On the first boundary problem for a hyperbolic equation in an arbitrary cylinder

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.