no, not if u saved.
its not like victini and he comes back after u go in the room
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Yes, you will get a second chance to catch Kyurem provided you defeated it and saved your game. You will need to challenge the Elite 4 and defeat all 4 trainers as well as the Champion, Alder and Alder's defeat will activate the regeneration system so that Kyurem is respawned in the Giant Chasm.
just keep walking
In Pokémon Emerald version I believe you have the option to fight and catch Rayqauza before the Pokémon League. In Pokémon Ruby and Sapphire Version you can only encounter and catch Rayquaza after you have defeated the Pokémon League.
if you've defeated the commader at valley wind works, then catch arceus and gith it to them or pay $100,000.
You will get as many chances as you need to catch Cobalion provided you keep on defeating it in battle. Every time you beat the Champion, it will respawn all Legendary Pokémon that were defeated in battle that weren't caught.