Communications in Applied Analysis 10 (2006), no. 2, 223–251 ESTIMATES OF SINGULAR SOLUTIONS OF PROTTER’S PROBLEM FOR THE 3-D HYPERBOLIC EQUATIONS Tzvetan D. Hristov 1 , Nedyu I. Popivanov 2 and Manfred Schneider 3 1,2 Department of Mathematics and Informatics University of Sofia 1164 Sofia, Bulgaria 1 tsvetan@fmi.uni-sofia.bg 2 nedyu@fmi.uni-sofia.bg 3 Department of Mathematics University of Karlsruhe Karlsruhe, Germany manfred.schneider@math.uni-karlsruhe.de Communicated by D.D. Bainov ABSTRACT: For 3-D wave equation M. Protter formulated (1952) some boundary value problems (BVP) which are three-dimensional analogues of the Darboux problems on the plane. Protter studied these problems in a 3-D domain Ω 0 , bounded by two characteristic cones Σ 1 and Σ 2,0 , and by a plane region Σ 0 . Now, 50 years later, it is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions. The reason of this fact had been discovered in the early 90-ties: the strong power-type singularity appears in the generalized solution on the characteristic cone Σ 2,0 . In the present paper we consider the case of the wave equation involving lower order terms and obtain some a priori estimates for the singular solutions of the third BVP. It is a strong power type singularity at the vertex O of the characteristic cone Σ 2,0 , which is isolated and does not propagate along the cone. AMS (MOS) Subject Classification: 35L05, 35L20, 35D05, 35A20 Received December 3, 2005 1083-2564 $03.50 c  Dynamic Publishers, Inc.