Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 01, pp. 1-26.
Title: On the singularities of 3-D Protter's problem for the wave equation
Authors: Myron K. Grammatikopoulos (Univ. of Ioannina, Greece)
Tzvetan D. Hristov (Bulgarian Academy of Sciences)
Nedyu I. Popivanov (Univ. of Sofia, Bulgaria)
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 $C^n(\bar{\Omega}_{0})$, for
which the corresponding unique generalized solution belongs to
$C^n(\bar{\Omega}_{0}\backslash O)$, 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.
Submitted October 30, 2000. Published January 1, 2001.
Math Subject Classifications: 35L05, 35L20, 35D05, 35A20.
Key Words: Wave equation; boundary-value problems; generalized solution;
singular solutions; propagation of singularities.