Screen Goes Black After Monitor is on for a while. (by 2 users)

Bad CCFLs in your LCD. CCFL stands for Cold cathode fluorescent lamps. These are the lamps that provide backlight for the LCD Monitor. These lamps do not last forever, and are a very common part to go bad. The circuit that provides the power for the CCFL is called the inverter, it might be going bad, but I still think it is the CCFL. Only trained techs should try and replace a CCFL or fix an Inverter. CCFLs do contain mercury, but only small amounts.

http://en.wikipedia.org/wiki/Cold_cathode

My advice, get a new LCD monitor and sell your bad one on ebay.