Radially Symmetric Solutions of
We investigate solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the pro...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2012/296591 |
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| _version_ | 1849467887886008320 |
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| author | William C. Troy Edward P. Krisner |
| author_facet | William C. Troy Edward P. Krisner |
| author_sort | William C. Troy |
| collection | DOAJ |
| description | We investigate solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, , such that as and as . Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies for infinitely many values . At the critical value there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime we prove the existence of a family of “super singular” sign changing singular solutions. |
| format | Article |
| id | doaj-art-e6be1ef4000e4aed8d91d0f767d8da63 |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-e6be1ef4000e4aed8d91d0f767d8da632025-08-20T03:26:00ZengWileyInternational Journal of Differential Equations1687-96431687-96512012-01-01201210.1155/2012/296591296591Radially Symmetric Solutions ofWilliam C. Troy0Edward P. Krisner1Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USADepartment of Mathematics, University of Pittsburgh at Greensburg, Greensburg, PA 15601, USAWe investigate solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, , such that as and as . Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies for infinitely many values . At the critical value there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime we prove the existence of a family of “super singular” sign changing singular solutions.http://dx.doi.org/10.1155/2012/296591 |
| spellingShingle | William C. Troy Edward P. Krisner Radially Symmetric Solutions of International Journal of Differential Equations |
| title | Radially Symmetric Solutions of |
| title_full | Radially Symmetric Solutions of |
| title_fullStr | Radially Symmetric Solutions of |
| title_full_unstemmed | Radially Symmetric Solutions of |
| title_short | Radially Symmetric Solutions of |
| title_sort | radially symmetric solutions of |
| url | http://dx.doi.org/10.1155/2012/296591 |
| work_keys_str_mv | AT williamctroy radiallysymmetricsolutionsof AT edwardpkrisner radiallysymmetricsolutionsof |