It's called Debris Flow.
On Day 353 after you complete the main mission objective (beating all the bosses in each specific area of Twilight Town) go up to the Clock Tower/Train Station and a cutscene will trigger.
This thing is bigger than all the other Wyvren bosses and you fight it in a small area making it harder to dodge it's attacks.
Killing this boss gives you 100% for that level, meaning it's the optional objective to complete.
It has 4 attacks but only one of them is hard to avoid; it shoots a series of fire balls out of it's mouth which home in on you exploding on impact.
Meaning you can't guard this attacks because the others will explode around you causing you to take damage and might I add they take down half your health bar if only 1 hits you (there's around 10 it shoots out).
I should also mention it has around 13 bars of health and your attacks barely do any damage to it.
I managed to get it down to 4 bars once before when it killed me with it's fire ball attack.