If the caster is in the radius of a harmful area of effect, such as a fireball spell, will the illusory duplicates exist after damage is taken or will they be destroyed?
The spell description explicitly states only attacks can destroy a duplicate.
The spell description of mirror image states:
A duplicate can be destroyed only by an attack that hits it. It ignores all other damage and effects.
Area of effect spells such as fireball are “other damage and effects”, not attacks.
No—they are only destroyed when you are targeted with an attack
The spell's description states, in relevant part:
Each time a creature targets you with an attack during the spell's duration, roll a d20 to determine whether the attack instead targets one of your duplicates.
Fireball is not an attack, because it uses a saving throw instead of an attack roll
The Sage Advice Compendium specifically addresses whether Fireball is an attack when describing its interaction with Uncanny Dodge (emphasis added):
A use of Uncanny Dodge works against only one attack, since it expends your reaction, and only if you can see the attacker. It works against attacks of all sorts, including spell attacks, but it is no help against a spell or other effect, such as fireball, that delivers its damage after a saving throw rather than after an attack roll.
This is pretty clear that it is not an attack, because it is unaffected by something that "works against attacks of all sorts, including spell attacks."
If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack roll, you're making an attack.
There's no attack roll, so there's no attack.
As additional (though unofficial) evidence, a directly on-point tweet from Jeremy Crawford:
Fireball is not an attack.
Side note: I had initially thought that, since Fireball targets a point, you wouldn't be "targeted" by it. However, this was incorrect:
Look carefully at the text of fireball: every creature affected is called a target.
The relevant text is:
A target takes 8d6 fire damage on a failed save, or half as much damage on a successful one.