  We consider the free additive convolution semigroup $\lbrace \mu^{\boxplus
t}:\,t\ge 1\rbrace$ and determine the local behavior of the density of
$\mu^{\boxplus t}$ at the endpoints and at any singular point of its support.
We then study the free additive convolution of two multi-cut probability
measures and show that its density decays either as a square root or as a cubic
root at any endpoints of its support. The probability measures considered in
this paper satisfy a power law behavior with exponents strictly between $-1$
and $1$ at the endpoints of their supports.
