Electron. J. Diff. Eqns., Vol. 2001(2001), No. 01, pp. 1-26.

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

Myron K. Grammatikopoulos, Tzvetan D. Hristov, & Nedyu I. Popivanov

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.

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Myron K. Grammatikopoulos
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: mgrammat@cc.uoi.gr
Tzvetan D. Hristov
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
1113 Sofia, Bulgaria
email: tzvetan@math.bas.bg
Nedyu I. Popivanov
Department of Mathematics and Informatics
University of Sofia
1164 Sofia, Bulgaria
email: nedyu@fmi.uni-sofia.bg

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