This is another slight spoilery question. The spoilery answer is that Yu does not stay in Inaba. He was only ever meant to stay in the town for a year while his parents were away. Once the year is up at the end of the main story, he goes back home to be with his immediate family. That said, Yu does return to Inaba canonically. This happens in Persona 4 Arena and its Ultimax follow-up, which is a direct sequel to the main game. In that game, he returns for the Golden Week holiday to see his friends and solve yet another mystery alongside the cast of Persona 3.