Exponential convergence for ultrafast diffusion equations with log-concave weights

TL;DR

This paper proves exponential convergence to equilibrium for a weighted ultrafast diffusion PDE with log-concave weights on the real line, extending previous results beyond compact settings and motivated by measure quantization.

Contribution

It establishes exponential convergence for a class of weighted ultrafast diffusion equations with log-concave weights, broadening the scope of prior compact setting results.

Findings

Proves exponential convergence to equilibrium.

Extends results to non-compact, real-line setting.

Connects diffusion PDEs with measure quantization.

Abstract

We study the asymptotic behavior of a weighted ultrafast diffusion PDE on the real line, with a log-concave and log-lipschitz weight, and prove exponential convergence to equilibrium. This result goes beyond the compact setting studied in [22]. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in [11].