Singular solutions to the Protter-Morawetz problem for Keldysh-type equations involving lower order terms.

The three-dimensional Protter-Morawetz problem for weakly hyperbolic equations of Keldysh type involving lower order terms is studied. Similar problem for Tricomi-type equations was proposed by M. Protter in connection with the Guderley-Morawetz plane problem that models the transonic flow phenomena. The considered Protter-Morawetz problem for Keldysh-type equations is not Fredholm in the frame of classical solvability, because it has infinite-dimensional co-kernel. In the present paper new nontrivial classical solutions to the homogeneous adjoint problem are found. Further, a generalized solution to the formulated problem is considered, in a special function space, for which existence and uniqueness theorems hold. Under some conditions on lower order terms smooth right-hand side functions are found, such that the corresponding generalized solutions have strong power type singularities. It is interesting that these singularities are isolated at only one boundary point, which makes this case different from the traditional case on the propagation of singularity. [ABSTRACT FROM AUTHOR]