Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation

In this article a uniqueness theorem for the classical solution of a mixed problem is proved under minimal assumptions on the coefficients of the differential operator for admitting the Fourier method of a hyperbolic second-order equation in an $(N+1)$-dimensional cylinder, whose cross section is a completely arbitrary bounded $N$-dimensional domain. Furthermore, it is proved that the classical solution of the indicated mixed problem, whenever it exists, belongs to the class $W^1_2$ and is the generalized solution from $W^1_2$ of the same problem.