The diffusive limit of Carleman-type models in the range of very fast diffusion equations.

We improve the existing results on the limiting behavior of the Cauchy problem for a class of Carleman-like models with power-type interaction rate in the diffusive scaling with data in the spaces Lp, 1 ≤ p ≤ ∞. The convergence result, which has been carefully established before for exponents of the interaction rate α ≤ 1, is extended here to the range of exponents 1 < α < 4/3. In addition, we discuss the problem of establishing a good theory in the still remaining range α ∈ [4/3, 2), by introducing a modified kinetic system which admits an explicit self-similar solution. The analysis of this solution clarifies the role of the exponent α ̄ = 4/3.