On analogue of the Trikomi problem for a third-order equation of mixed type

It is commonly known that equations of mixed type are partial differential equations that belong to different types in different parts of the domain under consideration. For example, the equation may belong to the elliptic type in one part of the domain and to the hyperbolic type in another one; these parts are separated by a transition line, at which the equation degenerates into parabolic or undefined. In 1923, the Italian mathematician F. Tricomi considered a boundary value problem for one equation of mixed elliptic-hyperbolic type (later named after him) in a domain bounded in the upper half-plane by the Lyapunov curve, and in the lower half-plane by the characteristics of the equation that emerge from the ends of this curve; the boundary conditions were then set on the curve and on one of the characteristics. The solution had to be continuous in the closure of the domain, continuously differentiable within it, and twice continuously differentiable in the upper (elliptic) and lower (hyperbolic) subdomains; for the first derivatives of the solution, singularities of integrable order were allowed near the ends of the curve. Tricomi proved the existence and uniqueness of the solution to the problem in the specified class; when proving the existence, he reduced the problem to a singular integral equation. The article studies an analogue of the Tricomi problem for a third-order hyperbolic-parabolic equation of mixed type with a spectral parameter. The uniqueness and existence of a solution to the problem are proved. The uniqueness of the solution to the problem is proved by the method of energy integrals, and the existence of the solution is proved by the method of reduction to the Fredholm integral equation of the second kind, the solvability of which follows from the uniqueness of the solution to the problem.