ESTIMATES FOR THE SINGULAR SOLUTIONS OF THE 3-D PROTTER’S PROBLEM

Abstract

For the wave equation we study boundary value problems, stated by Protter in 1952, as some three-dimensional analogues of Darboux problems on the plane. It is known that Protter's problems are not well posed and the solution may have singularity at the vertex $O$ of a characteristic cone, which is a part of the domain's boundary $\partial \Omega $. It is shown that for $n$ in $\mathbb{N}$ there exists a right-hand side smooth function from $C^{n}(\bar{\Omega})$, for which the corresponding unique generalized solution belongs to $C^{n}(\bar{\Omega}\backslash O)$, but it has a strong power-type singularity. It is isolated at the vertex $O$ and does not propagate along the cone. The present article gives some necessary and sufficient conditions for the existence of a fixed order singularity. It states some exact a priori estimates for the solution.