Myron. K. Grammatikopoulos, Tzvetan D. Hristov, & Nedyu I. Popivanov
In 1952, at a conference in New York, Protter formulated some boundary value problems for the wave equation, which are three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems) on the plane. Protter studied these problems in a 3-D domain , bounded by two characteristic cones and , and by a plane region . It is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the case of Dirichlet's and Neumann's conditions on : the strong power-type singularity appears in the generalized solution on the characteristic cone . In the present paper we consider the case of third boundary-value problem on and obtain the existence of many singular solutions for the wave equation involving lower order terms. Especifically, for Protter's problems in it is shown here that for any there exists a -function, for which the corresponding unique generalized solution belongs to and 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. For the wave equation without lower order terms, we presented the exact behavior of the singular solutions at the point .
Submitted May 27, 2002. Published January 2, 2003.
Math Subject Classifications: 35L05,35L20, 35D05, 35A20.
Key Words: Wave equation, boundary value problems, generalized solutions, 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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