Refined asymptotics for the Cauchy problem for the fast $p$-Laplace evolution equation

TL;DR

This paper investigates the long-term behavior of solutions to the fast $p$-Laplace evolution equation in various parameter ranges, providing new convergence rates, asymptotic analysis, and a novel entropy method applicable even without displacement convexity.

Contribution

It introduces a new entropy method for analyzing the $p$-Laplace equation, deriving explicit convergence rates and asymptotic behaviors across different fast diffusion regimes.

Findings

Convergence to self-similar profiles with explicit rates in the good fast diffusion range.

First asymptotic analysis near extinction time in the very fast diffusion range.

Identification of new critical exponents affecting qualitative behavior.

Abstract

Our focus is on the fast diffusion equation driven by the $p$-Laplacian operator, that is $\partial_t u=\Delta_p u$ with $1<p<2$, posed in the whole space $\mathbb{R}^N$, $N\geq 2$. The nonnegative solutions are expected to converge in time toward a stationary profile. While such convergence had been previously established for $p$ close to $2$, no quantitative rates were known, and the asymptotic behaviour remained poorly understood across the full fast diffusion range. In fact, the long time behaviour of solutions to the $p$-Laplace Cauchy problem drastically change in different subranges of the $p$. Some of them are analysed here for the first time. In this work, we provide the convergence rates for nonnegative, integrable solutions in the so-called good fast diffusion range, $p_c=\tfrac{2N}{N+1} <p<2$, where mass is conserved. We prove that solutions converge to a self-similar profile with matching mass, with explicit rates measured in relative error. Our constructive proof is based on a new entropy method that remains effective even when the entropy is not displacement convex -- where optimal transport techniques fail. In the very fast diffusion range $1<p<p_c$, we give the first asymptotic analysis near the extinction time. We uncover new critical exponents -- especially in high dimensions -- that give rise to markedly different qualitative behaviour depending on the value of $p$. We also establish convergence rates for the gradients of radial solutions in the good fast diffusion range, again measured in relative error. Finally, we analyze the structural properties required for the entropy method to apply, thereby opening a broader investigation into the basin of attraction of Barenblatt-type profiles, particularly in the singular case of $p$ close to $1$.