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Advances in Mathematical Physics
Volume 2017, Article ID 1571959, 16 pages
Research Article

Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations

1Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
2Faculty of Mathematics and Informatics, University of Sofia, 1164 Sofia, Bulgaria
3Faculty of Applied Mathematics and Informatics, Technical University of Sofia, 1000 Sofia, Bulgaria
4Faculty of Mathematics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany

Correspondence should be addressed to Nedyu Popivanov; gb.aifos-inu.imf@uyden

Received 29 April 2017; Accepted 16 August 2017; Published 16 October 2017

Academic Editor: Alexander Iomin

Copyright © 2017 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.


We study a boundary value problem for (3 + 1)-D weakly hyperbolic equations of Keldysh type (problem PK). The Keldysh-type equations are known in some specific applications in plasma physics, optics, and analysis on projective spaces. Problem PK is not well-posed since it has infinite-dimensional cokernel. Actually, this problem is analogous to a similar one proposed by M. Protter in 1952, but for Tricomi-type equations which, in part, are closely connected with transonic fluid dynamics. We consider a properly defined, in a special function space, generalized solution to problem PK for which existence and uniqueness theorems hold. It is known that it may have a strong power-type singularity at one boundary point even for very smooth right-hand sides of the equation. In the present paper we study the asymptotic behavior of the generalized solutions of problem PK at the singular point. There are given orthogonality conditions on the right-hand side of the equation, which are necessary and sufficient for the existence of a generalized solution with fixed order of singularity.