Sounds like a heat issue given the fact the system does start up cold.
One of the things I would do here is install a good fan & temp sensor monitor like [http://www.tunabellysoftware.com/tgpro/|Temperature Gauge]. Lets see what it tells us.
If your system is not stable enough to install the software you could try this: Put your system into a tight plastic bag (squeeze out as much air as you can) and place the system in your refrigerator for a few hours (not too long now). Does the system stay running longer? If it does see if you can install the monitoring software while the free version will give you some info the full version writes the info to a file so you can review it just when the system crashes.