In this paper, we study a problem with initial and boundary conditions for one class of high-order partial differential equations in several variables. The solution to the initial boundary value problem is constructed as the sum of a series in the system of eigenfunctions of the multidimensional spectral problem. The eigenvalues of the spectral problem are found and the corresponding system of eigenfunctions is constructed. It is shown that this system of eigenfunctions is complete and forms a Riesz basis in the Sobolev space. Based on the completeness of the system of eigenfunctions, a uniqueness theorem for the solution of the problem is proved. In the Sobolev classes, the existence of a regular solution to the stated initial-boundary value problem is proved.