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.