Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 24, pp. 1-17.
Title: Existence and uniqueness of classical solutions
to certain nonlinear integro-differential
Fokker-Planck type equations
Authors: Denis R. Akhmetov (Sobolev Inst. of Math., Novosibirsk, Russia)
Mikhail M. Lavrentiev, Jr. (Sobolev Inst. of Math., Novosibirsk, Russia)
Renato Spigler (Univ. di Roma Tre, Italy)
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.