All these seem to be answering the question ‘where’ and thus would be considered adverbial.
But I’d say, in 1, the phrase modified ‘animals’ since it’s immediately following that word. In 2, the prep phrase modifies ‘the boy’ for the same reason.
In 3, however, the prep phrase is clearly telling where they study, and is therefore adverbial, modifying ‘study.’
1. They study sea animals in Aantarctica.
[What does 'in Antarctica' modify? 'Sea animals' or 'study'? Are they in Antarctica now? Or though they are not in Antarctica, are they study sea animals at home or at a workplace.]
2. Look at the boy in the room.
[What about 'in the room'? Does 'in the room' modify 'the boy' or 'look'?]
3. They study math in the room.
['In the room' seems to modify 'study' here, isn't it?]
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