Inverse Problems for a Three-Dimensional Equation of Parabolic-Hyperbolic Type in Finding Time-Dependent Factors of the Right-Hand Sides

For an inhomogeneous three-dimensional equation of mixed parabolic-hyperbolic type in a rectangular parallelepiped, a direct initial-boundary problem is studied. A criterion for the uniqueness of a solution is established. The solution is constructed as the sum of an orthogonal series. In substantiating the convergence of the series, the problem of small denominators of two natural arguments arose. Estimates are established for the separation from zero of the small denominators with the corresponding asymptotics. These estimates made it possible to justify the convergence of the constructed series in the class of regular solutions of this equation. Based on the solution of the direct problem, three inverse problems were posed and studied to find the time-dependent factor of the right-hand side only from the parabolic or hyperbolic part of the equation, and when the factors from both sides of the equation are unknown at the same time. The solution of inverse problems is equivalently reduced to the solvability of loaded integral equations. Based on the theory of integral equations, the corresponding uniqueness and existence theorems are proved for solutions of the inverse problems posed. Moreover, the solutions of inverse problems are constructed in an explicit form - as the sum of orthogonal series.