Electron. J. Diff. Eqns., Vol. 2002(2002), No. 24, pp. 1-17.
Existence and uniqueness of classical solutions
to certain nonlinear integro-differential
Fokker-Planck type equations
Denis R. Akhmetov, Mikhail M. Lavrentiev, Jr., & Renato Spigler
Abstract:
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.
Submitted October 23, 2001. Published February 27, 2002.
Math Subject Classifications: 35K20, 35K60, 45K05.
Key Words: nonlinear integro-differential parabolic equations,
ultraparabolic equations, Fokker-Planck equation,
degenerate parabolic equations, regularization.
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Denis R. Akhmetov
Sobolev Institute of Mathematics,
4 Acad. Koptyug prosp., 630090 Novosibirsk, Russia
e-mail: adr@math.nsc.ru
http://www.math.nsc.ru/LBRT/d4/akhmetov.htm |
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Mikhail M. Lavrentiev, Jr.
Sobolev Institute of Mathematics,
4 Acad. Koptyug prosp., 630090 Novosibirsk, Russia
e-mail: mmlavr@nsu.ru
http://server.math.nsc.ru/conference/mml/misha.htm |
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Renato Spigler
Dipartimento di Matematica, Universita di ``Roma Tre",
1 Largo San Leonardo Murialdo, 00146 Rome, Italy
e-mail: spigler@mat.uniroma3.it
http://www.mat.uniroma3.it/dipartimento/membri/spigler_homepage.html
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