Yes, the old "was working when I last turned it on" seller comment...
If it is ok for the first few seconds, I would suspect a power supply issue (rising volts as it warms up) or maybe a capacitor problem. In my experience, if it's an IC it either works or it doesn't. Mind you a faulty PSU can soon blow up RAM and other chips.
Commodore power supplies are notorious for developing faults and blowing up chips...at least for the stand-alone PSUs on the C64s. I'm not sure about the SX-64 which has it's own built in PSU.
There are lots of diagnostics docs on the web for the C64s, if you can't find anyone to take a look at it.