Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 03, pp. 1-31.
Title: Singular solutions to Protter's problem for the 3-D wave equation
involving lower order terms
Authors: Myron. K. Grammatikopoulos (Univ. of Ioannina, Greece)
Tzvetan D. Hristov (Univ. of Sofia, Bulgaria)
Nedyu I. Popivanov (Univ. of Sofia, Bulgaria)
Abstract:
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. Especifica ally, 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.