I remember in SA2 or so that he lost his memory. I didn't understand anything that was going on in Sonic Heroes though.

TargetDummy : Yes on the sonic wiki it says he is indestructable witch i think is stupid.

Cjjjj : I dont see how it's stupid, Shadow was made by Gerald Robotnik while he was trying to find a way to cure his granddaughter Maria's fatal illness.
Shadow is indestructible and immortal because that's just part of his DNA, the way he was made.

Shadow survived because maybe sonic was fighting a clone of him and he was hidden somewhere in the ARK. Then he got to the earth when super sonic and the fake super shadow defeated finalhazard.

I think he survived, and here's why, but be warned, as it contains major spoilers.

It was kinda explained in sonic X. Dr. Gerald robotnik was working on an immortality project to make the ultimate lifeform, wich resulted in shadow. but, he never was able to make more "shadows" cause his project was shut down and everyone was arrested or killed (like Maria, who shadow has tried to avenge by being "evil"). This battle took place later in space when shadow and rouge have retrieved the chaos emeralds for dr. Robotnik/eggman to power up the "ARK" It turns out, after a failed attempt to power the ( what I believe is called) the "eclipse cannon", a recorded video of Dr. Gerald Robotnik in prison, telling that the ARK will crash to earth because of his daughter Maria being killed. sonic, knuckles and others (including eggman) go bring the master emerald to go and make the chaos emeralds stop working so that the ARK can stop plummeting to earth. They get to the room, but, a giant lizard ,named final hazard, is gaurding the chaos emeralds. So shadow and sonic go battle it while knuckles gets the master emerald to stop the chaos emeralds. Sonic and shadow, as it seems at the moment, kill final hazard. Knuckles is also able to stop the chaos emeralds too. They think all is well, until they found out final hazard wasn't killed, just teleported to the outside, pulling it to earth. Sonic and shadow go super form to be able to stop final hazard. They then go battle it. And at the end, shadow gains all memory and knows who he is. They then kill final hazard and get the ARK out of earths orbit. Sonic survives, but it seems, shadow sacrificed his life to help others, like Maria wanted. And it seemed, that what he was created for was done, and he didn't need to live anymore. So they made it look like he was killed off. But he had just fallen to earth and lost his memory. Leading to the game, "shadow the hedgehog".

I personally say that Sega was just being inconsistent, but out of your choises, I say Shadow survived.