Inverse Problem of Finding a Factor Depending on the Spatial Variables on the Right-Hand Side in a Parabolic-Hyperbolic Equation

Consider an inhomogeneous equation of the mixed parabolic-hyperbolic type in a rectangular parallelepiped, where the inhomogeneity is the product of two factors, one being a function depending only on the spatial variables and the other being a function depending only on time. For this equation, we study the inverse problem of finding the factor depending on the spatial variables. A criterion for the uniqueness of the solution is established. The solution is constructed as the sum of a series in an orthogonal function system. When justifying the convergence of the series, one encounters the problem of small denominators depending on two positive integer arguments. We obtain estimates guaranteeing the separation of the denominators from zero with an indication of the asymptotics. These estimates permit justifying the convergence of the series in the class of regular solutions. The stability of the solution under perturbations of the boundary functions is established.