<p>In the present paper, initial boundary value and inverse source problems with non-local boundary conditions are formulated and studied for a time- and space-degenerate second-order partial differential equation in a rectangular domain. First, the uniqueness of the solution to the direct problem is proved using the method of energy integrals. Then, by employing the method of separation of variables, a spectral problem is derived for an ordinary differential equation in the spatial variable. The existence of eigenvalues and eigenfunctions for this spectral problem is established by equivalently reducing it to a homogeneous Fredholm integral equation of the second kind with a symmetric kernel. Using the theory of integral equations, the existence of the eigenvalues and eigenfunctions is further confirmed. Next, the solution to the direct problem is constructed. Subsequently, the existence and uniqueness of the solution to the inverse source problem are proved. The solutions to the direct and inverse source problems are expressed as Fourier series expansions over the system of eigenfunctions of the spectral problem. The uniform convergence of these series is also proven.</p>