My Alpine MRP-M650 Amp is acting up. When I start
There are a lot of things that could cause this issue ...
1. Your amp requires more power than it gets when you start your car, and needs to charge up before it can work. If this is the case, you should be able to give your stereo about 5 minutes to warm up at low volume and it should come on. This happens a lot with amps with built-in capacitors (capacitors give you stunning, instant kick when working, but require a bunch of power to initialize). If you wire a bypass switch for the power to your amp and bass components, you can turn it on separate from the rest of the car and allow it to charge before you start your car (just be careful to turn it off when you are going to be away from your car for a while since this will allow your system to drain your battery).
2. Your amp might be set incorrectly for handling the system you are running. If your Ohms do not match up exactly (8 ohm speakers with a 4 ohm amp -- it's the "Impedance rating") then the system should not work at all, or should burn out the amp ... but with a jury-rigged system it can work. If this is the case, I recommend getting a new amp that matches your speakers.
3. Depending on the amp, there might be a volume-based cutoff that is working improperly. Turn on the car with the stereo at low volume. While the speakers are still not working, check to make sure that the amp has power, and seems to be functioning correctly. If everything seems like it's on the up and up, but the speakers are not working, try changing the volume-filter options on your amp.
4. I put this last because it seems like you know what you are doing with the system setup, but it is possible that you have a bad connection between the speakers and the amp, or between the stereo and the amp. There could be a loose connection somewhere (either inside the amp, between the amp and the speakers at the jack, or between the stereo and the amp at the jack). If this were the case, everything would look perfect from a visual inspection, but there could be a tiny space, or corrosion in a connection. Turning the stereo up to a high volume could then bridge this short circuit, and the electric force could magnetize the connection shut, allowing you to then turn the stereo down with the working connection. The best-case scenario here would be an external wiring issue, where you can clean any corrosion off the connections with an eraser, and reconnect everything tightly fixing the problem. The worst case would be a short WITHIN the amp, which would have to be repaired by someone with a lot of time and a circuit tester -- in which case you might be better off trading the amp in for a new one.