  We investigate an initial-boundary value problem for a time-fractional
subdiffusion equation with the Caputo derivatives on $N$-dimensional torus by
the classical Fourier method. Since our solution is established on the
eigenfunction expansion of elliptic operator, the method proposed in this
article can be used to an arbitrary domain and an elliptic operator with
variable coefficients. It should be noted that the conditions for the existence
of a solution to the initial-boundary value problem found in the article cannot
be weakened, and the article provides a corresponding example.
