But if the data provides die, so that LLMs may live, where will the LLMs get new data?

From other LLMs of course.

Which is the main issue I see with them, if no one publishes anything new, all you will get is whatever was there before, which may be incorrect or obsolete. If you want something new, an LLM can't discover it if no data source exists. So over time the LLM becomes useless for current information.