Boussinesq integro-differential equation with integral conditions and a small coefficient of mixed derivatives

In this paper, we prove the unique solvability of a nonlocal boundary-value problem for a high-order, three-dimensional, linear Boussinesq integro-differential equation with a degenerate kernel and general integral conditions and construct a solution in the form of a Fourier series. The absolute and uniform convergence of the resulting series and the possibility of term-by-term differentiation of the solution with respect to all variables are established. A criterion for the unique solvability of the boundary-value problem in the case of regular values of the parameter is obtained. For irregular values of the parameter, an infinite set of solutions is constructed in the form of a Fourier series.