On the singularities of 3-D Protter's problem for the wave equation

• Main Authors: Myron K. Grammatikopoulos, Tzvetan D. Hristov, Nedyu I. Popivanov
• Format: Article
• Language: English
• Published: Texas State University 2001-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: Wave equation; boundary-value problems; generalized solution; singular solutions; propagation of singularities.
• Online Access: http://ejde.math.txstate.edu/Volumes/2001/01/abstr.html

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.