Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem

In this paper we study the asymptotic behavior of a very fast diffusion PDE in 1D with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in \cite{CGI1}. We prove exponential convergence to equilibrium under minimal assumptions on the data, and we also provide sufficient conditions for $W_2$-stability of solutions.