Abstract and Applied Analysis
Volume 2004 (2004), Issue 4, Pages 315-335
doi:10.1155/S1085337504306111
New singular solutions of Protter's problem for the 3D wave equation
1Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
2Department of Mathematics and Informatics, University of Sofia, Sofia 1164, Bulgaria
Received 10 September 2002
Copyright © 2004 M. K. Grammatikopoulos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In 1952, for the wave equation,Protter formulated some boundary
value problems (BVPs), which are multidimensional
analogues of Darboux problems on the plane. He studied these
problems in a 3D domain Ω0, bounded by two
characteristic cones Σ1 and Σ2,0 and a
plane region Σ0. What is the situation around these
BVPs now after 50 years? It is well known that, for the infinite
number of smooth functions in the right-hand side of the
equation, these problems do not have classical solutions.
Popivanov and Schneider (1995) discovered the reason of this fact
for the cases of Dirichlet's or Neumann's conditions on Σ0. In the present paper, we consider the case of third BVP on
Σ0 and obtain the existence of many singular
solutions for the wave equation. Especially, for Protter's
problems in ℝ3, it is shown here that for any n∈ℕ there exists a Cn(Ω¯0) - right-hand
side function, for which the corresponding unique generalized
solution belongs to Cn(Ω¯0\O), but
has a strong power-type singularity of order n at the point
O. This singularity is isolated only at the vertex O of the
characteristic cone Σ2,0 and does not propagate along
the cone.