TY - JOUR
A2 - Covachev, Valery
AU - Dechevski, Lubomir
AU - Popivanov, Nedyu
AU - Popov, Todor
PY - 2012
DA - 2012/09/06
TI - Exact Asymptotic Expansion of Singular Solutions for the ()-D Protter Problem
SP - 278542
VL - 2012
AB - We study three-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in ℝ2. In contrast to the planar Darboux problem the three-dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of classical solutions. On the other hand, it is known that for smooth right-hand side functions there is a uniquely determined generalized solution that may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic light cone and does not propagate along the cone. The present paper describes asymptotic expansion of the generalized solutions in negative powers of the distance to this singular point. We derive necessary and sufficient conditions for existence of solutions with a fixed order of singularity and give a priori estimates for the singular solutions.
SN - 1085-3375
UR - https://doi.org/10.1155/2012/278542
DO - 10.1155/2012/278542
JF - Abstract and Applied Analysis
PB - Hindawi Publishing Corporation
KW -
ER -