This is the 551st most frequent Ukrainian word.
живуть
"Live" (third-person plural form).
The word "живуть" is used in its literal meaning to denote residing or dwelling in a place.
Вони живуть у великому місті.
They live in a big city.
Here, "живуть" is used metaphorically to indicate that hopes are alive or exist within their hearts.
Їхні надії живуть у їхніх серцях.
Their hopes live in their hearts.
In this context, "живуть" describes the continuation or preservation of traditions over time.