On the singularities of 3-D Protter's problem for the wave equation
Grammatikopoulos, Myron K.
Hristov, Tzvetan D.
Popivanov, Nedyu I.
Southwest Texas State University, Department of Mathematics
In this paper we study boundary-value problems for the wave equation, which are three-dimensional analogue of Darboux-problems (or of Cauchy-Goursat problems) on the plane. It is shown that for n in ℕ there exists a right hand side smooth function from Cn(Ω̅0), for which the corresponding unique generalized solution belongs to Cn(Ω̅0)O), and it has a strong power-type singularity at the point O. This singularity is isolated at the vertex O of the characteristic cone and does not propagate along the cone. In this paper we investigate the behavior of the singular solutions at the point O. Also, we study more general boundary-value problems and find that there exist an infinite number of smooth right-hand side functions for which the corresponding unique generalized solutions are singular. Some a priori estimates are also stated.
Wave equation, Boundary-value problems, Generalized solution, Singular solutions, Propagation of singularities
Grammatikopoulos, M. K., Hristov, T. D., & Popivanov, N. I. (2001). On the singularities of 3-D Protter's problem for the wave equation. <i>Electronic Journal of Differential Equations, 2001</i>(01), pp. 1-26.