I have not had that problem but I can tell you how I would trouble shoot the issue.
Since you have tried both the outlet in your car as well as the home charger, it would seem that the problem is most likely either the power cord or the unit itself. Check the most obvious problem first, and that would be by using the hard wire power line instead of the power cord that plugs into a cigarette lighter socket.
Be careful of your polarity, and connect the power leads directly to your car battery. If the unity will power up, the issue is with the plug in power cord. If that is the case, I would suspect that there is either an internal short or a break in the wiring. I have not seen any information that would lead me to believe that the plug/speaker unit has an internal fuse, but it would make sense that there could be one. Speaking for myself, I would unscrew the case and take a careful look.