Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)‎-D Protter-Morawetz Problem

Abstract EN

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied.

It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point.

This singularity is isolated at the vertex O of the boundary light characteristic cone and does not propagate along the bicharacteristics.

The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance to O .

Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.