New singular solutions for the (3+1)-D Protter problem

Abstract

For the nonhomogeneous wave equation with three space and one time variables we study a boundary value problem that can be regarded as a four-dimensional analogue of the Darboux problem in R². Unlike the planar Darboux problem, the R⁴-version is not well posed and has an infinite-dimensional cokernel. Therefore the problem is not Fredholm in the framework of classical solvability. On the other hand, it is known that for smooth right-hand side functions, there is a uniquely determined generalized solution that may have a strong power-type singularity at one boundary point. The singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. In the present article we announce new singular solutions with exponential growth.