Inverse problems for a degenerate mixed parabolic-hyperbolic equation on finding time-depending factors in right hand sides

Abstract

We consider direct and inverse problems on determining time-dependent factors in the right hand sides for a mixed parabolic-hyperbolic equation with a degenerate hyperbolic part in a rectangular area. As a preliminary, we study a direct initial boundary problem for this equation. By the method of spectral analysis we establish the uniqueness criterion for the solution and the solution is constructed as a sum over the system of the eigenfunctions of the corresponding one-dimensional Sturm-Liouville spectral problem. In justifying the convergence of the series, the problem of small denominators arises. Because of this, we prove the estimates for the distance from the zero to the small denominators with a corresponding asymptotics. These estimates allow us to justify the convergence of the constructed series in the class of regular solutions of this equation. On the base of the solution to the direct problem, we formulate and study three inverse problems on finding time-dependent factors in the right hand side only by the parabolic or hyperbolic part of the equation, and also as the factors in the both sides of the equation are unknown. Using the formula of solution to the direct initial boundary problem, the solution of inverse problems is equivalently reduced to the solvability of loaded integral equations. By means of the theory of integral equations, the corresponding theorems of uniqueness and the existence of solutions of the stated inverse problems are proved. At that, the solutions of inverse problems are constructed explicitly, as sums of orthogonal series.