A. Carpio, LONG-TIME BEHAVIOR FOR SOLUTIONS OF THE VLASOV-POISSON-FOKKER-PLANCK EQUATION, Mathematical methods in the applied sciences, 21(11), 1998, pp. 985-1014
We study the long-time behaviour of solutions of the Vlasov-Poisson-Fo
kker-Planck equation for initial data small enough and satisfying some
suitable integrability conditions. Our analysis relies on the study o
f the linearized problems with bounded potentials decaying fast enough
for large times. We obtain global bounds in time for the fundamental
solutions of such problems and their derivatives. This allows to get s
harp bounds for the decay of the difference between the solutions of t
he Vlasov-Poisson-Fokker-Planck equation and the solution of the free
equation with the same initial data. Thanks to these bounds, we get an
explicit form for the second term in the asymptotic expansion of the
solutions for large times. (C) 1998 B. G. Teubner Stuttgart-John Wiley
& Sons, Ltd.