On a mixed problem for a third order degenerating hyperbolic equation

The paper investigates a mixed boundary value problem for a third-order hyperbolic equation with order degeneration inside the domain In the positive part of the domain, the equation under consideration coincides with the Hallaire equation, which is a third-order hyperbolic equation, although it is commonly called an pseudoparabolic equation. In the negative part of the domain, it coincides with the degenerate hyperbolic equation of the first kind, the special case of the Bizadze-Lyskov equation. For the problem under study, a theorem on the existence and uniqueness of a regular solution is proved. The uniqueness of the solution is proved by the Tricomi method. Regarding the desired solution, the corresponding fundamental ratios have been found. Using the method of integral equations, the existence of a solution is equivalently reduced to the solvability of the Volterra integral equation of the second kind with respect the derivative of the desired solution. According to the general theory of Volterra integral equations of the second kind, the resulting equation is uniquely solvable in the class of regular functions. The solution to the problem can be stated explicitly as a solution to the mixed problem for the Hallaire equation in the positive part of the domain and as a solution to the Cauchy problem for the degenerate hyperbolic equation of the first kind in the negative part of the domain.