New singular solutions of Protter's problem for the $3$D wave equation

Abstract

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 ${ℝ}^3$, it is shown here that for any $n\in \mathbb{N}$ there exists a $C^n\left({\overline{\Omega }}_0\right)$ - right-hand side function, for which the corresponding unique generalized solution belongs to $C^n\left({\overline{\Omega }}_0\backslash O\right),$ 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.