On the exponential convergence to equilibrium for ultrafast diffusion equations

2025/09/09 by Yi C. Huang, Huang, Yi C., Xin T. Tong +1
Computer Science · Mathematics · Medicine · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2509.07382

openalex publication_date 2025/09/09 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

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 \citefathi2025 which are valid for Gaussian-excluded one-dimensional weights. This simplification allows us to extend their results to Gaussian measures in higher dimensions.

Citations

• Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods
• Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities
• Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem

Related

• Exponential convergence for ultrafast diffusion equations with log-concave weights
• Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods
• Sharp Extinction Rates for Fast Diffusion Equations on Generic Bounded\n Domains
• Exponential Convergence of Non-Linear Monotone SPDEs
• Trends to Equilibrium in Total Variation Distance