  The long-time behavior of the solutions for a non-isothermal model in
superfluidity is investigated. The model describes the transition between the
normal and the superfluid phase in liquid 4He by means of a non-linear
differential system, where the concentration of the superfluid phase satisfies
a non-isothermal Ginzburg-Landau equation. This system, which turns out to be
consistent with thermodynamical principles and whose well-posedness has been
recently proved, has been shown to admit a Lyapunov functional. This allows to
prove existence of the global attractor which consists of the unstable manifold
of the stationary solutions. Finally, by exploiting recent techniques of
semigroups theory, we prove the existence of an exponential attractor of finite
fractal dimension which contains the global attractor.
