Last one of these I worked on with that issue, it turned out to be a stuffed fan. First thing to check is for any thermal related problems that would be shutting down the system as a form of protection. May wish to also look into a replacement top case for it and see if that makes any difference failing a thermal issue. I'm aware that this machine is for parts, but I somehow can't help but think the logic board isn't the problem here. It wouldn't be my first guess anyway.