Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations
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 co...
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| Format: | Article |
| Language: | English |
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Wiley
2017-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2017/1571959 |
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| author | Nedyu Popivanov Tsvetan Hristov Aleksey Nikolov Manfred Schneider |
| author_facet | Nedyu Popivanov Tsvetan Hristov Aleksey Nikolov Manfred Schneider |
| author_sort | Nedyu Popivanov |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-49bc1b3cff8344fa8419131ca4da94c0 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-49bc1b3cff8344fa8419131ca4da94c02025-02-03T06:06:39ZengWileyAdvances in Mathematical Physics1687-91201687-91392017-01-01201710.1155/2017/15719591571959Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type EquationsNedyu Popivanov0Tsvetan Hristov1Aleksey Nikolov2Manfred Schneider3Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, 1113 Sofia, BulgariaFaculty of Mathematics and Informatics, University of Sofia, 1164 Sofia, BulgariaFaculty of Applied Mathematics and Informatics, Technical University of Sofia, 1000 Sofia, BulgariaFaculty of Mathematics, Karlsruhe Institute of Technology, 76131 Karlsruhe, GermanyWe 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.http://dx.doi.org/10.1155/2017/1571959 |
| spellingShingle | Nedyu Popivanov Tsvetan Hristov Aleksey Nikolov Manfred Schneider Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations Advances in Mathematical Physics |
| title | Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations |
| title_full | Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations |
| title_fullStr | Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations |
| title_full_unstemmed | Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations |
| title_short | Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations |
| title_sort | singular solutions to a 3 1 d protter morawetz problem for keldysh type equations |
| url | http://dx.doi.org/10.1155/2017/1571959 |
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