For this example, de is changed by the noun that follows it. Because it’s used for a possession purpose.

Le chat des enfants. Le chat de l’ enfant

You use des because there are multiple childrens, it’s their cat.
For the second one, usually, we would use du or de la.

Le chat du voisin.

Le chat de la voisine.

But, because enfant starts with a vowel, due to the prononciation, you have to write and say de l’.
Because it’s kinda hard to pronounce Le chat du enfant

For the second example, there’s no possession here, just description, so de won’t change.

Mon mode de vie. Une épidémie d’ obésité.

Une volonté de fer. Un mental *d’*acier.

And for the same reason, we use d’ instead of de, when it’s followed by a vowel.