Exponential convergence for ultrafast diffusion equations with log-concave weights

2025/07/17 by Fathi, Max, Iacobelli, Mikaela
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.13060

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].

Citations

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• On Sobolev regularity of mass transport and transportation inequalities
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• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

Related

• On the exponential convergence to equilibrium for ultrafast diffusion equations
• New Lipschitz estimates and long-time asymptotic behavior for porous medium and fast diffusion equations
• Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem
• Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods
• Exponential convergence for a convexifying equation and a non-autonomous gradient flow for global minimization