On uniqueness of quasi-regular solutions to Protter problem for Keldish type equations

Some three-dimensional boundary value problems for mixed type equations of second kind are studied. Such type problems, but for mixed type equations of first kind are stated by M. Protter in the fifties. For hyperbolic-elliptic equations they are multidimensional analogue of the classical two-dimensional Morawetz-Guderley transonic problem. For hyperbolic and weakly hyperbolic equations the Protter problems are 3D analogues of Darboux or Cauchy-Goursat plane problems. In this case, in contrast of well-posedness of 2D problems, the new problems are strongly ill-posed. In this paper are given similar statement of Protter problems for equations of Keldish type, involving lower order terms. It is shown that the new problems are also ill-posed. A notion of quasi-regular solution is given and sufficient conditions for uniqueness of such solutions are found. The dependence of lower order terms is also studied.