No , once it faints , it will not reappear , unless you save the game right before you make it faint .
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Mesprit does flee from the battles normally at the beginning. But if you manage to inflict damage on it then when you see it again it will still have the damage to it. The best way I found to catch Mesprit is to get a level 30-40 Pokemon that knows Mean Look, I evolved my gastly as i found my captured Haunters didnt know Mean Look. Then give your Pokemon that knows mean look a quick claw. Track down Mesprit using the Poketech app (forgot what it is called, my apologies >.
i have been searching for 2 weeks but still no answer so far but you might got mt cornet as may moonstones as pillars put diagla at front with stone and uxie mespirit azelf
If you make a legandarie faint you can not catch it. here is a tip if you are faceing a legandarie and all of your Pokemon faint but one. you should run away go heal your Pokemon and come back to where you saw the legandarie it will still be there
No once any Pokemon has fainted you can not capture it sorry
no i had to change mine back to a wep