Invariant manifolds and long-time asymptotics for the Vlasov-Poisson-Fokker-Planck equation

We study the large time behavior of small solutions to the Cauchy problem for the Vlasov-Poisson-Fokker-Planck equation, which is a degenerate parabolic equation with nonlocal nonlinearity. We construct finite dimensional invariant manifolds in a neighborhood of the origin in polynomially weighted Sobolev spaces, which enables us to compute systematically the long-time asymptotics for small solutions. To construct invariant manifolds, we make use of the "similarity variables" transformation as in C. E. Wayne's work in 1997, where invariant manifolds for parabolic equations in unbounded domains are constructed.