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.