On the exponential convergence to equilibrium for ultrafast diffusion equations

We propose a simple proof of the exponential convergence to equilibrium for ultrafast diffusion equations in Rⁿ. Our approach, based on the direct use of Poincaré inequality, gets rid of the optimal transport arguments used in {fathi2025} which are valid for Gaussian-excluded one-dimensional weights. This simplification allows us to extend their results to Gaussian measures in higher dimensions.