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

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.