Heavy damping occurs when the resistive force exceeds that of critical damping.
The object returns to and remains at its equilibrium position without being displaced to the other side (similar to critical damping) but it takes a longer time to do so than in the case of critical damping.
This may be counterintuitive as one would think that a more heavily damped system would return to zero at a faster rate. However, recall that the damping force acts in a direction that is the opposite of the velocity of the object.
This means that the force resists the motion of the object as it moves towards the equilibrium position. More time is therefore required for the object to reach equilibrium.
An example of a heavily damped system is a coiled spring mattress, where the springs slowly return to a new equilibrium position when a person lies on the mattress.