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

Multi-D Fast Diffusion Equation via Diffusive Scaling of Generalized Carleman Kinetic Equation