Publication TypeJournal Article
Year of Publication2006
AuthorsHristov, TD, Popivanov, NI, Schneider, M
Start Page223
Date Published2006
AMS35A20, 35D05, 35L05, 35L20

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 ${Σ_{2,0} }$. In the present paper we consider the case of the wave equation involving lower order terms and obtain some a priori estimates for the singular solutions of the third BVP. It is a strong power type singularity at the vertex ${O}$ of the characteristic cone ${ Σ_{2,0} }$, which is isolated and does not propagate along the cone.

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