Kinetic Approach to Long time Behavior of Linearized Fast Diffusion Equations

We show that the rate of convergence towards the self-similar solution of certain linearized versions of the fast diffusion equation can be related to the number of moments of the initial datum that are equal to the moments of the self-similar solution at a fixed time. As a consequence, we find an improved rate of convergence to self-similarity in terms of a Fourier based distance between two solutions. The results are based on the asymptotic equivalence of a collisional kinetic model of Boltzmann type with a linear Fokker-Planck equation with nonconstant coefficients, and make use of methods first applied to the reckoning of the rate of convergence towards equilibrium for the spatially homogeneous Boltzmann equation for Maxwell molecules.