogues of the problems of Darboux and Goursat is established and a theorem on the uniqueness of a solution of the Cauchy characteristic problem is proved.