The diffusive limit for Carleman-type kinetic models

We study the limiting behavior of the Cauchy problem for a class of Carleman-like models in the diffusive scaling with data in the spaces L, 1 ≤ p ≤ ∞. We show that, in the limit, the solution of such models converges towards the solution of a nonlinear diffusion equation with initial values determined by the data of the hyperbolic system. When the data belong to L, a condition of conservation of mass is needed to uniquely identify the solution in some cases, whereas the solution may disappear in the limit in other cases.