You need to find out why it shuts down. The power button is temporary and it signals the motherboard to power up the system; even if you hold it forever or short the power switch pins on the motherboard, the system should still shutdown after a few minutes until the issue has been resolved.
You cleaned the fans, but does the CPU or another component overheats? The CPU temp should be available in the BIOS. Does it shutdown when on a BIOS screen or only once the OS is loaded?