If you're reading this page, chance is that you've been googling for this issue: When you turn on your PS3 slim, it runs for a few seconds, and then the fan stops, and the power LED keeps flashing. You may also get an additional issue like the PS3 crashing immediately while playing a specific game, and then the power LED just keeps flashing nonstop.
Usually people attribute this to one of the following:
1. Aging PSU: Let me tell you that this is likely not the case. I've tried this and it didn't work for me. Usually the PSU can last a very long time (at least 10 years), so you probably don't want to waste your money on that.
2. Thermal issue: It's true that being too hot is probably problematic. However, even early slim models like my CECH-2001A typically run at 68-70°C. Unless your PS3 reaches 80°C, you should still be safe. Bear that in mind before disassembling to fix it.
3. CMOS battery: Did you know that this thing easily lasts a decade? Unlike PC motherboards, PS3 consumes very little power from the CMOS battery, so you can just forget about it.
For me the two biggest possible causes are the following:
1. Aging motherboard
2. The HDD
In my case, it's the combination of the two above. Basically the PS3 tries to boot the HDD, but it fails right at that and automatically shuts down the rest of the system. This problem can't be fixed, but it can be circumvented.
Recommended method: Remove the screw that holds the HDD tray in place, then remove the HDD and boot without it. If it still turns off, you either need a new PSU or a new PS3. If it doesn't automatically turn off, congrats, and let it run for a few minutes. After that, turn off your PS3, insert the HDD back in and then start over. At this point the entire board should be hot enough and should run normally.
Not recommended: Keep turning the PS3 off and on with the physical power button. While this can work, the PS3 will likely tell you that it didn't turn off properly and that it needs a system check for storage corruption. In other word, this method can cause damage to the HDD and therefore not advised.