He lives in the bottom left most house. After or before Shaymin, he gives you an Upgrade.
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professor rowan doesnt have a lab in eterna but oak stays in a house north west of eterna once u finish the national dex
He will apear in Eterna city in the house right from the gym
After you obtain the National Pokédex and visited the Pal Park, Professor Oak appears in the southernmost house in Eterna City.
He is in a house in the southwest corner of the city.
He isn't in Pal Park, he's in Eterna City