Geodesic
On the first boundary problem for a hyperbolic equation in an arbitrary cylinder
Matematičeskie zametki, Tome 4 (1968) no. 2, pp. 181-189 
A. F. Filippov. On the first boundary problem for a hyperbolic equation in an arbitrary cylinder. Matematičeskie zametki, Tome 4 (1968) no. 2, pp. 181-189. http://geodesic.mathdoc.fr/item/MZM_1968_4_2_a9/
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     author = {A. F. Filippov},
     title = {On the first boundary problem for a~hyperbolic equation in an arbitrary cylinder},
     journal = {Matemati\v{c}eskie zametki},
     pages = {181--189},
     year = {1968},
     volume = {4},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_4_2_a9/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A study is made of the solutions of a second-order hyperbolic equation which vanish on the boundary of an arbitrary domain in the space of the variables $x_1,\dots,x_n$ The degree of smoothness in the initial conditions, necessary and sufficient to guarantee the same degree of smoothness in the solution (considered as a function of $x_1,\dots,x_n$ for all $t$, is ascertained.