The Darboux Problems in R3 for a Class of Degenerating Hyperbolic Equations

Abstract

In this paper we investigate some boundary value problems for degenerate hyperbolic equations in R 3 which are the three-dimensional analogues of the Darboux-problems (or Cauchy-Goursat problems) in R 2. It is well known that the Darboux-problems in the plane are well posed, while the same is not true for the corresponding problems in R 3. It turns out that two of the considered homogeneous problems have an infinite number of classical solutions. This means that for the solvability of the adjoint problems, the function on the right hand side has to be orthogonal to all the infinitely many classical solutions of the homogeneous problems. We define appropriate generalized solution and special function spaces where uniqueness and existence theorems hold for the considered problems in R 3 and show that especially M. H. Protter′s problem in R 3 has solutions with strong singularities on one part of the boundary even for smooth functions on the righthand side.