On the solvability of a mixed problem for a fractional partial differential equation with delayed time argument and Laplace operators with nonlocal boundary conditions in Sobolev classes

In this paper, we study a problem with initial functions and boundary conditions for partial differential equations of fractional order with Laplace operators. The boundary conditions of the problem are nonlocal, and the solution is supposed to belong to one of Sobolev classes. The solution of the initial boundary value problem is constructed as the sum of a series of multidimensional spectral problem's eigenfunctions. The eigenvalues of the spectral problem are found and the corresponding system of eigenfunctions is constructed. It is shown that this system is complete and forms a Riesz basis in the subspaces of Sobolev spaces. Basing on the completeness of the eigenfunctions' system, the uniqueness theorem for the solution of the problem is proved. The existence of a regular solution of the initial boundary value problem is proved in Sobolev subspaces.