Once purchased, it does not expire but completing the Special Research tasks will be much easier during the event window today. That's a lot of work for an eight hour window but also some amazing rewards.

Do special research expire?

Most Special Research tasks are available once the player reaches level 5. They can only be completed once, and tasks cannot be deleted.

What's at the end of a mythical discovery?

The first Special Research quest is named 'A Mythical Discovery', and will see Professor Willow "take Trainers further into the world of Pokémon Go than ever before!", according to Niantic. At at the end? Players will be given Mew as a reward for its completion.