You either have a bad lamp or a bad inverter. If the screen has a
pinkish tint, it is probably a bad lamp. Sometimes one of the wires
comes loose from one end of the lamp, which also causes the inverter to
shut down because it senses an overcurrent condition. The strange part
of your problem is the length of time the lamp stays on before shutting
down. Usually a bad bulb or a defective inverter will cause shutdown
within a second or two, not a minute or so.

At any rate, if the problem is a bad inverter, they are pretty easy to
replace by removing the bezel around the LCD panel. It is located
inside the lid below the bottom of the panel, and plugs into the video
cable on one end, and the lamp plugs into the other end. You can get
the inverter on Ebay, similar to item #190261086678.

If the lamp is defective, it is not replaceable without taking the LCD
panel apart or doing some creative cutting and grinding, and the usual
solution is to replace the complete panel.