When Patronus Charm is spelled, Dementors escape. Its possible that many wizards could cast that spell from all directions to block all escape paths of Dementors. What would happen then?
Would Dementors vanish to null void? Or, something else?
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A Patronus is defined (by Prof. Lupin, in the third book), as a shield made of positive thoughts and memories. The Dementor would feed off of the Patronus in place of the user.
In the movie, it was seen that Dementors run away from Patronuses, or that they are hurt by it. That's not the same in the books.
So if a Dementor is "trapped" inside of a Patronus prison, he'd simply have more to feed on it.
There's plenty of speculation - though no definitive evidence (that I've seen - if there is, I'd love to see it) - that Dementors would fade away if removed permanently from the conditions in which they thrive and reproduce.
A Patronus is known to repel Dementors, so the most I think we could definitively say is that the Dementors would be trapped for as long as the sphere was intact.
According to JK Rowling herself, you cannot kill a Demontor. You can only reduce their speed of reproduction, by taking away the very thing that they thrive upon, fear misery and despair.
As already stated above, they would just be held there, imprisoned. They feed off a patronus, instead of the caster.