Uniqueness of the solution of the inverse problem for one model equation of mixed parabolic-hyperbolic type in the two-dimensional case

The study of inverse and direct problems for mixed-type equations is not as well developed as for classical types of equations. The problem considered in the article is relevant to practice. Mathematical models of various processes in the field of natural sciences can be reduced to equations combining parabolic and hyperbolic characteristics. This paper considers the inverse problem for such a model equation in two dimensions in a parallelepiped. We apply a spectral technique based on representing the unknown functions and initial conditions as Fourier series, which, when substituted into the original formulation, leads to a system of ordinary differential equations. By solving these equations, we obtain a condition for the solution’s uniqueness to the inverse problem, and prove the uniqueness of the original problem. The results are presented in the form of a theorem.