Cockroaches can live without their heads for a long time before they starve to death. Like other animals they do not bleed to death after being decapitated, their necks gets sealed by clotting. Cockroaches do not need their brain to breathe. They have small holes all over their body called the spiracles which is a kind of an inlet for air. These spiracles directly transfer air to all the critical tissues in their bodies.
Not just that they can also perform basic activities like moving and standing despite of being decapitated. They have clumps of ganglia which is a kind of nerve tissue agglomerate. These clumps make their body capable of performing basic bodily functions. Cockroaches can live for weeks on one meal and after being decapitated also they can survive for a long time before they die of starvation.
My answer is:Cockroaches can live without their heads for a long time before they starve to death. Like other animals they do not bleed to death after beingdecapitated, their necks gets sealed by clotting. Cockroaches do not need their brain to breathe.