Solutions with Power Type or Exponential Singularity for (3+1)-D Protter-Morawetz Problems

Transonic flows in fluid dynamics are modelled by mixed-type equations on the plane. On the other hand, M. H. Protter proposed multi-dimensional analogues of the classical planar Guderley-Morawetz problem for equations of hyperbolic-elliptic type. We study the Protter-Morawetz problems in the hyperbolic part of the domain. Unlike the two-dimensional analogues, for smooth right-hand sides, in the frame of classical solvability, the four-dimensional problem is not well-posed and not Fredholm. Alternatively, the uniquely determined generalized solutions may have singularities at one boundary point. We study the growth of the generalized solution at the point of singularity.