Electron. J. Diff. Eqns., Vol. 2003(2003), No. 03, pp. 1-31.

Singular solutions to Protter's problem for the 3-D wave equation involving lower order terms

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 $\Omega_0$, bounded by two characteristic cones $\Sigma_1$ and $\Sigma_{2,0}$, and by a plane region $\Sigma_0$. 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 $\Sigma_0$: the strong power-type singularity appears in the generalized solution on the characteristic cone $\Sigma_{2,0}$. In the present paper we consider the case of third boundary-value problem on $\Sigma_0$ and obtain the existence of many singular solutions for the wave equation involving lower order terms. Especifically, for Protter's problems in $\mathbb{R}^{3}$ it is shown here that for any $n\in N$ there exists a $C^{n}(\bar{\Omega}_0)$-function, for which the corresponding unique generalized solution belongs to $C^{n}(\bar{\Omega}_0\backslash O)$ and has a strong power type singularity at the point $O$. This singularity is isolated at the vertex $O$ of the characteristic cone $\Sigma_{2,0}$. 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 $O$.

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
email: mgrammat@cc.uoi.gr
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
email: nedyu@fmi.uni-sofia.bg

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