Asymptotic expansions of singular solutions for (3+1)-D Protter problems

Four-dimensional boundary value problems for the nonhomogeneous wave equation are studied, which are analogues of Darboux problems in the plane. The smoothness of the right-hand side function of the wave equation is decisive for the behavior of the solution of the boundary value problem. It is shown that for each n∈N there exists such a right-hand side function from Cn, for which the uniquely determined generalized solution has a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. The present article describes asymptotic expansions of the generalized solutions in negative powers of the distance to this singularity point. Some necessary and sufficient conditions for existence of regular solutions, or solutions with fixed order of singularity, are derived and additionally some a priori estimates for the singular solutions are given.