The problem with the missing Goursat condition at the boundary of the domain for a degenerate hyperbolic equation with a singular coefficient

Abstract

The work is devoted to the formulation and study of the solvability for a problem with missing conditions on the characteristic boundary of the domain and an analogue of the Frankl condition on the segment of the degeneracy for a hyperbolic equation. The difference between this problem and known local and nonlocal problems is that, firstly, a hyperbolic equation is taken with arbitrary positive power degeneracy and singular coefficients on the part of the boundary, and secondly, the characteristic boundary of the domain is arbitrarily divided into two pieces and the value of the desired function is set on the first piece, and the second piece is freed from the boundary condition and this missing Goursat condition is replaced by an analogue of the Frankl condition on the degeneracy segment, and the value of an unknown function on another characteristic boundary of the domain is also considered to be known. The conditions for the coefficients of the equation and the data of the formulated problem, ensuring the validity of the uniqueness theorem are found. The theorem of the existence of a solution to the problem is proved by reducing to the problem of solving a non-standard singular integral equation with a non-Fredholm integral operator in the non-characteristic part of the equation, the kernel of which has an isolated first-order singularity. Applying the Carleman regularization method to the received equation, the Wiener-Hopf integral equation is obtained. It is proved that the index of the Wiener-Hopf equation is zero, therefore it is uniquely reduced to the Fredholm integral equation of the second kind, the solvability of which follows from the uniqueness of the problem’s solution.