  We study the long time behaviour of solutions for the weakly damped forced
Kawahara equation on the torus. More precisely, we prove the existence of a
global attractor in $L^2$, to which as time passes all solutions draw closer.
In fact, we show that the global attractor turns out to lie in a smoother space
$H^2$ and be bounded therein. Further, we give an upper bound of the size of
the attractor in $H^2$ that depends only on the damping parameter and the norm
of the forcing term.
