New singular solutions of Protter's problem for the3D wave equation

Abstract

In 1952, for the wave equation,Protter formulated some boundary value problems (BVPs), which are multidimensional analogues of Darboux problems on the plane. He studied these problems in a3D domainΩ0,bounded by two characteristic conesΣ1andΣ2,0and a plane regionΣ0. What is the situation around these BVPs now after 50 years? It is well known that, for the infinite number of smooth functions in the right-hand side of the equation, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the cases of Dirichlet's or Neumann's conditions onΣ0. In the present paper, we consider the case of third BVP onΣ0and obtain the existence of many singular solutions for the wave equation. Especially, for Protter's problems inℝ3, it is shown here that for anyn∈ℕthere exists aCn(Ω¯0)- right-hand side function, for which the corresponding unique generalized solution belongs toCn(Ω¯0\O),but has a strong power-type singularity of ordernat the pointO. This singularity is isolated only at the vertexOof the characteristic coneΣ2,0and does not propagate along the cone.