Gellerstedt–Moiseev Problem with Data on Parallel Characteristics in the Unbounded Domain for a Mixed Type Equation with Singular Coefficients

Abstract

In this work, in an unbounded domain, which consists of a half-plane \(y>0\) and a characteristic triangle for \(y<0\), a degenerate equation of elliptic-hyperbolic type with singular coefficients is considered for the lower terms of the equation. The correctness of the Gellerstedt–Moiseev (\(GM\)) problem is studied for data on the part of the boundary and internal characteristics parallel to it. When studying the \(GM\) problem in the half-plane \(y>0\), the integral representation of the solution of the Dirichlet problem is used. In the characteristic triangle the Darboux formula, which gives an integral representation of the solution to the modified Cauchy problem with data on the segment \([-1,1]\) of the \(y=0\) axis, is used. To prove the uniqueness of the solution to the problem, a combined method of the extremum principle (for a specially constructed finite domain \(D_{R}\)) and the passing to the limit from the finite domain \(D_{R}\) to the unbounded domain \(D\) are used. Using the Dirichlet and Darboux formulas the existence of the solution to the \(GM\) problem is equivalently reduced to the study of the system of non-standard singular integral equations, which the non-characteristic parts contain non-Fredholm operators with kernels that have isolated first-order singularities. Using the Carleman’s method, i.e., temporarily assuming the non-characteristic parts of these equations as known functions, the regularization of these equations are carried out. From the obtained two relations, one of the unknown function is explicitly expressed through the second one and this makes it possible to reduce this system to the Wiener–Hopf integral equation, which belongs to the class of singular integral equations. It has been proved that the index of this equation is equal to zero. By solving this equation a second kind Fredholm integral equation is obtained. The uniquely solvability of this equation follows from the uniqueness of the solution of the \(GM\) problem.