On one minimal hyperbolic Gellerstedt operator with internal degeneracy

The minimal operators generated by overdetermined boundary value problems for differential equations are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. The study of overdetermined boundary value problems is closely related to the theory of correct restrictions and extensions and the construction of minimal differential operators. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, including minimal operators, which is reflected in the study of problems, including for hyperbolic equations and systems arising in physics, geophysics, seismic tomography, medicine and many other practical areas. Thus, the study of minimal operators is of both theoretical and applied interest. In this paper, a criterion for the invertibility of the minimal hyperbolic Gellerstedt operator with internal degeneracy is established. The proof is based on the Gellerstedt potential, properties of solutions to the Goursat problem, and properties of special functions. It should be noted that the Gellerstedt differential operator has numerous applications in transonic gas dynamics, the theory of infinitesimal surface bends, the instantaneous theory of shells with curvature of variable sign, magnetodynamics and hydrodynamics, and the conditions of invertibility of minimal operators imposed on the right side of the initial equation are widely used in the study of the so-called source problem, which arises in a variety of applications inverse problems of mathematical physics.