Electronic Journal of Differential Equations (Jan 2001)
On the singularities of 3-D Protter's problem for the wave equation
Abstract
In this paper we study boundary-value problems for the wave equation, which are three-dimensional analogue of Darboux-problems (or of Cauchy-Goursat problems) on the plane. It is shown that for $n$ in $mathbb{N}$ there exists a right hand side smooth function from $C^n(ar{Omega}_{0})$, for which the corresponding unique generalized solution belongs to $C^n(ar{Omega}_{0}ackslash O)$, and it has a strong power-type singularity at the point $O$. This singularity is isolated at the vertex $O$ of the characteristic cone and does not propagate along the cone. In this paper we investigate the behavior of the singular solutions at the point $O$. Also, we study more general boundary-value problems and find that there exist an infinite number of smooth right-hand side functions for which the corresponding unique generalized solutions are singular. Some a priori estimates are also stated.
