If the warp drive is shut down either by its destruction or just plain turning it off, what happens? Does the vessel stop? Continue coasting forward? If so how would the time dilation factor at speeds several thousand times the speed of light?
I *think* that once you have created the Alcubierre spacetime, nothing physically needs to be done to maintain it. That is, the bubble will continue coasting with whatever velocity (with respect to distant observers) it had originally. The only thing is you will no longer be able to stop.
Although, the Alcubierre solution is static, so it's not entirely clear what would happen if you tried to modify it.