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

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