Abstract
In this paper we investigate some boundary value problems for the wave equation, which are the three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems) on the plane. It is well known that for an infinite number of smooth functions in the right-hand side these problems do not have a classical solution. We define an appropriate generalized solution, for which the existence and uniqueness theorems are proved. But some of these " solutions" have a strong power singularity at one point of the boundary. It is interesting that this singularity is only at the vertex of the characteristic cone and does not propagate along the cone. Some weight a priori estimates are proved.
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