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

M.K. Grammatikopoulos1 , N.I. Popivanov2 and T.P. Popov2

1Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
 2Department of Mathematics and Informatics, University of Sofia, Sofia 1164, Bulgaria

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.