Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 149, pp. 1-20.

Title: Exact behavior of singular solutions to Protter's problem 
       with lower order terms

Authors: Aleksey Nikolov (Univ. of Sofia, Bulgaria)
         Nedyu Popivanov (Univ. of Sofia, Bulgaria)

Abstract:
 For the (2+1)-D wave equation  Protter formulated (1952)  some boundary
 value problems which are  three-dimensional analogues of the Darboux
 problems on the plane. Protter studied these problems  in a 3-D domain,
 bounded by two characteristic cones  and by a planar region.  Now  it is
 well known that, for an infinite number of smooth functions  in the
 right-hand side, these problems do not have classical solutions, because of
 the strong  power-type singularity which appears in the generalized
 solution. In the present paper we consider  the wave equation involving
 lower order terms  and obtain new a priori estimates describing the exact
 behavior of  singular solutions of the third boundary value problem.
 According to the new estimates their singularity is of the same order as in
 case of the wave equation without lower order terms.

Submitted May 8, 2012. Published August 29, 2012.
Math Subject Classifications: 35L05, 35L20, 35D05, 35A20.
Key Words: Wave equation; boundary value problems; generalized solutions;
           singular solutions; propagation of singularities.