  In this paper, we investigate generalized Carleman kinetic equation for
n$\ge$2 and prove convergence towards the solution of equation with fast
diffusion or porous medium type, $u_t=\Delta u^m$ ($0\le m\le2$), in its
diffusive hydrodynamic limit. Using comparison principle of system combined
with fixed speed propagation property of transport equation, we create a new
barrier argument for this hyperbolic system. It is crucial to construct
explicit local sub and solution of system and this is done by employing an
ansatz from second order asymptotic expansion. This allow us prove diffusive
limit toward subcritical FDE, which is thought to be difficult with previous
method due to the lack of mass conservation. Moreover, we can also prove
convergence with growing initial data in slow diffusion range (including
$n=1$), which was also unknown before.
