Well-posedness of an initial-boundary value problem for a degenerate partial differential equation of high even order

In the present paper, an initial-boundary problem for a degenerate partial differential equation of high even order in a rectangle is formulated and investigated. Using the method of separation of variables, a spectral problem for an ordinary differential equation is obtained. By constructing the Green's function, the spectral problem is equivalently reduced to the second kind Fredholm integral equation with a symmetric kernel. Based on the theory of integral equations with symmetric kernel, the existence of eigenvalues and a system of eigenfunctions of the spectral problem were established. Moreover, with the help of the derived integral equation and Mercer's theorem, the uniform convergence of some bilinear series involving the obtained eigenfunctions has been proved. The solution to the problem under consideration is presented as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniqueness of the solution is proved using the energy integral method. To demonstrate the continuous dependence of the solution on the given data, appropriate estimates are derived.

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