Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations
A nonlinear Fokker-Planck type ultraparabolic integro-differential equation is studied. It arises from the statistical description of the dynamical behavior of populations of infinitely many (nonlinearly coupled) random oscillators subject to ``mean-field'' interaction. A regularized parab...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Texas State University
2002-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2002/24/abstr.html |
| _version_ | 1818395250138284032 |
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| author | Denis R. Akhmetov Mikhail M. Lavrentiev Jr Renato Spigler |
| author_facet | Denis R. Akhmetov Mikhail M. Lavrentiev Jr Renato Spigler |
| author_sort | Denis R. Akhmetov |
| collection | DOAJ |
| description | A nonlinear Fokker-Planck type ultraparabolic integro-differential equation is studied. It arises from the statistical description of the dynamical behavior of populations of infinitely many (nonlinearly coupled) random oscillators subject to ``mean-field'' interaction. A regularized parabolic equation with bounded coefficients is first considered, where a small spatial diffusion is incorporated in the model equation and the unbounded coefficients of the original equation are replaced by a special ``bounding" function. Estimates, uniform in the regularization parameters, allow passing to the limit, which identifies a classical solution to the original problem. Existence and uniqueness of classical solutions are then established in a special class of functions decaying in the velocity variable. |
| first_indexed | 2024-12-14T06:14:07Z |
| format | Article |
| id | doaj.art-2ab597f80f8349ef971e6f7b8bebfd19 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-14T06:14:07Z |
| publishDate | 2002-02-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-2ab597f80f8349ef971e6f7b8bebfd192022-12-21T23:14:05ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912002-02-01200224117Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equationsDenis R. AkhmetovMikhail M. Lavrentiev JrRenato SpiglerA nonlinear Fokker-Planck type ultraparabolic integro-differential equation is studied. It arises from the statistical description of the dynamical behavior of populations of infinitely many (nonlinearly coupled) random oscillators subject to ``mean-field'' interaction. A regularized parabolic equation with bounded coefficients is first considered, where a small spatial diffusion is incorporated in the model equation and the unbounded coefficients of the original equation are replaced by a special ``bounding" function. Estimates, uniform in the regularization parameters, allow passing to the limit, which identifies a classical solution to the original problem. Existence and uniqueness of classical solutions are then established in a special class of functions decaying in the velocity variable.http://ejde.math.txstate.edu/Volumes/2002/24/abstr.htmlnonlinear integro-differential parabolic equationsultraparabolic equationsFokker-Planck equationdegenerate parabolic equationsregularization. |
| spellingShingle | Denis R. Akhmetov Mikhail M. Lavrentiev Jr Renato Spigler Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations Electronic Journal of Differential Equations nonlinear integro-differential parabolic equations ultraparabolic equations Fokker-Planck equation degenerate parabolic equations regularization. |
| title | Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations |
| title_full | Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations |
| title_fullStr | Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations |
| title_full_unstemmed | Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations |
| title_short | Existence and uniqueness of classical solutions to certain nonlinear integro-differential Fokker-Planck type equations |
| title_sort | existence and uniqueness of classical solutions to certain nonlinear integro differential fokker planck type equations |
| topic | nonlinear integro-differential parabolic equations ultraparabolic equations Fokker-Planck equation degenerate parabolic equations regularization. |
| url | http://ejde.math.txstate.edu/Volumes/2002/24/abstr.html |
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