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Abstract and Applied Analysis
Volume 2014, Article ID 260287, 19 pages
http://dx.doi.org/10.1155/2014/260287
Research Article

Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem

1Department of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria
2Department of Mathematics, City University of New York, College of Staten Island, Staten Island, NY 10314, USA

Received 28 February 2014; Accepted 11 May 2014; Published 14 October 2014

Academic Editor: Donal O’Regan

Copyright © 2014 Nedyu Popivanov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertex of the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance to . Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.