Bit more info please. How long does it run before it switches off? Is the monitor powered by a power brick of some sort? When it goes off, how do you attempt to get it working again? Does the PC sleep, or only the monitor?
The general thought I am now having is - power problem, self-defending power-off. Obvious test is to use the monitor attached to something else, not your PC. Then you know if it's the monitor or Windows, It could even be a loose wire in the monitor signal connector or power supply cord.