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.
Highlights
In 1952, at a conference of the American Mathematical Society in New York, Protter introduced some boundary value problems (BVPs) for the 3D wave equation u ≡ ux1x1 + ux2x2 − utt = f in a domain Ω0 ⊂ R3. These problems are three-dimensional analogous of the Darboux problems on the plane
In order to avoid an infinite number of necessary conditions in the frame of classical solvability, Popivanov and Schneider in [22, 23] gave definitions of a generalized solution of problem (P2) with an eventual singularity on the characteristic cone Σ2,0, or only at its vertex O
Considering problems (P1) and (P2), Popivanov and Schneider [22] announced the existence of singular solutions for both wave and degenerate hyperbolic equations
Summary
For Protter’s problems in R3, it is shown here that for any n ∈ N there exists a Cn(Ω 0) - right-hand side function, for which the corresponding unique generalized solution belongs to Cn(Ω 0\O), but has a strong power-type singularity of order n at the point O. In order to avoid an infinite number of necessary conditions in the frame of classical solvability, Popivanov and Schneider in [22, 23] gave definitions of a generalized solution of problem (P2) with an eventual singularity on the characteristic cone Σ2,0, or only at its vertex O.
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