Geodesic
Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation
Matematičeskie zametki, Tome 17 (1975) no. 1, pp. 91-101.

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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. 
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     author = {V. A. Il'in},
     title = {Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {91--101},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_1_a10/}
}
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V. A. Il'in. Proof of uniqueness and membership in $W^1_2$ of the classical solution of a mixed problem for a self-adjoint hyperbolic equation. Matematičeskie zametki, Tome 17 (1975) no. 1, pp. 91-101. http://geodesic.mathdoc.fr/item/MZM_1975_17_1_a10/