Geodesic
On the singularities of 3-D Protter's problem for the wave equation
Electronic journal of differential equations, Tome 2001 (2001)
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 $\mathbb{N}$ there exists a right hand side smooth function from $C^n(\bar{\Omega}_{0})$, for which the corresponding unique generalized solution belongs to $C^n(\bar{\Omega}_{0}\backslash O)$, 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.
Classification : 35L05, 35L20, 35D05, 35A20
Keywords: wave equation, boundary-value problems, generalized solution, singular solutions, propagation of singularities 
@article{EJDE_2001__2001__a177,
     author = {Grammatikopoulos,  Myron K. and Hristov,  Tzvetan D. and Popivanov,  Nedyu I.},
     title = {On the singularities of {3-D} {Protter's} problem for the wave equation},
     journal = {Electronic journal of differential equations},
     year = {2001},
     volume = {2001},
     zbl = {0973.35010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a177/}
}
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%A Popivanov,  Nedyu I.
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%J Electronic journal of differential equations
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%F EJDE_2001__2001__a177
Grammatikopoulos,  Myron K.; Hristov,  Tzvetan D.; Popivanov,  Nedyu I. On the singularities of 3-D Protter's problem for the wave equation. Electronic journal of differential equations, Tome 2001 (2001). http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a177/