It sounds like the CPU(s) could be overheating during boot.
Is this the liquid cooled model?
1. Identify if it is a software issue first; Try to safe boot by holding shift at startup and/or boot from a known working OS (like install disc) - reinstall the OS if it appears stable on an alt system.
2. Download and burn Apple Hardware Test (Mactracker states AHT v2.5.2 suits the A1047):
https://archive.org/download/aht-powerpc/AHT/PowerMac G5/
3. Boot into AHT and run the full test for memory, temperature sensors, fans, water pump(s) etc
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Some hardware housekeeping to help improve the cooling system;
1. Blow out the years of dust to clean the fans (use a can of air, air compressor, even a vacuum cleaner or garden blower - be careful). Blast air through the power supply at the bottom of the case as it can get pretty thick down there.
2. Try removing and reseating the RAM modules - sometimes RAM connections need to be reseated in the G5, as it gets pretty hot at the contacts.
3. If it is an LCS model (Dual 2.5GHz), you might want to dismantle the cooling system, flush out the radiator grills, clean the pump(s) and add fresh coolant to the system. This would be a good time to replace existing tubing and clamps with new to protect it from potential coolant leaks.
4. While you’re in there, remove the CPUs and separate them from the heatsinks/LCS to clean out the old thermal paste and apply fresh Arctic Silver or similar.
Working on the G5s can be a bit of an undertaking, but follow through with directions from ifixit.com and search the forum threads here for a heap of G5 maintenance advice.
Edit: Grab the
Power Mac G5 Apple Service Manual from Tim:
https://tim.id.au/blog/tims-laptop-service-manuals/