Pull out your DMM and check voltage on the remote turn on wire and the 12 volt positive. Also check your fuse up front for corrosion and physically pull the fuse out to see if it is in tact. You may have voltage drop at the fuse block if it is partially broken. Also check your ground wire and location. Sanded metal and a solid spot with thick sheet metal.
Same thing happened to a buddy and I just glanced at the fuse and it "looked" fine but after trying new grounds, new turn on, switching impedance on the sub, I decided to pull out my trusty old DMM and it showed that I only had 9 volts on the power wire. When he would start his car, the fuse would just contact enough so it would read 12 volts and allow the amp to play for a couple seconds then immediately go into protect again dropping down to 9 volts.
If it's not a bad ground and you have 12 volts at remote turn on and power wire, then idk whats the deal with your amp.