In 1952, for the wave equation, Protter formulated some boundary
value problems (BVPs), which are multidimensional
analogues of Darboux problems on the plane. He studied these
problems in a $3$ D domain $\Omega _{0},$ bounded by two
characteristic cones $\Sigma _{1}$ and $\Sigma _{2,0}$ and a
plane region $\Sigma _{0}$ . What is the situation around these
BVPs now after 50 years? It is well known that, for the infinite
number of smooth functions in the right-hand side of the
equation, these problems do not have classical solutions.
Popivanov and Schneider (1995) discovered the reason of this fact
for the cases of Dirichlet's or Neumann's conditions on $\Sigma _{0}$ . In the present paper, we consider the case of third BVP on
$\Sigma _{0}$ and obtain the existence of many singular
solutions for the wave equation. Especially, for Protter's
problems in $\mathbb{R}^{3}$ , it is shown here that for any $n\in \mathbb{N}$ there exists a $C^{n}(\bar{\Omega}_{0})$ - right-hand
side function, for which the corresponding unique generalized
solution belongs to $C^{n}(\bar{\Omega}_{0}\backslash O),$ but
has a strong power-type singularity of order $n$ at the point
$O$ . This singularity is isolated only at the vertex $O$ of the
characteristic cone $\Sigma _{2,0}$ and does not propagate along
the cone.