  In this paper, we first establish a criterion based on contractive function
for the existence of polynomial attractors. This criterion only involves some
rather weak compactness associated with the repeated limit inferior and
requires no compactness, which makes it suitable for critical cases. Then by
this abstract theorem, we verify the existence of a polynomial attractor and
estimate its attractive speed for the wave equation with nonlocal weak damping,
anti-damping and critical nonlinearity.
