# ESTIMATES OF SINGULAR SOLUTIONS OF PROTTER’S PROBLEM FOR THE 3-D HYPERBOLIC EQUATIONS

 Title ESTIMATES OF SINGULAR SOLUTIONS OF PROTTER’S PROBLEM FOR THE 3-D HYPERBOLIC EQUATIONS Publication Type Journal Article Year of Publication 2006 Authors Hristov, TD, Popivanov, NI, Schneider, M Volume 10 Issue 2 Start Page 223 Pagination 30 Date Published 2006 ISSN 1083-2564 AMS 35A20, 35D05, 35L05, 35L20 Abstract For 3-D wave equation M. Protter formulated (1952) some boundary value problems (BVP) which are three-dimensional analogues of the Darboux problems on the plane. Protter studied these problems in a 3-D domain ${ Ω_0}$, bounded by two characteristic cones ${Σ_1}$ and ${Σ_{2,0} }$, and by a plane region ${Σ_0}$. Now, 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. The reason of this fact had been discovered in the early 90-ties: the strong power-type singularity appears in the generalized solution on the characteristic cone ${Σ_{2,0} }$. 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