We present
a new necessary and sufficient condition to verify
the asymptotic compactness
of an evolution equation
defined in an unbounded domain,
which involves the Littlewood-Paley projection operators.
We then use this condition to
prove the existence of an attractor
for the damped \bbme in the phase space $H^1({\bf R})$
by showing the solutions are point dissipative and asymptotically
compact. Moreover the attractor is in fact smoother and it belongs to $H^{3/2-\ve}$ for every $\ve>0$.