What do cockroaches live under?
Cockroaches prefer dark places, usually damp. In the home, they will hide/breed behind walls, under kitchen/bathroom sinks, bathtubs, etc. They will also be in pantries, near stoves, or any place that will provide a meal.
Adult cockroaches live for about a year, but have already lived for a year from egg stage, to adult. So the actual lifespan is about 2 years. Unfortunately, like fleas, cockroaches are difficult to kill, which is why the little blighters have been around for many millions of years, (they can even survive radiation) and are amongst the most successful insects to inhabit the planet.