Начально-краевая задача с граничным условием третьего рода для дифференциального уравнения высокого четного порядка вырождающегося на границе

In this work, we formulate and investigate an initial-boundary value problem with third-type
boundary conditions for a degenerate partial differential equation of high even order in a rectangular
domain. Using the Fourier method based on the separation of variables, we derive a spectral
problem for an ordinary differential equation. By applying the Green’s function method, this problem
is equivalently reduced to a Fredholm integral equation of the second kind with a symmetric
kernel, from which the existence of eigenvalues and a system of eigenfunctions for the spectral
problem follows. Using the obtained integral equation and Mercer’s theorem, we prove the uniform
convergence of certain bilinear series depending on the found eigenfunctions. The order of the Fourier
coefficients is established. The solution to the studied problem is expressed as a Fourier series in terms
of the eigenfunctions of the spectral problem. The uniqueness of the solution is proved using the
energy integrals method. An estimate for the solution is obtained, implying its continuous dependence
on the given functions.