Stock battery? Could be on it's way out so the car is shutting down consumer electronics to keep the car running. If so could also be a charging system issue. The over-run clutch on the alternator pulley could be bad, or the alternator bad. Common issues for VW. If you have a multimeter hook it up to the battery and check voltages. A healthy battery is 12.6 volts, but can often read lower depending if some computers are still awake (anything under 12.2 even when the computers are awake is troubling, but could just need a recharge). When the car is running at idle you should see 13v+, and if you rev up the car 14v+. Also check with all electrical loads on. Blower motor on full, and high beams on. At idle it should still read 13v+ and 14v+ when you rev it up. Recently diagnosed an 08 A3 with a failing alternator. Wasn't bad enough to set the EPS (battery) light yet. To turn off that light the car only needs to see 12.5v I think (when running).
Not sure what VW's cut off is, but BMW's will continue running (after shutting off all consumer electronics, and reducing power to things like the electronic power steering rack), until the battery reaches 8.6v. The last thing to lose power and force the car to shut off is the ABS module / pump, due to safety.
If it is just a blower motor issue, it could be the final stage/resistor for the blower motor, or the blower motor itself failing. I'm assuming our cars use a final stage to control the blower motor speed (most do). One way to test this is to put the car on full fan on defrost/fog and see if the fan still dies. That is the default fail state for the resistor, as for safety it is important to be able to defog the windshield. Normal indicator for the final stage failing is you only get fan speed on max. Since your fan starts out at max and slowly fades out, the resistor is likely good. So a blower motor or charging system issue is more likely.
Remote chance it is a issue with the blend door flaps or motors blocking off the air flow. That is normally fixed through an update, unless it is a mechanical or electrical issue. A few people have had issues with blend doors, mainly making noise.