## Abstract and Applied Analysis

### New singular solutions of Protter's problem for the $3$D wave equation

#### 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 $3$D domain $\Omega _{0},$ bounded by two characteristic cones $\Sigma _{1}$ and $\Sigma _{2,0}$ and a plane region $\Sigma _{0}$. 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 $\Sigma _{0}$. In the present paper, we consider the case of third BVP on $\Sigma _{0}$ and obtain the existence of many singular solutions for the wave equation. Especially, for Protter's problems in $\mathbb{R}^{3}$, it is shown here that for any $n\in \mathbb{N}$ there exists a $C^{n}(\bar{\Omega}_{0})$ - right-hand side function, for which the corresponding unique generalized solution belongs to $C^{n}(\bar{\Omega}_{0}\backslash O),$ 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 $\Sigma _{2,0}$ and does not propagate along the cone.

#### Article information

Source
Abstr. Appl. Anal., Volume 2004, Number 4 (2004), 315-335.

Dates
First available in Project Euclid: 4 May 2004

https://projecteuclid.org/euclid.aaa/1083679181

Digital Object Identifier
doi:10.1155/S1085337504306111

Mathematical Reviews number (MathSciNet)
MR2064144

Zentralblatt MATH identifier
1115.35078

Subjects
Primary: 35L05: Wave equation 35L20
Secondary: 35D05 35A20 33C05 33C90

#### Citation

Grammatikopoulos, M. K.; Popivanov, N. I.; Popov, T. P. New singular solutions of Protter's problem for the $3$D wave equation. Abstr. Appl. Anal. 2004 (2004), no. 4, 315--335. doi:10.1155/S1085337504306111. https://projecteuclid.org/euclid.aaa/1083679181