  We find a general expression for the free energy of $G(\phi)=\phi^{2k}$
generalized 2D Yang-Mills theories in the strong ($A>A_c$) region at large $N$.
We also show that in this region, the density function of Young tableau of
these models is a three-cut problem. In the specific $\phi^6$ model, we show
that the theory has a third order phase transition, like $\phi^2$ (YM_2) and
$\phi^4$ models. We have some comments for $k \geq 4$ cases. At the end, we
study the phase structure of $\phi^2 + g \phi^4$ model for $g \leq A/4$ region.
