SMS reminder after Apple shuts down.

If you mean to send a message to the turned-off Apple phone, will you receive a reminder that the phone is turned off? That certainly won't happen. At present, mobile phones do not have this function, and operators do not have this service.

If you mean the Apple phone that is turned off, will you be prompted with new news? That's not right either. Apple doesn't have this function, but you can still receive your text messages after booting.

Also, the success of texting has nothing to do with whether the other party is turned off.