In this talk I would like to present some recent results on the asymptotic behavior of a very fast diffusion PDE with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures. I prove exponential convergence to equilibrium under minimal assumptions on the data, and I also provide sufficient conditions for W2-stability of solutions. Moreover, I will present a work in progress with Filippo Santambrogio and Francesco Saverio Patacchini where we use the JKO scheme to relax the hypotheses of my previous convergence result.