Geodesic
Singular Solutions of Protter’s Problem for a Class of 3-D Hyperbolic Equations
Mathematics and Education in Mathematics, Tome 41 (2012) no. 1, pp. 191-196.

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For 3-D wave equation M. Protter formulated (1952) some boundary value problems which are three-dimensional analogues of the Darboux problems on the plane. Protter studied these problems in a 3-D domain, bounded by two characteristic cones and by a plane region. Now, more than 50 years later, it is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions and the generalized solution have a strong power type singularity at the vertex of the characteristic cone, which is isolated and does not propagate along the cone. In the present paper we consider the third boundary value problem for the wave equation involving lower order terms with a right-hand function of the form of trigonometric polynomial and give a new upper estimate of possible singularity of the solutions. It is interesting that the solutions of the considered problem have the same order of possible singularity as the solutions of the wave equation without lower order terms. *2000 Mathematics Subject Classification: 35L05, 35L20, 35D05, 35A20.
Keywords: Wave Equation, Boundary Value Problems, Generalized Solutions, Singular Solutions, Propagation of Singularities 
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Popivanov, Nedyu; Nikolov, Aleksey. Singular Solutions of Protter’s Problem for a Class of 3-D Hyperbolic Equations. Mathematics and Education in Mathematics, Tome 41 (2012) no. 1, pp. 191-196. http://geodesic.mathdoc.fr/item/MEM_2012_41_1_a18/