Geodesic
Singular Solutions with Exponential Growth for the (3+1)-D Wave Equation
Mathematics and Education in Mathematics, Tome 40 (2011) no. 1, pp. 200-206.

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Four-dimensional boundary value problems for the nonhomogeneous wave equation are studied. They were proposed by M. Protter as multidimensional analogues of Darboux problems in the plane. It is known that the unique generalized solution may have a strong power-type singularity at only one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. Another aspect is that the problem is not Fredholm, since it has infinite-dimensional cokernel. Some known results suggest that the solution may have at most exponential growth, but the question whether such solutions really exist was still open. We show that the answer is positive and construct generalized solution of Protter problem with exponential singularity. *2000 Mathematics Subject Classification: 35L05, 35L20, 35D05, 35D10, 35C10.
Keywords: Boundary Value Problems, Generalized Solution, Semi-Fredholm Solvability, Special Functions, Wave Equation 
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Popivanov, Nedyu; Popov, Todor; Scherer, Rudolf. Singular Solutions with Exponential Growth for the (3+1)-D Wave Equation. Mathematics and Education in Mathematics, Tome 40 (2011) no. 1, pp. 200-206. http://geodesic.mathdoc.fr/item/MEM_2011_40_1_a18/