Boundary value problem for a third-order degenerate equation

The article deals with a boundary value problem in a rectangular area for a third-order degenerate equation with minor terms.
The study of such equations is caused by both a theoretical and applied interest (known as VT (viscous transonic) — the equation can be found in gas dynamics).
Imposing some restrictions on the coefficients of lower derivatives and using the method of energy integrals, the unique solvability of the problem is demonstrated. The solution of the problem is sought by separation of variables (Fourier method), thus two one-dimensional boundary value problems for ordinary differential equations are obtained.
According to the variable y we have the problem on eigenvalues and eigenfunctions for a second-order degenerate equation. The eigenvalues and eigenfunctions are found. Eigenfunctions are the first-order Bessel functions. In order to obtain some necessary estimates the spectral problem reduces to an integral equation by constructing the Green's function. Hereafter, Bessel inequality is used. The possibility of expansion of boundary functions in the system of eigenfunctions is also shown.
In order to obtain the necessary a priori estimates for the solution of one-dimensional boundary value problem with respect to the variable x and its derivatives, the problem reduces to a second-order Fredholm integral equation, with the help of Green's function. The estimates of Green's function and its derivatives are obtained. Fredholm equation is solved by the method of successive approximations, and the necessary estimates for this solution and its derivatives are obtained.
The formal solution of the boundary value problem is obtained in the form of an infinite series in eigenfunctions. In order to prove the uniform convergence of the last series composed of the partial derivatives, first using the Cauchy–Bunyakovsky inequality, the series consisting of two variables is decomposed into two one-dimensional series, and then all of the obtained estimates mentioned above and estimates for the Fourier coefficients are used.