On a Boundary Value Problem for a Loaded Parabolic-Hyperbolic Equation of the Second Kind Degenerating Inside the Domain

Abstract

This paper studies an analog of the Tricomi problem for a loaded parabolic-hyperbolic equation of the second kind, which degenerates inside the domain. In proving the existence and uniqueness theorem for a classical solution to the Tricomi-type problem, a general representation of the solution to the loaded parabolic-hyperbolic equation degenerating within the domain is derived. The uniqueness of the solution is established using the extremum principle and the energy integral method. The existence of the solution is equivalently reduced to integral equations of the second kind, specifically Volterra and Fredholm equations, which remain relatively unexplored. Furthermore, a class of prescribed functions is determined to ensure the solvability of the obtained integral equations.