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We consider the modeling multidimensional hyperbolic-parabolic equation in the cylindrical area of Euclidean space and formulate the mixed problem with non-homogeneous boundary conditions for it. We show the unique solvability of the problem for the class of continuously differentiable functions and give a way to construct its explicit classical solution.

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.

In early works the author studied the Dirichlet and Poincaré problems for multidimensional hyperbolic equations, which shows the well-posedness of these problems in cylindrical domains, significantly dependent on the height of the considered cylindrical domain. Here a multidimensional region inside a characteristic cone is considered, in which the Dirichlet and Poincaré problems have unique solutions for one class of hyperbolic equations.

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.

This paper proves the unique solvability of the local boundary value problem in a cylindric domain for the multi-dimensional wave equation, which is the generalization of the Dirichlet and Poincare problems. We also obtain the criterion for the uniqueness of the regular solution.

This paper proves the unique solvability of the local boundary value problem in a cylindric domain for the multi-dimensional wave equation, which is the generalization of the Dirichlet and Poincare problems. We also obtain the criterion for the uniqueness of the regular solution.