Singular Solutions to Protter's Problem for the 3-D Wave Equation Involving Lower Order Terms

Grammatikopoulos, Myron K.
Hristov, Tzvetan D.
Popivanov, Nedyu I.
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Southwest Texas State University, Department of Mathematics
In 1952, at a conference in New York, Protter formulated some boundary value problems for the wave equation, which are three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems) on the plane. Protter studied these problems in a 3-D domain Ω0, bounded by two characteristic cones Σ1 and Σ2,0, and by a plane region Σ0. It is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the case of Dirichlet's and Neumann's conditions on Σ0: the strong power-type singularity appears in the generalized solution on the characteristic cone Σ2,0. In the present paper we consider the case of third boundary-value problem on Σ0 and obtain the existence of many singular solutions for the wave equation involving lower order terms. Especifica ally, for Protter's problems in ℝ3 it is shown here that for any n ∈ N there exists a Cn (ˉΩ0)-function, for which the corresponding unique generalized solution belongs to Cn (ˉΩ0\O) and has a strong power type singularity at the point O. This singularity is isolated at the vertex O of the characteristic cone Σ2,0 and does not propagate along the cone. For the wave equation without lower order terms, we presented the exact behavior of the singular solutions at the point O.
Wave equation, Boundary value problems, Generalized solutions, Singular solutions, Propagation of singularities
Grammatikopoulos, M. K., Hristov, T. D., & Popivanov, N. I. (2003). Singular solutions to Protter's problem for the 3-D wave equation involving lower order terms. <i>Electronic Journal of Differential Equations, 2003</i>(03), pp. 1-31.