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 in there exists a right hand side smooth function from , for which the corresponding unique generalized solution belongs to , and it has a strong power-type singularity at the point . This singularity is isolated at the vertex 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 . 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
| Tzvetan D. Hristov |
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
1113 Sofia, Bulgaria
| Nedyu I. Popivanov |
Department of Mathematics and Informatics
University of Sofia
1164 Sofia, Bulgaria
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