When I slam my car door sometimes it will come on
heater not blowing on 2003 Cadillac DeVille
If the blower motor is intermittent I had the same problem on my '03 - remove panel under dash on passenger side - look up and see the blower motor plug - all I had to do was to wiggle the wires around and the motor started to run continuously - changed the motor cable location slightly so that stress was different on the motor wiring at the connector - been working great ever since.
2 more answers
It sounds like you have an elictrical short in your wiring
Sometimes there is a bad connection or an exposed wire/connection. If you hit with hand directly under the glove compartment (a little to the left) and it starts, then most likely connection problem.