Electron. J. Diff. Eqns., Vol. 2003(2003), No. 03, pp. 131.
Singular solutions to Protter's problem for the 3D wave equation
involving lower order terms
Myron. K. Grammatikopoulos, Tzvetan D. Hristov, & Nedyu I. Popivanov
Abstract:
In 1952, at a conference in New York, Protter formulated
some boundary value problems for the wave equation, which are
threedimensional analogues of the Darboux problems (or CauchyGoursat
problems) on the plane. Protter studied these problems
in a 3D 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 righthand 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
powertype singularity appears in the generalized solution on the
characteristic cone
.
In the present paper we consider the
case of third boundaryvalue 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
email: mgrammat@cc.uoi.gr 

Tzvetan D. Hristov
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
1113 Sofia, Bulgaria
email:tsvetan@fmi.unisofia.bg 

Nedyu I. Popivanov
Department of Mathematics and Informatics
University of Sofia
1164 Sofia, Bulgaria
email: nedyu@fmi.unisofia.bg 
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