Integral representation of singular solutions to BVP for the wave equation

Abstract

We consider the Protter problem for the four-dimensional wave equation, where the boundary conditions are posed on a characteristic surface and on a non-characteristic one. In particular, we consider a case when the right-hand side of the equation is of the form of harmonic polynomial. This problem is known to be ill-posed, because its adjoint homogeneous problem has infinitely many nontrivial classical solutions. The solutions of the Protter problem may have strong power type singularity isolated at one boundary point. Bounded solutions are possible only if the right-hand side of the equation is orthogonal to all the classical solutions of the adjoint homogeneous problem, which in fact is a necessary but not sufficient condition for the classical solvability of the problem. In this paper we offer an explicit integral form of the solutions of the problem, which is more simple than the known so far. Additionally, we give a condition on the coefficients of the harmonic polynomial to obtain not only bounded but also continuous solution.